3.5.36 \(\int (g+h x) (a+b \log (c (d (e+f x)^p)^q))^3 \, dx\) [436]

Optimal. Leaf size=306 \[ \frac {6 a b^2 (f g-e h) p^2 q^2 x}{f}-\frac {6 b^3 (f g-e h) p^3 q^3 x}{f}-\frac {3 b^3 h p^3 q^3 (e+f x)^2}{8 f^2}+\frac {6 b^3 (f g-e h) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}+\frac {3 b^2 h p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2}-\frac {3 b (f g-e h) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}-\frac {3 b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2}+\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2} \]

[Out]

6*a*b^2*(-e*h+f*g)*p^2*q^2*x/f-6*b^3*(-e*h+f*g)*p^3*q^3*x/f-3/8*b^3*h*p^3*q^3*(f*x+e)^2/f^2+6*b^3*(-e*h+f*g)*p
^2*q^2*(f*x+e)*ln(c*(d*(f*x+e)^p)^q)/f^2+3/4*b^2*h*p^2*q^2*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^2-3*b*(-e*h
+f*g)*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2-3/4*b*h*p*q*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/f^2+(-
e*h+f*g)*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f^2+1/2*h*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/f^2

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Rubi [A]
time = 0.38, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341, 2495} \begin {gather*} \frac {3 b^2 h p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2}+\frac {6 a b^2 p^2 q^2 x (f g-e h)}{f}-\frac {3 b p q (e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}+\frac {(e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}-\frac {3 b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2}+\frac {6 b^3 p^2 q^2 (e+f x) (f g-e h) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}-\frac {3 b^3 h p^3 q^3 (e+f x)^2}{8 f^2}-\frac {6 b^3 p^3 q^3 x (f g-e h)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

(6*a*b^2*(f*g - e*h)*p^2*q^2*x)/f - (6*b^3*(f*g - e*h)*p^3*q^3*x)/f - (3*b^3*h*p^3*q^3*(e + f*x)^2)/(8*f^2) +
(6*b^3*(f*g - e*h)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f^2 + (3*b^2*h*p^2*q^2*(e + f*x)^2*(a + b*Log[c
*(d*(e + f*x)^p)^q]))/(4*f^2) - (3*b*(f*g - e*h)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f^2 - (3*b*
h*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*f^2) + ((f*g - e*h)*(e + f*x)*(a + b*Log[c*(d*(e + f*
x)^p)^q])^3)/f^2 + (h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(2*f^2)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx &=\text {Subst}\left (\int (g+h x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\int \left (\frac {(f g-e h) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{f}+\frac {h (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{f}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {h \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {h \text {Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h) \text {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2}-\text {Subst}\left (\frac {(3 b h p q) \text {Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 b (f g-e h) p q) \text {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {3 b (f g-e h) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}-\frac {3 b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2}+\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2}+\text {Subst}\left (\frac {\left (3 b^2 h p^2 q^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (6 b^2 (f g-e h) p^2 q^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {6 a b^2 (f g-e h) p^2 q^2 x}{f}-\frac {3 b^3 h p^3 q^3 (e+f x)^2}{8 f^2}+\frac {3 b^2 h p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2}-\frac {3 b (f g-e h) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}-\frac {3 b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2}+\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2}+\text {Subst}\left (\frac {\left (6 b^3 (f g-e h) p^2 q^2\right ) \text {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {6 a b^2 (f g-e h) p^2 q^2 x}{f}-\frac {6 b^3 (f g-e h) p^3 q^3 x}{f}-\frac {3 b^3 h p^3 q^3 (e+f x)^2}{8 f^2}+\frac {6 b^3 (f g-e h) p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}+\frac {3 b^2 h p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2}-\frac {3 b (f g-e h) p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}-\frac {3 b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2}+\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 231, normalized size = 0.75 \begin {gather*} \frac {8 (f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+4 h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-24 b (f g-e h) p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )-3 b h p q \left (2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{8 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

(8*(f*g - e*h)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3 + 4*h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])
^3 - 24*b*(f*g - e*h)*p*q*((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 - 2*b*p*q*(f*(a - b*p*q)*x + b*(e + f*
x)*Log[c*(d*(e + f*x)^p)^q])) - 3*b*h*p*q*(2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + b*p*q*(b*f*p*q*x
*(2*e + f*x) - 2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))))/(8*f^2)

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (h x +g \right ) \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

[Out]

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (316) = 632\).
time = 0.32, size = 770, normalized size = 2.52 \begin {gather*} \frac {1}{2} \, b^{3} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} - 3 \, a^{2} b f g p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} - \frac {3}{4} \, a^{2} b f h p q {\left (\frac {f x^{2} - 2 \, x e}{f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}}\right )} + \frac {3}{2} \, a b^{2} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + b^{3} g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + \frac {3}{2} \, a^{2} b h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + 3 \, a b^{2} g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac {1}{2} \, a^{3} h x^{2} + 3 \, a^{2} b g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - 3 \, {\left (2 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f}\right )} a b^{2} g - {\left (3 \, f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} - {\left (\frac {{\left (e \log \left (f x + e\right )^{3} + 3 \, e \log \left (f x + e\right )^{2} - 6 \, f x + 6 \, e \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{2}} - \frac {3 \, {\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} p q \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{f^{2}}\right )} f p q\right )} b^{3} g - \frac {3}{4} \, {\left (2 \, f p q {\left (\frac {f x^{2} - 2 \, x e}{f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - \frac {{\left (f^{2} x^{2} - 6 \, f x e + 2 \, e^{2} \log \left (f x + e\right )^{2} + 6 \, e^{2} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{2}}\right )} a b^{2} h - \frac {1}{8} \, {\left (6 \, f p q {\left (\frac {f x^{2} - 2 \, x e}{f^{2}} + \frac {2 \, e^{2} \log \left (f x + e\right )}{f^{3}}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + {\left (\frac {{\left (3 \, f^{2} x^{2} + 4 \, e^{2} \log \left (f x + e\right )^{3} - 42 \, f x e + 18 \, e^{2} \log \left (f x + e\right )^{2} + 42 \, e^{2} \log \left (f x + e\right )\right )} p^{2} q^{2}}{f^{3}} - \frac {6 \, {\left (f^{2} x^{2} - 6 \, f x e + 2 \, e^{2} \log \left (f x + e\right )^{2} + 6 \, e^{2} \log \left (f x + e\right )\right )} p q \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{f^{3}}\right )} f p q\right )} b^{3} h + a^{3} g x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="maxima")

[Out]

1/2*b^3*h*x^2*log(((f*x + e)^p*d)^q*c)^3 - 3*a^2*b*f*g*p*q*(x/f - e*log(f*x + e)/f^2) - 3/4*a^2*b*f*h*p*q*((f*
x^2 - 2*x*e)/f^2 + 2*e^2*log(f*x + e)/f^3) + 3/2*a*b^2*h*x^2*log(((f*x + e)^p*d)^q*c)^2 + b^3*g*x*log(((f*x +
e)^p*d)^q*c)^3 + 3/2*a^2*b*h*x^2*log(((f*x + e)^p*d)^q*c) + 3*a*b^2*g*x*log(((f*x + e)^p*d)^q*c)^2 + 1/2*a^3*h
*x^2 + 3*a^2*b*g*x*log(((f*x + e)^p*d)^q*c) - 3*(2*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*c) +
(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*p^2*q^2/f)*a*b^2*g - (3*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(((
f*x + e)^p*d)^q*c)^2 - ((e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e))*p^2*q^2/f^2 - 3*(e*
log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*p*q*log(((f*x + e)^p*d)^q*c)/f^2)*f*p*q)*b^3*g - 3/4*(2*f*p*q*((f*x
^2 - 2*x*e)/f^2 + 2*e^2*log(f*x + e)/f^3)*log(((f*x + e)^p*d)^q*c) - (f^2*x^2 - 6*f*x*e + 2*e^2*log(f*x + e)^2
+ 6*e^2*log(f*x + e))*p^2*q^2/f^2)*a*b^2*h - 1/8*(6*f*p*q*((f*x^2 - 2*x*e)/f^2 + 2*e^2*log(f*x + e)/f^3)*log(
((f*x + e)^p*d)^q*c)^2 + ((3*f^2*x^2 + 4*e^2*log(f*x + e)^3 - 42*f*x*e + 18*e^2*log(f*x + e)^2 + 42*e^2*log(f*
x + e))*p^2*q^2/f^3 - 6*(f^2*x^2 - 6*f*x*e + 2*e^2*log(f*x + e)^2 + 6*e^2*log(f*x + e))*p*q*log(((f*x + e)^p*d
)^q*c)/f^3)*f*p*q)*b^3*h + a^3*g*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1776 vs. \(2 (316) = 632\).
time = 0.43, size = 1776, normalized size = 5.80 \begin {gather*} \frac {4 \, {\left (b^{3} f^{2} h p^{3} q^{3} x^{2} + 2 \, b^{3} f^{2} g p^{3} q^{3} x + 2 \, b^{3} f g p^{3} q^{3} e - b^{3} h p^{3} q^{3} e^{2}\right )} \log \left (f x + e\right )^{3} + 4 \, {\left (b^{3} f^{2} h x^{2} + 2 \, b^{3} f^{2} g x\right )} \log \left (c\right )^{3} + 4 \, {\left (b^{3} f^{2} h q^{3} x^{2} + 2 \, b^{3} f^{2} g q^{3} x\right )} \log \left (d\right )^{3} - {\left (3 \, b^{3} f^{2} h p^{3} q^{3} - 6 \, a b^{2} f^{2} h p^{2} q^{2} + 6 \, a^{2} b f^{2} h p q - 4 \, a^{3} f^{2} h\right )} x^{2} + 6 \, {\left (7 \, b^{3} f h p^{3} q^{3} - 6 \, a b^{2} f h p^{2} q^{2} + 2 \, a^{2} b f h p q\right )} x e - 6 \, {\left ({\left (b^{3} f^{2} h p^{3} q^{3} - 2 \, a b^{2} f^{2} h p^{2} q^{2}\right )} x^{2} + 4 \, {\left (b^{3} f^{2} g p^{3} q^{3} - a b^{2} f^{2} g p^{2} q^{2}\right )} x - {\left (3 \, b^{3} h p^{3} q^{3} - 2 \, a b^{2} h p^{2} q^{2}\right )} e^{2} - 2 \, {\left (b^{3} f h p^{3} q^{3} x - 2 \, b^{3} f g p^{3} q^{3} + 2 \, a b^{2} f g p^{2} q^{2}\right )} e - 2 \, {\left (b^{3} f^{2} h p^{2} q^{2} x^{2} + 2 \, b^{3} f^{2} g p^{2} q^{2} x + 2 \, b^{3} f g p^{2} q^{2} e - b^{3} h p^{2} q^{2} e^{2}\right )} \log \left (c\right ) - 2 \, {\left (b^{3} f^{2} h p^{2} q^{3} x^{2} + 2 \, b^{3} f^{2} g p^{2} q^{3} x + 2 \, b^{3} f g p^{2} q^{3} e - b^{3} h p^{2} q^{3} e^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right )^{2} + 6 \, {\left (2 \, b^{3} f h p q x e - {\left (b^{3} f^{2} h p q - 2 \, a b^{2} f^{2} h\right )} x^{2} - 4 \, {\left (b^{3} f^{2} g p q - a b^{2} f^{2} g\right )} x\right )} \log \left (c\right )^{2} + 6 \, {\left (2 \, b^{3} f h p q^{3} x e - {\left (b^{3} f^{2} h p q^{3} - 2 \, a b^{2} f^{2} h q^{2}\right )} x^{2} - 4 \, {\left (b^{3} f^{2} g p q^{3} - a b^{2} f^{2} g q^{2}\right )} x + 2 \, {\left (b^{3} f^{2} h q^{2} x^{2} + 2 \, b^{3} f^{2} g q^{2} x\right )} \log \left (c\right )\right )} \log \left (d\right )^{2} - 8 \, {\left (6 \, b^{3} f^{2} g p^{3} q^{3} - 6 \, a b^{2} f^{2} g p^{2} q^{2} + 3 \, a^{2} b f^{2} g p q - a^{3} f^{2} g\right )} x + 6 \, {\left ({\left (b^{3} f^{2} h p^{3} q^{3} - 2 \, a b^{2} f^{2} h p^{2} q^{2} + 2 \, a^{2} b f^{2} h p q\right )} x^{2} + 2 \, {\left (b^{3} f^{2} h p q x^{2} + 2 \, b^{3} f^{2} g p q x + 2 \, b^{3} f g p q e - b^{3} h p q e^{2}\right )} \log \left (c\right )^{2} + 2 \, {\left (b^{3} f^{2} h p q^{3} x^{2} + 2 \, b^{3} f^{2} g p q^{3} x + 2 \, b^{3} f g p q^{3} e - b^{3} h p q^{3} e^{2}\right )} \log \left (d\right )^{2} + 4 \, {\left (2 \, b^{3} f^{2} g p^{3} q^{3} - 2 \, a b^{2} f^{2} g p^{2} q^{2} + a^{2} b f^{2} g p q\right )} x - {\left (7 \, b^{3} h p^{3} q^{3} - 6 \, a b^{2} h p^{2} q^{2} + 2 \, a^{2} b h p q\right )} e^{2} + 2 \, {\left (4 \, b^{3} f g p^{3} q^{3} - 4 \, a b^{2} f g p^{2} q^{2} + 2 \, a^{2} b f g p q - {\left (3 \, b^{3} f h p^{3} q^{3} - 2 \, a b^{2} f h p^{2} q^{2}\right )} x\right )} e - 2 \, {\left ({\left (b^{3} f^{2} h p^{2} q^{2} - 2 \, a b^{2} f^{2} h p q\right )} x^{2} + 4 \, {\left (b^{3} f^{2} g p^{2} q^{2} - a b^{2} f^{2} g p q\right )} x - {\left (3 \, b^{3} h p^{2} q^{2} - 2 \, a b^{2} h p q\right )} e^{2} - 2 \, {\left (b^{3} f h p^{2} q^{2} x - 2 \, b^{3} f g p^{2} q^{2} + 2 \, a b^{2} f g p q\right )} e\right )} \log \left (c\right ) - 2 \, {\left ({\left (b^{3} f^{2} h p^{2} q^{3} - 2 \, a b^{2} f^{2} h p q^{2}\right )} x^{2} + 4 \, {\left (b^{3} f^{2} g p^{2} q^{3} - a b^{2} f^{2} g p q^{2}\right )} x - {\left (3 \, b^{3} h p^{2} q^{3} - 2 \, a b^{2} h p q^{2}\right )} e^{2} - 2 \, {\left (b^{3} f h p^{2} q^{3} x - 2 \, b^{3} f g p^{2} q^{3} + 2 \, a b^{2} f g p q^{2}\right )} e - 2 \, {\left (b^{3} f^{2} h p q^{2} x^{2} + 2 \, b^{3} f^{2} g p q^{2} x + 2 \, b^{3} f g p q^{2} e - b^{3} h p q^{2} e^{2}\right )} \log \left (c\right )\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 6 \, {\left ({\left (b^{3} f^{2} h p^{2} q^{2} - 2 \, a b^{2} f^{2} h p q + 2 \, a^{2} b f^{2} h\right )} x^{2} - 2 \, {\left (3 \, b^{3} f h p^{2} q^{2} - 2 \, a b^{2} f h p q\right )} x e + 4 \, {\left (2 \, b^{3} f^{2} g p^{2} q^{2} - 2 \, a b^{2} f^{2} g p q + a^{2} b f^{2} g\right )} x\right )} \log \left (c\right ) + 6 \, {\left ({\left (b^{3} f^{2} h p^{2} q^{3} - 2 \, a b^{2} f^{2} h p q^{2} + 2 \, a^{2} b f^{2} h q\right )} x^{2} - 2 \, {\left (3 \, b^{3} f h p^{2} q^{3} - 2 \, a b^{2} f h p q^{2}\right )} x e + 2 \, {\left (b^{3} f^{2} h q x^{2} + 2 \, b^{3} f^{2} g q x\right )} \log \left (c\right )^{2} + 4 \, {\left (2 \, b^{3} f^{2} g p^{2} q^{3} - 2 \, a b^{2} f^{2} g p q^{2} + a^{2} b f^{2} g q\right )} x + 2 \, {\left (2 \, b^{3} f h p q^{2} x e - {\left (b^{3} f^{2} h p q^{2} - 2 \, a b^{2} f^{2} h q\right )} x^{2} - 4 \, {\left (b^{3} f^{2} g p q^{2} - a b^{2} f^{2} g q\right )} x\right )} \log \left (c\right )\right )} \log \left (d\right )}{8 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="fricas")

[Out]

1/8*(4*(b^3*f^2*h*p^3*q^3*x^2 + 2*b^3*f^2*g*p^3*q^3*x + 2*b^3*f*g*p^3*q^3*e - b^3*h*p^3*q^3*e^2)*log(f*x + e)^
3 + 4*(b^3*f^2*h*x^2 + 2*b^3*f^2*g*x)*log(c)^3 + 4*(b^3*f^2*h*q^3*x^2 + 2*b^3*f^2*g*q^3*x)*log(d)^3 - (3*b^3*f
^2*h*p^3*q^3 - 6*a*b^2*f^2*h*p^2*q^2 + 6*a^2*b*f^2*h*p*q - 4*a^3*f^2*h)*x^2 + 6*(7*b^3*f*h*p^3*q^3 - 6*a*b^2*f
*h*p^2*q^2 + 2*a^2*b*f*h*p*q)*x*e - 6*((b^3*f^2*h*p^3*q^3 - 2*a*b^2*f^2*h*p^2*q^2)*x^2 + 4*(b^3*f^2*g*p^3*q^3
- a*b^2*f^2*g*p^2*q^2)*x - (3*b^3*h*p^3*q^3 - 2*a*b^2*h*p^2*q^2)*e^2 - 2*(b^3*f*h*p^3*q^3*x - 2*b^3*f*g*p^3*q^
3 + 2*a*b^2*f*g*p^2*q^2)*e - 2*(b^3*f^2*h*p^2*q^2*x^2 + 2*b^3*f^2*g*p^2*q^2*x + 2*b^3*f*g*p^2*q^2*e - b^3*h*p^
2*q^2*e^2)*log(c) - 2*(b^3*f^2*h*p^2*q^3*x^2 + 2*b^3*f^2*g*p^2*q^3*x + 2*b^3*f*g*p^2*q^3*e - b^3*h*p^2*q^3*e^2
)*log(d))*log(f*x + e)^2 + 6*(2*b^3*f*h*p*q*x*e - (b^3*f^2*h*p*q - 2*a*b^2*f^2*h)*x^2 - 4*(b^3*f^2*g*p*q - a*b
^2*f^2*g)*x)*log(c)^2 + 6*(2*b^3*f*h*p*q^3*x*e - (b^3*f^2*h*p*q^3 - 2*a*b^2*f^2*h*q^2)*x^2 - 4*(b^3*f^2*g*p*q^
3 - a*b^2*f^2*g*q^2)*x + 2*(b^3*f^2*h*q^2*x^2 + 2*b^3*f^2*g*q^2*x)*log(c))*log(d)^2 - 8*(6*b^3*f^2*g*p^3*q^3 -
6*a*b^2*f^2*g*p^2*q^2 + 3*a^2*b*f^2*g*p*q - a^3*f^2*g)*x + 6*((b^3*f^2*h*p^3*q^3 - 2*a*b^2*f^2*h*p^2*q^2 + 2*
a^2*b*f^2*h*p*q)*x^2 + 2*(b^3*f^2*h*p*q*x^2 + 2*b^3*f^2*g*p*q*x + 2*b^3*f*g*p*q*e - b^3*h*p*q*e^2)*log(c)^2 +
2*(b^3*f^2*h*p*q^3*x^2 + 2*b^3*f^2*g*p*q^3*x + 2*b^3*f*g*p*q^3*e - b^3*h*p*q^3*e^2)*log(d)^2 + 4*(2*b^3*f^2*g*
p^3*q^3 - 2*a*b^2*f^2*g*p^2*q^2 + a^2*b*f^2*g*p*q)*x - (7*b^3*h*p^3*q^3 - 6*a*b^2*h*p^2*q^2 + 2*a^2*b*h*p*q)*e
^2 + 2*(4*b^3*f*g*p^3*q^3 - 4*a*b^2*f*g*p^2*q^2 + 2*a^2*b*f*g*p*q - (3*b^3*f*h*p^3*q^3 - 2*a*b^2*f*h*p^2*q^2)*
x)*e - 2*((b^3*f^2*h*p^2*q^2 - 2*a*b^2*f^2*h*p*q)*x^2 + 4*(b^3*f^2*g*p^2*q^2 - a*b^2*f^2*g*p*q)*x - (3*b^3*h*p
^2*q^2 - 2*a*b^2*h*p*q)*e^2 - 2*(b^3*f*h*p^2*q^2*x - 2*b^3*f*g*p^2*q^2 + 2*a*b^2*f*g*p*q)*e)*log(c) - 2*((b^3*
f^2*h*p^2*q^3 - 2*a*b^2*f^2*h*p*q^2)*x^2 + 4*(b^3*f^2*g*p^2*q^3 - a*b^2*f^2*g*p*q^2)*x - (3*b^3*h*p^2*q^3 - 2*
a*b^2*h*p*q^2)*e^2 - 2*(b^3*f*h*p^2*q^3*x - 2*b^3*f*g*p^2*q^3 + 2*a*b^2*f*g*p*q^2)*e - 2*(b^3*f^2*h*p*q^2*x^2
+ 2*b^3*f^2*g*p*q^2*x + 2*b^3*f*g*p*q^2*e - b^3*h*p*q^2*e^2)*log(c))*log(d))*log(f*x + e) + 6*((b^3*f^2*h*p^2*
q^2 - 2*a*b^2*f^2*h*p*q + 2*a^2*b*f^2*h)*x^2 - 2*(3*b^3*f*h*p^2*q^2 - 2*a*b^2*f*h*p*q)*x*e + 4*(2*b^3*f^2*g*p^
2*q^2 - 2*a*b^2*f^2*g*p*q + a^2*b*f^2*g)*x)*log(c) + 6*((b^3*f^2*h*p^2*q^3 - 2*a*b^2*f^2*h*p*q^2 + 2*a^2*b*f^2
*h*q)*x^2 - 2*(3*b^3*f*h*p^2*q^3 - 2*a*b^2*f*h*p*q^2)*x*e + 2*(b^3*f^2*h*q*x^2 + 2*b^3*f^2*g*q*x)*log(c)^2 + 4
*(2*b^3*f^2*g*p^2*q^3 - 2*a*b^2*f^2*g*p*q^2 + a^2*b*f^2*g*q)*x + 2*(2*b^3*f*h*p*q^2*x*e - (b^3*f^2*h*p*q^2 - 2
*a*b^2*f^2*h*q)*x^2 - 4*(b^3*f^2*g*p*q^2 - a*b^2*f^2*g*q)*x)*log(c))*log(d))/f^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 991 vs. \(2 (299) = 598\).
time = 3.17, size = 991, normalized size = 3.24 \begin {gather*} \begin {cases} a^{3} g x + \frac {a^{3} h x^{2}}{2} - \frac {3 a^{2} b e^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f^{2}} + \frac {3 a^{2} b e g \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {3 a^{2} b e h p q x}{2 f} - 3 a^{2} b g p q x + 3 a^{2} b g x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {3 a^{2} b h p q x^{2}}{4} + \frac {3 a^{2} b h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2} + \frac {9 a b^{2} e^{2} h p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f^{2}} - \frac {3 a b^{2} e^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2 f^{2}} - \frac {6 a b^{2} e g p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {3 a b^{2} e g \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} - \frac {9 a b^{2} e h p^{2} q^{2} x}{2 f} + \frac {3 a b^{2} e h p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + 6 a b^{2} g p^{2} q^{2} x - 6 a b^{2} g p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + 3 a b^{2} g x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {3 a b^{2} h p^{2} q^{2} x^{2}}{4} - \frac {3 a b^{2} h p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2} + \frac {3 a b^{2} h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2} - \frac {21 b^{3} e^{2} h p^{2} q^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{4 f^{2}} + \frac {9 b^{3} e^{2} h p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{4 f^{2}} - \frac {b^{3} e^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3}}{2 f^{2}} + \frac {6 b^{3} e g p^{2} q^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - \frac {3 b^{3} e g p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} + \frac {b^{3} e g \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3}}{f} + \frac {21 b^{3} e h p^{3} q^{3} x}{4 f} - \frac {9 b^{3} e h p^{2} q^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2 f} + \frac {3 b^{3} e h p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2 f} - 6 b^{3} g p^{3} q^{3} x + 6 b^{3} g p^{2} q^{2} x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - 3 b^{3} g p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + b^{3} g x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3} - \frac {3 b^{3} h p^{3} q^{3} x^{2}}{8} + \frac {3 b^{3} h p^{2} q^{2} x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{4} - \frac {3 b^{3} h p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{4} + \frac {b^{3} h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{3}}{2} & \text {for}\: f \neq 0 \\\left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right )^{3} \left (g x + \frac {h x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*ln(c*(d*(f*x+e)**p)**q))**3,x)

[Out]

Piecewise((a**3*g*x + a**3*h*x**2/2 - 3*a**2*b*e**2*h*log(c*(d*(e + f*x)**p)**q)/(2*f**2) + 3*a**2*b*e*g*log(c
*(d*(e + f*x)**p)**q)/f + 3*a**2*b*e*h*p*q*x/(2*f) - 3*a**2*b*g*p*q*x + 3*a**2*b*g*x*log(c*(d*(e + f*x)**p)**q
) - 3*a**2*b*h*p*q*x**2/4 + 3*a**2*b*h*x**2*log(c*(d*(e + f*x)**p)**q)/2 + 9*a*b**2*e**2*h*p*q*log(c*(d*(e + f
*x)**p)**q)/(2*f**2) - 3*a*b**2*e**2*h*log(c*(d*(e + f*x)**p)**q)**2/(2*f**2) - 6*a*b**2*e*g*p*q*log(c*(d*(e +
f*x)**p)**q)/f + 3*a*b**2*e*g*log(c*(d*(e + f*x)**p)**q)**2/f - 9*a*b**2*e*h*p**2*q**2*x/(2*f) + 3*a*b**2*e*h
*p*q*x*log(c*(d*(e + f*x)**p)**q)/f + 6*a*b**2*g*p**2*q**2*x - 6*a*b**2*g*p*q*x*log(c*(d*(e + f*x)**p)**q) + 3
*a*b**2*g*x*log(c*(d*(e + f*x)**p)**q)**2 + 3*a*b**2*h*p**2*q**2*x**2/4 - 3*a*b**2*h*p*q*x**2*log(c*(d*(e + f*
x)**p)**q)/2 + 3*a*b**2*h*x**2*log(c*(d*(e + f*x)**p)**q)**2/2 - 21*b**3*e**2*h*p**2*q**2*log(c*(d*(e + f*x)**
p)**q)/(4*f**2) + 9*b**3*e**2*h*p*q*log(c*(d*(e + f*x)**p)**q)**2/(4*f**2) - b**3*e**2*h*log(c*(d*(e + f*x)**p
)**q)**3/(2*f**2) + 6*b**3*e*g*p**2*q**2*log(c*(d*(e + f*x)**p)**q)/f - 3*b**3*e*g*p*q*log(c*(d*(e + f*x)**p)*
*q)**2/f + b**3*e*g*log(c*(d*(e + f*x)**p)**q)**3/f + 21*b**3*e*h*p**3*q**3*x/(4*f) - 9*b**3*e*h*p**2*q**2*x*l
og(c*(d*(e + f*x)**p)**q)/(2*f) + 3*b**3*e*h*p*q*x*log(c*(d*(e + f*x)**p)**q)**2/(2*f) - 6*b**3*g*p**3*q**3*x
+ 6*b**3*g*p**2*q**2*x*log(c*(d*(e + f*x)**p)**q) - 3*b**3*g*p*q*x*log(c*(d*(e + f*x)**p)**q)**2 + b**3*g*x*lo
g(c*(d*(e + f*x)**p)**q)**3 - 3*b**3*h*p**3*q**3*x**2/8 + 3*b**3*h*p**2*q**2*x**2*log(c*(d*(e + f*x)**p)**q)/4
- 3*b**3*h*p*q*x**2*log(c*(d*(e + f*x)**p)**q)**2/4 + b**3*h*x**2*log(c*(d*(e + f*x)**p)**q)**3/2, Ne(f, 0)),
((a + b*log(c*(d*e**p)**q))**3*(g*x + h*x**2/2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2717 vs. \(2 (316) = 632\).
time = 5.70, size = 2717, normalized size = 8.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="giac")

[Out]

(f*x + e)*b^3*g*p^3*q^3*log(f*x + e)^3/f + 1/2*(f*x + e)^2*b^3*h*p^3*q^3*log(f*x + e)^3/f^2 - (f*x + e)*b^3*h*
p^3*q^3*e*log(f*x + e)^3/f^2 - 3*(f*x + e)*b^3*g*p^3*q^3*log(f*x + e)^2/f - 3/4*(f*x + e)^2*b^3*h*p^3*q^3*log(
f*x + e)^2/f^2 + 3*(f*x + e)*b^3*h*p^3*q^3*e*log(f*x + e)^2/f^2 + 3*(f*x + e)*b^3*g*p^2*q^3*log(f*x + e)^2*log
(d)/f + 3/2*(f*x + e)^2*b^3*h*p^2*q^3*log(f*x + e)^2*log(d)/f^2 - 3*(f*x + e)*b^3*h*p^2*q^3*e*log(f*x + e)^2*l
og(d)/f^2 + 6*(f*x + e)*b^3*g*p^3*q^3*log(f*x + e)/f + 3/4*(f*x + e)^2*b^3*h*p^3*q^3*log(f*x + e)/f^2 - 6*(f*x
+ e)*b^3*h*p^3*q^3*e*log(f*x + e)/f^2 + 3*(f*x + e)*b^3*g*p^2*q^2*log(f*x + e)^2*log(c)/f + 3/2*(f*x + e)^2*b
^3*h*p^2*q^2*log(f*x + e)^2*log(c)/f^2 - 3*(f*x + e)*b^3*h*p^2*q^2*e*log(f*x + e)^2*log(c)/f^2 - 6*(f*x + e)*b
^3*g*p^2*q^3*log(f*x + e)*log(d)/f - 3/2*(f*x + e)^2*b^3*h*p^2*q^3*log(f*x + e)*log(d)/f^2 + 6*(f*x + e)*b^3*h
*p^2*q^3*e*log(f*x + e)*log(d)/f^2 + 3*(f*x + e)*b^3*g*p*q^3*log(f*x + e)*log(d)^2/f + 3/2*(f*x + e)^2*b^3*h*p
*q^3*log(f*x + e)*log(d)^2/f^2 - 3*(f*x + e)*b^3*h*p*q^3*e*log(f*x + e)*log(d)^2/f^2 - 6*(f*x + e)*b^3*g*p^3*q
^3/f - 3/8*(f*x + e)^2*b^3*h*p^3*q^3/f^2 + 6*(f*x + e)*b^3*h*p^3*q^3*e/f^2 + 3*(f*x + e)*a*b^2*g*p^2*q^2*log(f
*x + e)^2/f + 3/2*(f*x + e)^2*a*b^2*h*p^2*q^2*log(f*x + e)^2/f^2 - 3*(f*x + e)*a*b^2*h*p^2*q^2*e*log(f*x + e)^
2/f^2 - 6*(f*x + e)*b^3*g*p^2*q^2*log(f*x + e)*log(c)/f - 3/2*(f*x + e)^2*b^3*h*p^2*q^2*log(f*x + e)*log(c)/f^
2 + 6*(f*x + e)*b^3*h*p^2*q^2*e*log(f*x + e)*log(c)/f^2 + 6*(f*x + e)*b^3*g*p^2*q^3*log(d)/f + 3/4*(f*x + e)^2
*b^3*h*p^2*q^3*log(d)/f^2 - 6*(f*x + e)*b^3*h*p^2*q^3*e*log(d)/f^2 + 6*(f*x + e)*b^3*g*p*q^2*log(f*x + e)*log(
c)*log(d)/f + 3*(f*x + e)^2*b^3*h*p*q^2*log(f*x + e)*log(c)*log(d)/f^2 - 6*(f*x + e)*b^3*h*p*q^2*e*log(f*x + e
)*log(c)*log(d)/f^2 - 3*(f*x + e)*b^3*g*p*q^3*log(d)^2/f - 3/4*(f*x + e)^2*b^3*h*p*q^3*log(d)^2/f^2 + 3*(f*x +
e)*b^3*h*p*q^3*e*log(d)^2/f^2 + (f*x + e)*b^3*g*q^3*log(d)^3/f + 1/2*(f*x + e)^2*b^3*h*q^3*log(d)^3/f^2 - (f*
x + e)*b^3*h*q^3*e*log(d)^3/f^2 - 6*(f*x + e)*a*b^2*g*p^2*q^2*log(f*x + e)/f - 3/2*(f*x + e)^2*a*b^2*h*p^2*q^2
*log(f*x + e)/f^2 + 6*(f*x + e)*a*b^2*h*p^2*q^2*e*log(f*x + e)/f^2 + 6*(f*x + e)*b^3*g*p^2*q^2*log(c)/f + 3/4*
(f*x + e)^2*b^3*h*p^2*q^2*log(c)/f^2 - 6*(f*x + e)*b^3*h*p^2*q^2*e*log(c)/f^2 + 3*(f*x + e)*b^3*g*p*q*log(f*x
+ e)*log(c)^2/f + 3/2*(f*x + e)^2*b^3*h*p*q*log(f*x + e)*log(c)^2/f^2 - 3*(f*x + e)*b^3*h*p*q*e*log(f*x + e)*l
og(c)^2/f^2 + 6*(f*x + e)*a*b^2*g*p*q^2*log(f*x + e)*log(d)/f + 3*(f*x + e)^2*a*b^2*h*p*q^2*log(f*x + e)*log(d
)/f^2 - 6*(f*x + e)*a*b^2*h*p*q^2*e*log(f*x + e)*log(d)/f^2 - 6*(f*x + e)*b^3*g*p*q^2*log(c)*log(d)/f - 3/2*(f
*x + e)^2*b^3*h*p*q^2*log(c)*log(d)/f^2 + 6*(f*x + e)*b^3*h*p*q^2*e*log(c)*log(d)/f^2 + 3*(f*x + e)*b^3*g*q^2*
log(c)*log(d)^2/f + 3/2*(f*x + e)^2*b^3*h*q^2*log(c)*log(d)^2/f^2 - 3*(f*x + e)*b^3*h*q^2*e*log(c)*log(d)^2/f^
2 + 6*(f*x + e)*a*b^2*g*p^2*q^2/f + 3/4*(f*x + e)^2*a*b^2*h*p^2*q^2/f^2 - 6*(f*x + e)*a*b^2*h*p^2*q^2*e/f^2 +
6*(f*x + e)*a*b^2*g*p*q*log(f*x + e)*log(c)/f + 3*(f*x + e)^2*a*b^2*h*p*q*log(f*x + e)*log(c)/f^2 - 6*(f*x + e
)*a*b^2*h*p*q*e*log(f*x + e)*log(c)/f^2 - 3*(f*x + e)*b^3*g*p*q*log(c)^2/f - 3/4*(f*x + e)^2*b^3*h*p*q*log(c)^
2/f^2 + 3*(f*x + e)*b^3*h*p*q*e*log(c)^2/f^2 - 6*(f*x + e)*a*b^2*g*p*q^2*log(d)/f - 3/2*(f*x + e)^2*a*b^2*h*p*
q^2*log(d)/f^2 + 6*(f*x + e)*a*b^2*h*p*q^2*e*log(d)/f^2 + 3*(f*x + e)*b^3*g*q*log(c)^2*log(d)/f + 3/2*(f*x + e
)^2*b^3*h*q*log(c)^2*log(d)/f^2 - 3*(f*x + e)*b^3*h*q*e*log(c)^2*log(d)/f^2 + 3*(f*x + e)*a*b^2*g*q^2*log(d)^2
/f + 3/2*(f*x + e)^2*a*b^2*h*q^2*log(d)^2/f^2 - 3*(f*x + e)*a*b^2*h*q^2*e*log(d)^2/f^2 + 3*(f*x + e)*a^2*b*g*p
*q*log(f*x + e)/f + 3/2*(f*x + e)^2*a^2*b*h*p*q*log(f*x + e)/f^2 - 3*(f*x + e)*a^2*b*h*p*q*e*log(f*x + e)/f^2
- 6*(f*x + e)*a*b^2*g*p*q*log(c)/f - 3/2*(f*x + e)^2*a*b^2*h*p*q*log(c)/f^2 + 6*(f*x + e)*a*b^2*h*p*q*e*log(c)
/f^2 + (f*x + e)*b^3*g*log(c)^3/f + 1/2*(f*x + e)^2*b^3*h*log(c)^3/f^2 - (f*x + e)*b^3*h*e*log(c)^3/f^2 + 6*(f
*x + e)*a*b^2*g*q*log(c)*log(d)/f + 3*(f*x + e)^2*a*b^2*h*q*log(c)*log(d)/f^2 - 6*(f*x + e)*a*b^2*h*q*e*log(c)
*log(d)/f^2 - 3*(f*x + e)*a^2*b*g*p*q/f - 3/4*(f*x + e)^2*a^2*b*h*p*q/f^2 + 3*(f*x + e)*a^2*b*h*p*q*e/f^2 + 3*
(f*x + e)*a*b^2*g*log(c)^2/f + 3/2*(f*x + e)^2*a*b^2*h*log(c)^2/f^2 - 3*(f*x + e)*a*b^2*h*e*log(c)^2/f^2 + 3*(
f*x + e)*a^2*b*g*q*log(d)/f + 3/2*(f*x + e)^2*a^2*b*h*q*log(d)/f^2 - 3*(f*x + e)*a^2*b*h*q*e*log(d)/f^2 + 3*(f
*x + e)*a^2*b*g*log(c)/f + 3/2*(f*x + e)^2*a^2*b*h*log(c)/f^2 - 3*(f*x + e)*a^2*b*h*e*log(c)/f^2 + (f*x + e)*a
^3*g/f + 1/2*(f*x + e)^2*a^3*h/f^2 - (f*x + e)*a^3*h*e/f^2

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Mupad [B]
time = 0.83, size = 651, normalized size = 2.13 \begin {gather*} x\,\left (\frac {4\,a^3\,e\,h+4\,a^3\,f\,g+18\,b^3\,e\,h\,p^3\,q^3-24\,b^3\,f\,g\,p^3\,q^3-12\,a^2\,b\,f\,g\,p\,q-12\,a\,b^2\,e\,h\,p^2\,q^2+24\,a\,b^2\,f\,g\,p^2\,q^2}{4\,f}-\frac {e\,h\,\left (4\,a^3-6\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-3\,b^3\,p^3\,q^3\right )}{4\,f}\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (\frac {x\,\left (\frac {6\,b^2\,\left (a\,e\,h+a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {3\,b^2\,e\,h\,\left (2\,a-b\,p\,q\right )}{f}\right )}{2}-\frac {3\,e\,\left (2\,a\,b^2\,e\,h-4\,a\,b^2\,f\,g-3\,b^3\,e\,h\,p\,q+4\,b^3\,f\,g\,p\,q\right )}{4\,f^2}+\frac {3\,b^2\,h\,x^2\,\left (2\,a-b\,p\,q\right )}{4}\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^3\,\left (\frac {b^3\,h\,x^2}{2}-\frac {e\,\left (b^3\,e\,h-2\,b^3\,f\,g\right )}{2\,f^2}+b^3\,g\,x\right )+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (x^2\,\left (6\,a^2\,b\,f\,g+\frac {3\,b\,e\,h\,\left (2\,a^2-2\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{2}-9\,b^3\,e\,h\,p^2\,q^2+12\,b^3\,f\,g\,p^2\,q^2+6\,a\,b^2\,e\,h\,p\,q-12\,a\,b^2\,f\,g\,p\,q\right )+\frac {3\,e\,x\,\left (2\,a^2\,b\,f\,g-3\,b^3\,e\,h\,p^2\,q^2+4\,b^3\,f\,g\,p^2\,q^2+2\,a\,b^2\,e\,h\,p\,q-4\,a\,b^2\,f\,g\,p\,q\right )}{f}+\frac {3\,b\,f\,h\,x^3\,\left (2\,a^2-2\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{2}\right )}{2\,e+2\,f\,x}+\frac {h\,x^2\,\left (4\,a^3-6\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-3\,b^3\,p^3\,q^3\right )}{8}-\frac {\ln \left (e+f\,x\right )\,\left (6\,h\,a^2\,b\,e^2\,p\,q-12\,f\,g\,a^2\,b\,e\,p\,q-18\,h\,a\,b^2\,e^2\,p^2\,q^2+24\,f\,g\,a\,b^2\,e\,p^2\,q^2+21\,h\,b^3\,e^2\,p^3\,q^3-24\,f\,g\,b^3\,e\,p^3\,q^3\right )}{4\,f^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)*(a + b*log(c*(d*(e + f*x)^p)^q))^3,x)

[Out]

x*((4*a^3*e*h + 4*a^3*f*g + 18*b^3*e*h*p^3*q^3 - 24*b^3*f*g*p^3*q^3 - 12*a^2*b*f*g*p*q - 12*a*b^2*e*h*p^2*q^2
+ 24*a*b^2*f*g*p^2*q^2)/(4*f) - (e*h*(4*a^3 - 3*b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 6*a^2*b*p*q))/(4*f)) + log(c*(
d*(e + f*x)^p)^q)^2*((x*((6*b^2*(a*e*h + a*f*g - b*f*g*p*q))/f - (3*b^2*e*h*(2*a - b*p*q))/f))/2 - (3*e*(2*a*b
^2*e*h - 4*a*b^2*f*g - 3*b^3*e*h*p*q + 4*b^3*f*g*p*q))/(4*f^2) + (3*b^2*h*x^2*(2*a - b*p*q))/4) + log(c*(d*(e
+ f*x)^p)^q)^3*((b^3*h*x^2)/2 - (e*(b^3*e*h - 2*b^3*f*g))/(2*f^2) + b^3*g*x) + (log(c*(d*(e + f*x)^p)^q)*(x^2*
(6*a^2*b*f*g + (3*b*e*h*(2*a^2 + b^2*p^2*q^2 - 2*a*b*p*q))/2 - 9*b^3*e*h*p^2*q^2 + 12*b^3*f*g*p^2*q^2 + 6*a*b^
2*e*h*p*q - 12*a*b^2*f*g*p*q) + (3*e*x*(2*a^2*b*f*g - 3*b^3*e*h*p^2*q^2 + 4*b^3*f*g*p^2*q^2 + 2*a*b^2*e*h*p*q
- 4*a*b^2*f*g*p*q))/f + (3*b*f*h*x^3*(2*a^2 + b^2*p^2*q^2 - 2*a*b*p*q))/2))/(2*e + 2*f*x) + (h*x^2*(4*a^3 - 3*
b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 6*a^2*b*p*q))/8 - (log(e + f*x)*(21*b^3*e^2*h*p^3*q^3 - 18*a*b^2*e^2*h*p^2*q^2
+ 6*a^2*b*e^2*h*p*q - 24*b^3*e*f*g*p^3*q^3 + 24*a*b^2*e*f*g*p^2*q^2 - 12*a^2*b*e*f*g*p*q))/(4*f^2)

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