3.5.30 \(\int (g+h x) (a+b \log (c (d (e+f x)^p)^q))^2 \, dx\) [430]

Optimal. Leaf size=211 \[ -\frac {2 a b (f g-e h) p q x}{f}+\frac {2 b^2 (f g-e h) p^2 q^2 x}{f}+\frac {b^2 h p^2 q^2 (e+f x)^2}{4 f^2}-\frac {2 b^2 (f g-e h) p q (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}-\frac {b h p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^2}+\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2} \]

[Out]

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

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Rubi [A]
time = 0.28, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {(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 {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 f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^2}-\frac {2 a b p q x (f g-e h)}{f}-\frac {2 b^2 p q (e+f x) (f g-e h) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^2}+\frac {b^2 h p^2 q^2 (e+f x)^2}{4 f^2}+\frac {2 b^2 p^2 q^2 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])^2,x]

[Out]

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

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Mathematica [A]
time = 0.07, size = 164, normalized size = 0.78 \begin {gather*} \frac {4 (f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+2 h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-8 b (f g-e h) 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 )+b h 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 )}{4 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2,x]

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (h x +g \right ) \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}\, dx\]

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))^2,x)

[Out]

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2,x)

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Maxima [A]
time = 0.29, size = 365, normalized size = 1.73 \begin {gather*} -2 \, a b f g p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} - \frac {1}{2} \, a 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 {1}{2} \, b^{2} h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + a b h x^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + b^{2} g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + \frac {1}{2} \, a^{2} h x^{2} + 2 \, a b g x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) - {\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 )} b^{2} g - \frac {1}{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 )} b^{2} h + a^{2} 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (218) = 436\).
time = 0.37, size = 646, normalized size = 3.06 \begin {gather*} \frac {{\left (b^{2} f^{2} h p^{2} q^{2} - 2 \, a b f^{2} h p q + 2 \, a^{2} f^{2} h\right )} x^{2} - 2 \, {\left (3 \, b^{2} f h p^{2} q^{2} - 2 \, a b f h p q\right )} x e + 2 \, {\left (b^{2} f^{2} h p^{2} q^{2} x^{2} + 2 \, b^{2} f^{2} g p^{2} q^{2} x + 2 \, b^{2} f g p^{2} q^{2} e - b^{2} h p^{2} q^{2} e^{2}\right )} \log \left (f x + e\right )^{2} + 2 \, {\left (b^{2} f^{2} h x^{2} + 2 \, b^{2} f^{2} g x\right )} \log \left (c\right )^{2} + 2 \, {\left (b^{2} f^{2} h q^{2} x^{2} + 2 \, b^{2} f^{2} g q^{2} x\right )} \log \left (d\right )^{2} + 4 \, {\left (2 \, b^{2} f^{2} g p^{2} q^{2} - 2 \, a b f^{2} g p q + a^{2} f^{2} g\right )} x - 2 \, {\left ({\left (b^{2} f^{2} h p^{2} q^{2} - 2 \, a b f^{2} h p q\right )} x^{2} + 4 \, {\left (b^{2} f^{2} g p^{2} q^{2} - a b f^{2} g p q\right )} x - {\left (3 \, b^{2} h p^{2} q^{2} - 2 \, a b h p q\right )} e^{2} - 2 \, {\left (b^{2} f h p^{2} q^{2} x - 2 \, b^{2} f g p^{2} q^{2} + 2 \, a b f g p q\right )} e - 2 \, {\left (b^{2} f^{2} h p q x^{2} + 2 \, b^{2} f^{2} g p q x + 2 \, b^{2} f g p q e - b^{2} h p q e^{2}\right )} \log \left (c\right ) - 2 \, {\left (b^{2} f^{2} h p q^{2} x^{2} + 2 \, b^{2} f^{2} g p q^{2} x + 2 \, b^{2} f g p q^{2} e - b^{2} h p q^{2} e^{2}\right )} \log \left (d\right )\right )} \log \left (f x + e\right ) + 2 \, {\left (2 \, b^{2} f h p q x e - {\left (b^{2} f^{2} h p q - 2 \, a b f^{2} h\right )} x^{2} - 4 \, {\left (b^{2} f^{2} g p q - a b f^{2} g\right )} x\right )} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} f h p q^{2} x e - {\left (b^{2} f^{2} h p q^{2} - 2 \, a b f^{2} h q\right )} x^{2} - 4 \, {\left (b^{2} f^{2} g p q^{2} - a b f^{2} g q\right )} x + 2 \, {\left (b^{2} f^{2} h q x^{2} + 2 \, b^{2} f^{2} g q x\right )} \log \left (c\right )\right )} \log \left (d\right )}{4 \, 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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (202) = 404\).
time = 1.41, size = 466, normalized size = 2.21 \begin {gather*} \begin {cases} a^{2} g x + \frac {a^{2} h x^{2}}{2} - \frac {a b e^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f^{2}} + \frac {2 a b e g \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {a b e h p q x}{f} - 2 a b g p q x + 2 a b g x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} - \frac {a b h p q x^{2}}{2} + a b h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + \frac {3 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 {b^{2} e^{2} h \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2 f^{2}} - \frac {2 b^{2} e g p q \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + \frac {b^{2} e g \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{f} - \frac {3 b^{2} e h p^{2} q^{2} x}{2 f} + \frac {b^{2} e h p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} + 2 b^{2} g p^{2} q^{2} x - 2 b^{2} g p q x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} + b^{2} g x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2} + \frac {b^{2} h p^{2} q^{2} x^{2}}{4} - \frac {b^{2} h p q x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{2} + \frac {b^{2} h x^{2} \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}^{2}}{2} & \text {for}\: f \neq 0 \\\left (a + b \log {\left (c \left (d e^{p}\right )^{q} \right )}\right )^{2} \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))**2,x)

[Out]

Piecewise((a**2*g*x + a**2*h*x**2/2 - a*b*e**2*h*log(c*(d*(e + f*x)**p)**q)/f**2 + 2*a*b*e*g*log(c*(d*(e + f*x
)**p)**q)/f + a*b*e*h*p*q*x/f - 2*a*b*g*p*q*x + 2*a*b*g*x*log(c*(d*(e + f*x)**p)**q) - a*b*h*p*q*x**2/2 + a*b*
h*x**2*log(c*(d*(e + f*x)**p)**q) + 3*b**2*e**2*h*p*q*log(c*(d*(e + f*x)**p)**q)/(2*f**2) - b**2*e**2*h*log(c*
(d*(e + f*x)**p)**q)**2/(2*f**2) - 2*b**2*e*g*p*q*log(c*(d*(e + f*x)**p)**q)/f + b**2*e*g*log(c*(d*(e + f*x)**
p)**q)**2/f - 3*b**2*e*h*p**2*q**2*x/(2*f) + b**2*e*h*p*q*x*log(c*(d*(e + f*x)**p)**q)/f + 2*b**2*g*p**2*q**2*
x - 2*b**2*g*p*q*x*log(c*(d*(e + f*x)**p)**q) + b**2*g*x*log(c*(d*(e + f*x)**p)**q)**2 + b**2*h*p**2*q**2*x**2
/4 - b**2*h*p*q*x**2*log(c*(d*(e + f*x)**p)**q)/2 + b**2*h*x**2*log(c*(d*(e + f*x)**p)**q)**2/2, Ne(f, 0)), ((
a + b*log(c*(d*e**p)**q))**2*(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. 1014 vs. \(2 (218) = 436\).
time = 4.94, size = 1014, normalized size = 4.81 \begin {gather*} \frac {{\left (f x + e\right )} b^{2} g p^{2} q^{2} \log \left (f x + e\right )^{2}}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h p^{2} q^{2} \log \left (f x + e\right )^{2}}{2 \, f^{2}} - \frac {{\left (f x + e\right )} b^{2} h p^{2} q^{2} e \log \left (f x + e\right )^{2}}{f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} g p^{2} q^{2} \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )}^{2} b^{2} h p^{2} q^{2} \log \left (f x + e\right )}{2 \, f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} h p^{2} q^{2} e \log \left (f x + e\right )}{f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} g p q^{2} \log \left (f x + e\right ) \log \left (d\right )}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h p q^{2} \log \left (f x + e\right ) \log \left (d\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} h p q^{2} e \log \left (f x + e\right ) \log \left (d\right )}{f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} g p^{2} q^{2}}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h p^{2} q^{2}}{4 \, f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} h p^{2} q^{2} e}{f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} g p q \log \left (f x + e\right ) \log \left (c\right )}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h p q \log \left (f x + e\right ) \log \left (c\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} h p q e \log \left (f x + e\right ) \log \left (c\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} g p q^{2} \log \left (d\right )}{f} - \frac {{\left (f x + e\right )}^{2} b^{2} h p q^{2} \log \left (d\right )}{2 \, f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} h p q^{2} e \log \left (d\right )}{f^{2}} + \frac {{\left (f x + e\right )} b^{2} g q^{2} \log \left (d\right )^{2}}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h q^{2} \log \left (d\right )^{2}}{2 \, f^{2}} - \frac {{\left (f x + e\right )} b^{2} h q^{2} e \log \left (d\right )^{2}}{f^{2}} + \frac {2 \, {\left (f x + e\right )} a b g p q \log \left (f x + e\right )}{f} + \frac {{\left (f x + e\right )}^{2} a b h p q \log \left (f x + e\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} a b h p q e \log \left (f x + e\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} g p q \log \left (c\right )}{f} - \frac {{\left (f x + e\right )}^{2} b^{2} h p q \log \left (c\right )}{2 \, f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} h p q e \log \left (c\right )}{f^{2}} + \frac {2 \, {\left (f x + e\right )} b^{2} g q \log \left (c\right ) \log \left (d\right )}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h q \log \left (c\right ) \log \left (d\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} b^{2} h q e \log \left (c\right ) \log \left (d\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} a b g p q}{f} - \frac {{\left (f x + e\right )}^{2} a b h p q}{2 \, f^{2}} + \frac {2 \, {\left (f x + e\right )} a b h p q e}{f^{2}} + \frac {{\left (f x + e\right )} b^{2} g \log \left (c\right )^{2}}{f} + \frac {{\left (f x + e\right )}^{2} b^{2} h \log \left (c\right )^{2}}{2 \, f^{2}} - \frac {{\left (f x + e\right )} b^{2} h e \log \left (c\right )^{2}}{f^{2}} + \frac {2 \, {\left (f x + e\right )} a b g q \log \left (d\right )}{f} + \frac {{\left (f x + e\right )}^{2} a b h q \log \left (d\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} a b h q e \log \left (d\right )}{f^{2}} + \frac {2 \, {\left (f x + e\right )} a b g \log \left (c\right )}{f} + \frac {{\left (f x + e\right )}^{2} a b h \log \left (c\right )}{f^{2}} - \frac {2 \, {\left (f x + e\right )} a b h e \log \left (c\right )}{f^{2}} + \frac {{\left (f x + e\right )} a^{2} g}{f} + \frac {{\left (f x + e\right )}^{2} a^{2} h}{2 \, f^{2}} - \frac {{\left (f x + e\right )} a^{2} h e}{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))^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.46, size = 302, normalized size = 1.43 \begin {gather*} x\,\left (\frac {2\,a^2\,e\,h+2\,a^2\,f\,g-2\,b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-4\,a\,b\,f\,g\,p\,q}{2\,f}-\frac {e\,h\,\left (2\,a^2-2\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{2\,f}\right )+\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (\frac {b\,h\,\left (2\,a-b\,p\,q\right )\,x^2}{2}+\left (\frac {2\,b\,\left (a\,e\,h+a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {b\,e\,h\,\left (2\,a-b\,p\,q\right )}{f}\right )\,x\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (\frac {b^2\,h\,x^2}{2}-\frac {e\,\left (b^2\,e\,h-2\,b^2\,f\,g\right )}{2\,f^2}+b^2\,g\,x\right )+\frac {\ln \left (e+f\,x\right )\,\left (3\,h\,b^2\,e^2\,p^2\,q^2-4\,f\,g\,b^2\,e\,p^2\,q^2-2\,a\,h\,b\,e^2\,p\,q+4\,a\,f\,g\,b\,e\,p\,q\right )}{2\,f^2}+\frac {h\,x^2\,\left (2\,a^2-2\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{4} \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))^2,x)

[Out]

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

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