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

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

[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)

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Rubi [A]  time = 0.534299, antiderivative size = 306, normalized size of antiderivative = 1., 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 = {2401, 2389, 2296, 2295, 2390, 2305, 2304, 2445} \[ \frac{3 b^2 h p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{4 f^2}+\frac{6 a b^2 p^2 q^2 x (f g-e h)}{f}-\frac{3 b p q (e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}+\frac{(e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^2}-\frac{3 b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 f^2}+\frac{6 b^3 p^2 q^2 (e+f x) (f g-e h) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}-\frac{3 b^3 h p^3 q^3 (e+f x)^2}{8 f^2}-\frac{6 b^3 p^3 q^3 x (f g-e h)}{f} \]

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 2401

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 2389

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 2296

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 2295

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

Rule 2390

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 2305

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 2304

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 2445

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 &=\operatorname{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 )\\ &=\operatorname{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 )\\ &=\operatorname{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 )+\operatorname{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 )\\ &=\operatorname{Subst}\left (\frac{h \operatorname{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 )+\operatorname{Subst}\left (\frac{(f g-e h) \operatorname{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}-\operatorname{Subst}\left (\frac{(3 b h p q) \operatorname{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 )-\operatorname{Subst}\left (\frac{(3 b (f g-e h) p q) \operatorname{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}+\operatorname{Subst}\left (\frac{\left (3 b^2 h p^2 q^2\right ) \operatorname{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 )+\operatorname{Subst}\left (\frac{\left (6 b^2 (f g-e h) p^2 q^2\right ) \operatorname{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}+\operatorname{Subst}\left (\frac{\left (6 b^3 (f g-e h) p^2 q^2\right ) \operatorname{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*}

Mathematica [A]  time = 0.123059, size = 231, normalized size = 0.75 \[ \frac{8 (e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-24 b p q (f g-e h) \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f x (a-b p q)+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+4 h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-3 b h p q \left (2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{8 f^2} \]

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 = 0.279, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{3}\, dx \end{align*}

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]  time = 1.20498, size = 988, normalized size = 3.23 \begin{align*} \text{result too large to display} \end{align*}

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*(2*e
^2*log(f*x + e)/f^3 + (f*x^2 - 2*e*x)/f^2) + 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*(2*e^
2*log(f*x + e)/f^3 + (f*x^2 - 2*e*x)/f^2)*log(((f*x + e)^p*d)^q*c) - (f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x
+ 6*e^2*log(f*x + e))*p^2*q^2/f^2)*a*b^2*h - 1/8*(6*f*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^2 - 2*e*x)/f^2)*log(
((f*x + e)^p*d)^q*c)^2 + ((4*e^2*log(f*x + e)^3 + 3*f^2*x^2 + 18*e^2*log(f*x + e)^2 - 42*e*f*x + 42*e^2*log(f*
x + e))*p^2*q^2/f^3 - 6*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 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]  time = 2.36837, size = 3510, normalized size = 11.47 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 31.6548, size = 2756, normalized size = 9.01 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.45472, size = 3668, normalized size = 11.99 \begin{align*} \text{result too large to display} \end{align*}

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