3.28 \(\int \frac{(c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x}))}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=89 \[ -\frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)}-\frac{B i^3 (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)} \]

[Out]

-(B*i^3*(c + d*x)^4)/(16*(b*c - a*d)*g^5*(a + b*x)^4) - (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/(4*(b*c - a*d)*g^5*(a + b*x)^4)

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Rubi [B]  time = 0.72228, antiderivative size = 373, normalized size of antiderivative = 4.19, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^3 i^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^5 (a+b x)}-\frac{3 d^2 i^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 g^5 (a+b x)^2}-\frac{d i^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^5 (a+b x)^3}-\frac{i^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^4 g^5 (a+b x)^4}-\frac{3 B d^2 i^3 (b c-a d)}{8 b^4 g^5 (a+b x)^2}-\frac{B d^4 i^3 \log (a+b x)}{4 b^4 g^5 (b c-a d)}+\frac{B d^4 i^3 \log (c+d x)}{4 b^4 g^5 (b c-a d)}-\frac{B d i^3 (b c-a d)^2}{4 b^4 g^5 (a+b x)^3}-\frac{B i^3 (b c-a d)^3}{16 b^4 g^5 (a+b x)^4}-\frac{B d^3 i^3}{4 b^4 g^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5,x]

[Out]

-(B*(b*c - a*d)^3*i^3)/(16*b^4*g^5*(a + b*x)^4) - (B*d*(b*c - a*d)^2*i^3)/(4*b^4*g^5*(a + b*x)^3) - (3*B*d^2*(
b*c - a*d)*i^3)/(8*b^4*g^5*(a + b*x)^2) - (B*d^3*i^3)/(4*b^4*g^5*(a + b*x)) - (B*d^4*i^3*Log[a + b*x])/(4*b^4*
(b*c - a*d)*g^5) - ((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^4*g^5*(a + b*x)^4) - (d*(b*c
- a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^5*(a + b*x)^3) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(2*b^4*g^5*(a + b*x)^2) - (d^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^5*(a
+ b*x)) + (B*d^4*i^3*Log[c + d*x])/(4*b^4*(b*c - a*d)*g^5)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(28 c+28 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac{21952 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^5}+\frac{65856 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}+\frac{65856 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}+\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}\right ) \, dx\\ &=\frac{\left (21952 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^5}+\frac{\left (65856 d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^5}+\frac{\left (65856 d (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^5}+\frac{\left (21952 (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^3 g^5}\\ &=-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (21952 B d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (32928 B d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (21952 B d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (5488 B (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^5}\\ &=-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (21952 B d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (32928 B d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (21952 B d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (5488 B (b c-a d)^4\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^5}\\ &=-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (21952 B d^3 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^5}+\frac{\left (32928 B d^2 (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^4 g^5}+\frac{\left (21952 B d (b c-a d)^3\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^4 g^5}+\frac{\left (5488 B (b c-a d)^4\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^4 g^5}\\ &=-\frac{1372 B (b c-a d)^3}{b^4 g^5 (a+b x)^4}-\frac{5488 B d (b c-a d)^2}{b^4 g^5 (a+b x)^3}-\frac{8232 B d^2 (b c-a d)}{b^4 g^5 (a+b x)^2}-\frac{5488 B d^3}{b^4 g^5 (a+b x)}-\frac{5488 B d^4 \log (a+b x)}{b^4 (b c-a d) g^5}-\frac{5488 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^4}-\frac{21952 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^3}-\frac{32928 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)^2}-\frac{21952 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{5488 B d^4 \log (c+d x)}{b^4 (b c-a d) g^5}\\ \end{align*}

Mathematica [B]  time = 0.495376, size = 427, normalized size = 4.8 \[ -\frac{i^3 \left (-24 a^2 A b^2 d^4 x^2-16 a^3 A b d^4 x-4 a^4 A d^4+4 B \left (-6 a^2 b^2 d^4 x^2-4 a^3 b d^4 x-a^4 d^4-4 a b^3 d^4 x^3+b^4 c \left (4 c^2 d x+c^3+6 c d^2 x^2+4 d^3 x^3\right )\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )-24 a^2 b^2 B d^4 x^2 \log (c+d x)-6 a^2 b^2 B d^4 x^2-16 a^3 b B d^4 x \log (c+d x)-4 a^3 b B d^4 x-4 a^4 B d^4 \log (c+d x)-a^4 B d^4-16 a A b^3 d^4 x^3-16 a b^3 B d^4 x^3 \log (c+d x)-4 a b^3 B d^4 x^3+4 B d^4 (a+b x)^4 \log (a+b x)+24 A b^4 c^2 d^2 x^2+16 A b^4 c^3 d x+4 A b^4 c^4+16 A b^4 c d^3 x^3+6 b^4 B c^2 d^2 x^2+4 b^4 B c^3 d x+b^4 B c^4+4 b^4 B c d^3 x^3-4 b^4 B d^4 x^4 \log (c+d x)\right )}{16 b^4 g^5 (a+b x)^4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5,x]

[Out]

-(i^3*(4*A*b^4*c^4 + b^4*B*c^4 - 4*a^4*A*d^4 - a^4*B*d^4 + 16*A*b^4*c^3*d*x + 4*b^4*B*c^3*d*x - 16*a^3*A*b*d^4
*x - 4*a^3*b*B*d^4*x + 24*A*b^4*c^2*d^2*x^2 + 6*b^4*B*c^2*d^2*x^2 - 24*a^2*A*b^2*d^4*x^2 - 6*a^2*b^2*B*d^4*x^2
+ 16*A*b^4*c*d^3*x^3 + 4*b^4*B*c*d^3*x^3 - 16*a*A*b^3*d^4*x^3 - 4*a*b^3*B*d^4*x^3 + 4*B*d^4*(a + b*x)^4*Log[a
+ b*x] + 4*B*(-(a^4*d^4) - 4*a^3*b*d^4*x - 6*a^2*b^2*d^4*x^2 - 4*a*b^3*d^4*x^3 + b^4*c*(c^3 + 4*c^2*d*x + 6*c
*d^2*x^2 + 4*d^3*x^3))*Log[(e*(a + b*x))/(c + d*x)] - 4*a^4*B*d^4*Log[c + d*x] - 16*a^3*b*B*d^4*x*Log[c + d*x]
- 24*a^2*b^2*B*d^4*x^2*Log[c + d*x] - 16*a*b^3*B*d^4*x^3*Log[c + d*x] - 4*b^4*B*d^4*x^4*Log[c + d*x]))/(16*b^
4*(b*c - a*d)*g^5*(a + b*x)^4)

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Maple [B]  time = 0.052, size = 406, normalized size = 4.6 \begin{align*}{\frac{{e}^{4}d{i}^{3}Aa}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}-{\frac{{e}^{4}{i}^{3}Abc}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}+{\frac{{e}^{4}d{i}^{3}Ba}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}-{\frac{{e}^{4}{i}^{3}Bbc}{4\, \left ( ad-bc \right ) ^{2}{g}^{5}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}+{\frac{{e}^{4}d{i}^{3}Ba}{16\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}}-{\frac{{e}^{4}{i}^{3}Bbc}{16\, \left ( ad-bc \right ) ^{2}{g}^{5}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x)

[Out]

1/4*e^4*d*i^3/(a*d-b*c)^2/g^5*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/4*e^4*i^3/(a*d-b*c)^2/g^5*A/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c+1/4*e^4*d*i^3/(a*d-b*c)^2/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e
/d+(a*d-b*c)*e/d/(d*x+c))*a-1/4*e^4*i^3/(a*d-b*c)^2/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-
b*c)*e/d/(d*x+c))*b*c+1/16*e^4*d*i^3/(a*d-b*c)^2/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/16*e^4*i^3/(a
*d-b*c)^2/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c

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Maxima [B]  time = 1.99735, size = 4194, normalized size = 47.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/48*B*d^3*i^3*(12*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^8*g^5*
x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5*x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) + (25*a^3*b^3*c^3 - 23*a^4*b^2*c^2*
d + 13*a^5*b*c*d^2 - 3*a^6*d^3 + 12*(4*b^6*c^3 - 6*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^3 + 6*(18*a*
b^5*c^3 - 22*a^2*b^4*c^2*d + 13*a^3*b^3*c*d^2 - 3*a^4*b^2*d^3)*x^2 + 4*(22*a^2*b^4*c^3 - 23*a^3*b^3*c^2*d + 13
*a^4*b^2*c*d^2 - 3*a^5*b*d^3)*x)/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^5*x^4 + 4*(a*b
^10*c^3 - 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^4*b^7*d^3)*g^5*x^3 + 6*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*
b^7*c*d^2 - a^5*b^6*d^3)*g^5*x^2 + 4*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b^5*d^3)*g^5*x + (
a^4*b^7*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*c*d^2 - a^7*b^4*d^3)*g^5) + 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^
2*b*c*d^3 - a^3*d^4)*log(b*x + a)/((b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^
4)*g^5) - 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*log(d*x + c)/((b^8*c^4 - 4*a*b^7*c^3*d
+ 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*g^5)) - 1/48*B*c*d^2*i^3*(12*(6*b^2*x^2 + 4*a*b*x + a^2)*
log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^
4*b^3*g^5) + (13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2
+ a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a*b^4*c^3
- 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7
*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^3 - 3*a
^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a
^6*b^4*d^3)*g^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2*d^2 -
4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3
*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2
*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5)) - 1/48*B*c^2*d*i^3*(12*(4*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x
+ c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) + (7*a*b^3*c^3 - 33*
a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d
^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*
b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^
5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 -
3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^
7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^
3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^
2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/48*B*c^3*i^3*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a
^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((
b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*
c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(
a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*
b^2*c*d^2 - a^7*b*d^3)*g^5) - 12*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b
^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2
- 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^
3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/4*(4*b*x + a)*A*c^2*d*i^3/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^
2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*(6*b^2*x^2 + 4*a*b*x + a^2)*A*c*d^2*i^3/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^
3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) - 1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*A*d^3
*i^3/(b^8*g^5*x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5*x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) - 1/4*A*c^3*i^3/(b^5*
g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [B]  time = 0.542133, size = 710, normalized size = 7.98 \begin{align*} -\frac{4 \,{\left ({\left (4 \, A + B\right )} b^{4} c d^{3} -{\left (4 \, A + B\right )} a b^{3} d^{4}\right )} i^{3} x^{3} + 6 \,{\left ({\left (4 \, A + B\right )} b^{4} c^{2} d^{2} -{\left (4 \, A + B\right )} a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 4 \,{\left ({\left (4 \, A + B\right )} b^{4} c^{3} d -{\left (4 \, A + B\right )} a^{3} b d^{4}\right )} i^{3} x +{\left ({\left (4 \, A + B\right )} b^{4} c^{4} -{\left (4 \, A + B\right )} a^{4} d^{4}\right )} i^{3} + 4 \,{\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x + B b^{4} c^{4} i^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{16 \,{\left ({\left (b^{9} c - a b^{8} d\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c - a^{2} b^{7} d\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c - a^{3} b^{6} d\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c - a^{4} b^{5} d\right )} g^{5} x +{\left (a^{4} b^{5} c - a^{5} b^{4} d\right )} g^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/16*(4*((4*A + B)*b^4*c*d^3 - (4*A + B)*a*b^3*d^4)*i^3*x^3 + 6*((4*A + B)*b^4*c^2*d^2 - (4*A + B)*a^2*b^2*d^
4)*i^3*x^2 + 4*((4*A + B)*b^4*c^3*d - (4*A + B)*a^3*b*d^4)*i^3*x + ((4*A + B)*b^4*c^4 - (4*A + B)*a^4*d^4)*i^3
+ 4*(B*b^4*d^4*i^3*x^4 + 4*B*b^4*c*d^3*i^3*x^3 + 6*B*b^4*c^2*d^2*i^3*x^2 + 4*B*b^4*c^3*d*i^3*x + B*b^4*c^4*i^
3)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c - a*b^8*d)*g^5*x^4 + 4*(a*b^8*c - a^2*b^7*d)*g^5*x^3 + 6*(a^2*b^7*c -
a^3*b^6*d)*g^5*x^2 + 4*(a^3*b^6*c - a^4*b^5*d)*g^5*x + (a^4*b^5*c - a^5*b^4*d)*g^5)

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Sympy [B]  time = 157.452, size = 864, normalized size = 9.71 \begin{align*} - \frac{B d^{4} i^{3} \log{\left (x + \frac{- \frac{B a^{2} d^{6} i^{3}}{a d - b c} + \frac{2 B a b c d^{5} i^{3}}{a d - b c} + B a d^{5} i^{3} - \frac{B b^{2} c^{2} d^{4} i^{3}}{a d - b c} + B b c d^{4} i^{3}}{2 B b d^{5} i^{3}} \right )}}{4 b^{4} g^{5} \left (a d - b c\right )} + \frac{B d^{4} i^{3} \log{\left (x + \frac{\frac{B a^{2} d^{6} i^{3}}{a d - b c} - \frac{2 B a b c d^{5} i^{3}}{a d - b c} + B a d^{5} i^{3} + \frac{B b^{2} c^{2} d^{4} i^{3}}{a d - b c} + B b c d^{4} i^{3}}{2 B b d^{5} i^{3}} \right )}}{4 b^{4} g^{5} \left (a d - b c\right )} - \frac{4 A a^{3} d^{3} i^{3} + 4 A a^{2} b c d^{2} i^{3} + 4 A a b^{2} c^{2} d i^{3} + 4 A b^{3} c^{3} i^{3} + B a^{3} d^{3} i^{3} + B a^{2} b c d^{2} i^{3} + B a b^{2} c^{2} d i^{3} + B b^{3} c^{3} i^{3} + x^{3} \left (16 A b^{3} d^{3} i^{3} + 4 B b^{3} d^{3} i^{3}\right ) + x^{2} \left (24 A a b^{2} d^{3} i^{3} + 24 A b^{3} c d^{2} i^{3} + 6 B a b^{2} d^{3} i^{3} + 6 B b^{3} c d^{2} i^{3}\right ) + x \left (16 A a^{2} b d^{3} i^{3} + 16 A a b^{2} c d^{2} i^{3} + 16 A b^{3} c^{2} d i^{3} + 4 B a^{2} b d^{3} i^{3} + 4 B a b^{2} c d^{2} i^{3} + 4 B b^{3} c^{2} d i^{3}\right )}{16 a^{4} b^{4} g^{5} + 64 a^{3} b^{5} g^{5} x + 96 a^{2} b^{6} g^{5} x^{2} + 64 a b^{7} g^{5} x^{3} + 16 b^{8} g^{5} x^{4}} + \frac{\left (- B a^{3} d^{3} i^{3} - B a^{2} b c d^{2} i^{3} - 4 B a^{2} b d^{3} i^{3} x - B a b^{2} c^{2} d i^{3} - 4 B a b^{2} c d^{2} i^{3} x - 6 B a b^{2} d^{3} i^{3} x^{2} - B b^{3} c^{3} i^{3} - 4 B b^{3} c^{2} d i^{3} x - 6 B b^{3} c d^{2} i^{3} x^{2} - 4 B b^{3} d^{3} i^{3} x^{3}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{4 a^{4} b^{4} g^{5} + 16 a^{3} b^{5} g^{5} x + 24 a^{2} b^{6} g^{5} x^{2} + 16 a b^{7} g^{5} x^{3} + 4 b^{8} g^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)

[Out]

-B*d**4*i**3*log(x + (-B*a**2*d**6*i**3/(a*d - b*c) + 2*B*a*b*c*d**5*i**3/(a*d - b*c) + B*a*d**5*i**3 - B*b**2
*c**2*d**4*i**3/(a*d - b*c) + B*b*c*d**4*i**3)/(2*B*b*d**5*i**3))/(4*b**4*g**5*(a*d - b*c)) + B*d**4*i**3*log(
x + (B*a**2*d**6*i**3/(a*d - b*c) - 2*B*a*b*c*d**5*i**3/(a*d - b*c) + B*a*d**5*i**3 + B*b**2*c**2*d**4*i**3/(a
*d - b*c) + B*b*c*d**4*i**3)/(2*B*b*d**5*i**3))/(4*b**4*g**5*(a*d - b*c)) - (4*A*a**3*d**3*i**3 + 4*A*a**2*b*c
*d**2*i**3 + 4*A*a*b**2*c**2*d*i**3 + 4*A*b**3*c**3*i**3 + B*a**3*d**3*i**3 + B*a**2*b*c*d**2*i**3 + B*a*b**2*
c**2*d*i**3 + B*b**3*c**3*i**3 + x**3*(16*A*b**3*d**3*i**3 + 4*B*b**3*d**3*i**3) + x**2*(24*A*a*b**2*d**3*i**3
+ 24*A*b**3*c*d**2*i**3 + 6*B*a*b**2*d**3*i**3 + 6*B*b**3*c*d**2*i**3) + x*(16*A*a**2*b*d**3*i**3 + 16*A*a*b*
*2*c*d**2*i**3 + 16*A*b**3*c**2*d*i**3 + 4*B*a**2*b*d**3*i**3 + 4*B*a*b**2*c*d**2*i**3 + 4*B*b**3*c**2*d*i**3)
)/(16*a**4*b**4*g**5 + 64*a**3*b**5*g**5*x + 96*a**2*b**6*g**5*x**2 + 64*a*b**7*g**5*x**3 + 16*b**8*g**5*x**4)
+ (-B*a**3*d**3*i**3 - B*a**2*b*c*d**2*i**3 - 4*B*a**2*b*d**3*i**3*x - B*a*b**2*c**2*d*i**3 - 4*B*a*b**2*c*d*
*2*i**3*x - 6*B*a*b**2*d**3*i**3*x**2 - B*b**3*c**3*i**3 - 4*B*b**3*c**2*d*i**3*x - 6*B*b**3*c*d**2*i**3*x**2
- 4*B*b**3*d**3*i**3*x**3)*log(e*(a + b*x)/(c + d*x))/(4*a**4*b**4*g**5 + 16*a**3*b**5*g**5*x + 24*a**2*b**6*g
**5*x**2 + 16*a*b**7*g**5*x**3 + 4*b**8*g**5*x**4)

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Giac [B]  time = 1.32788, size = 776, normalized size = 8.72 \begin{align*} -\frac{B d^{4} \log \left (b x + a\right )}{4 \,{\left (b^{5} c g^{5} i - a b^{4} d g^{5} i\right )}} + \frac{B d^{4} \log \left (d x + c\right )}{4 \,{\left (b^{5} c g^{5} i - a b^{4} d g^{5} i\right )}} + \frac{{\left (4 \, B b^{3} d^{3} i x^{3} + 6 \, B b^{3} c d^{2} i x^{2} + 6 \, B a b^{2} d^{3} i x^{2} + 4 \, B b^{3} c^{2} d i x + 4 \, B a b^{2} c d^{2} i x + 4 \, B a^{2} b d^{3} i x + B b^{3} c^{3} i + B a b^{2} c^{2} d i + B a^{2} b c d^{2} i + B a^{3} d^{3} i\right )} \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left (b^{8} g^{5} x^{4} + 4 \, a b^{7} g^{5} x^{3} + 6 \, a^{2} b^{6} g^{5} x^{2} + 4 \, a^{3} b^{5} g^{5} x + a^{4} b^{4} g^{5}\right )}} + \frac{16 \, A b^{3} d^{3} i x^{3} + 20 \, B b^{3} d^{3} i x^{3} + 24 \, A b^{3} c d^{2} i x^{2} + 30 \, B b^{3} c d^{2} i x^{2} + 24 \, A a b^{2} d^{3} i x^{2} + 30 \, B a b^{2} d^{3} i x^{2} + 16 \, A b^{3} c^{2} d i x + 20 \, B b^{3} c^{2} d i x + 16 \, A a b^{2} c d^{2} i x + 20 \, B a b^{2} c d^{2} i x + 16 \, A a^{2} b d^{3} i x + 20 \, B a^{2} b d^{3} i x + 4 \, A b^{3} c^{3} i + 5 \, B b^{3} c^{3} i + 4 \, A a b^{2} c^{2} d i + 5 \, B a b^{2} c^{2} d i + 4 \, A a^{2} b c d^{2} i + 5 \, B a^{2} b c d^{2} i + 4 \, A a^{3} d^{3} i + 5 \, B a^{3} d^{3} i}{16 \,{\left (b^{8} g^{5} x^{4} + 4 \, a b^{7} g^{5} x^{3} + 6 \, a^{2} b^{6} g^{5} x^{2} + 4 \, a^{3} b^{5} g^{5} x + a^{4} b^{4} g^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

-1/4*B*d^4*log(b*x + a)/(b^5*c*g^5*i - a*b^4*d*g^5*i) + 1/4*B*d^4*log(d*x + c)/(b^5*c*g^5*i - a*b^4*d*g^5*i) +
1/4*(4*B*b^3*d^3*i*x^3 + 6*B*b^3*c*d^2*i*x^2 + 6*B*a*b^2*d^3*i*x^2 + 4*B*b^3*c^2*d*i*x + 4*B*a*b^2*c*d^2*i*x
+ 4*B*a^2*b*d^3*i*x + B*b^3*c^3*i + B*a*b^2*c^2*d*i + B*a^2*b*c*d^2*i + B*a^3*d^3*i)*log((b*x + a)/(d*x + c))/
(b^8*g^5*x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5*x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) + 1/16*(16*A*b^3*d^3*i*x^3
+ 20*B*b^3*d^3*i*x^3 + 24*A*b^3*c*d^2*i*x^2 + 30*B*b^3*c*d^2*i*x^2 + 24*A*a*b^2*d^3*i*x^2 + 30*B*a*b^2*d^3*i*
x^2 + 16*A*b^3*c^2*d*i*x + 20*B*b^3*c^2*d*i*x + 16*A*a*b^2*c*d^2*i*x + 20*B*a*b^2*c*d^2*i*x + 16*A*a^2*b*d^3*i
*x + 20*B*a^2*b*d^3*i*x + 4*A*b^3*c^3*i + 5*B*b^3*c^3*i + 4*A*a*b^2*c^2*d*i + 5*B*a*b^2*c^2*d*i + 4*A*a^2*b*c*
d^2*i + 5*B*a^2*b*c*d^2*i + 4*A*a^3*d^3*i + 5*B*a^3*d^3*i)/(b^8*g^5*x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5*x^2
+ 4*a^3*b^5*g^5*x + a^4*b^4*g^5)