3.11 \(\int (a g+b g x)^2 (c i+d i x)^2 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=337 \[ \frac{b^2 g^2 i^2 (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^3}+\frac{g^2 i^2 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 d^3}-\frac{b g^2 i^2 (c+d x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^3}-\frac{B g^2 i^2 x (b c-a d)^4}{30 b^2 d^2}-\frac{B g^2 i^2 (b c-a d)^5 \log \left (\frac{a+b x}{c+d x}\right )}{30 b^3 d^3}-\frac{B g^2 i^2 (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac{B g^2 i^2 (c+d x)^2 (b c-a d)^3}{60 b d^3}+\frac{B g^2 i^2 (c+d x)^3 (b c-a d)^2}{10 d^3}-\frac{b B g^2 i^2 (c+d x)^4 (b c-a d)}{20 d^3} \]

[Out]

-(B*(b*c - a*d)^4*g^2*i^2*x)/(30*b^2*d^2) - (B*(b*c - a*d)^3*g^2*i^2*(c + d*x)^2)/(60*b*d^3) + (B*(b*c - a*d)^
2*g^2*i^2*(c + d*x)^3)/(10*d^3) - (b*B*(b*c - a*d)*g^2*i^2*(c + d*x)^4)/(20*d^3) - (B*(b*c - a*d)^5*g^2*i^2*Lo
g[(a + b*x)/(c + d*x)])/(30*b^3*d^3) + ((b*c - a*d)^2*g^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])
)/(3*d^3) - (b*(b*c - a*d)*g^2*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^3) + (b^2*g^2*i^2*(c
+ d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^3) - (B*(b*c - a*d)^5*g^2*i^2*Log[c + d*x])/(30*b^3*d^3)

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Rubi [A]  time = 0.50839, antiderivative size = 296, normalized size of antiderivative = 0.88, number of steps used = 14, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 43} \[ \frac{d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^3}+\frac{g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^3}+\frac{d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^3}+\frac{B g^2 i^2 x (b c-a d)^4}{30 b^2 d^2}-\frac{B g^2 i^2 (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac{B g^2 i^2 (a+b x)^2 (b c-a d)^3}{60 b^3 d}-\frac{B g^2 i^2 (a+b x)^3 (b c-a d)^2}{10 b^3}-\frac{B d g^2 i^2 (a+b x)^4 (b c-a d)}{20 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^2*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(B*(b*c - a*d)^4*g^2*i^2*x)/(30*b^2*d^2) - (B*(b*c - a*d)^3*g^2*i^2*(a + b*x)^2)/(60*b^3*d) - (B*(b*c - a*d)^2
*g^2*i^2*(a + b*x)^3)/(10*b^3) - (B*d*(b*c - a*d)*g^2*i^2*(a + b*x)^4)/(20*b^3) + ((b*c - a*d)^2*g^2*i^2*(a +
b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3) + (d*(b*c - a*d)*g^2*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b
*x))/(c + d*x)]))/(2*b^3) + (d^2*g^2*i^2*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^3) - (B*(b*c -
a*d)^5*g^2*i^2*Log[c + d*x])/(30*b^3*d^3)

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (11 c+11 d x)^2 (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac{(-b c+a d)^2 g^2 (11 c+11 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}-\frac{2 b (b c-a d) g^2 (11 c+11 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{11 d^2}+\frac{b^2 g^2 (11 c+11 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{121 d^2}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int (11 c+11 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{121 d^2}-\frac{\left (2 b (b c-a d) g^2\right ) \int (11 c+11 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{11 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int (11 c+11 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}\\ &=\frac{121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac{121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac{121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac{\left (b^2 B g^2\right ) \int \frac{161051 (b c-a d) (c+d x)^4}{a+b x} \, dx}{6655 d^3}+\frac{\left (b B (b c-a d) g^2\right ) \int \frac{14641 (b c-a d) (c+d x)^3}{a+b x} \, dx}{242 d^3}-\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{1331 (b c-a d) (c+d x)^2}{a+b x} \, dx}{33 d^3}\\ &=\frac{121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac{121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac{121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac{\left (121 b^2 B (b c-a d) g^2\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^3}+\frac{\left (121 b B (b c-a d)^2 g^2\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{2 d^3}-\frac{\left (121 B (b c-a d)^3 g^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 d^3}\\ &=\frac{121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac{121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac{121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac{\left (121 b^2 B (b c-a d) g^2\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^3}+\frac{\left (121 b B (b c-a d)^2 g^2\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{2 d^3}-\frac{\left (121 B (b c-a d)^3 g^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 d^3}\\ &=-\frac{121 B (b c-a d)^4 g^2 x}{30 b^2 d^2}-\frac{121 B (b c-a d)^3 g^2 (c+d x)^2}{60 b d^3}+\frac{121 B (b c-a d)^2 g^2 (c+d x)^3}{10 d^3}-\frac{121 b B (b c-a d) g^2 (c+d x)^4}{20 d^3}-\frac{121 B (b c-a d)^5 g^2 \log (a+b x)}{30 b^3 d^3}+\frac{121 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac{121 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac{121 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}\\ \end{align*}

Mathematica [A]  time = 0.250358, size = 362, normalized size = 1.07 \[ \frac{g^2 i^2 \left (12 d^5 (a+b x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+30 d^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+20 d^3 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+10 B (b c-a d)^3 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )-5 B (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+B (b c-a d) \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )\right )}{60 b^3 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^2*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]

[Out]

(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 30*d^4*(b*c - a*d)*(a + b*x)
^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 12*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 10*B*(b*c
- a*d)^3*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) - 5*B*(b*c - a*d)^2*(6*b*d*(b*
c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + B*(b*c -
a*d)*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a + b
*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x])))/(60*b^3*d^3)

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Maple [B]  time = 0.191, size = 6116, normalized size = 18.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^2*(d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

result too large to display

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Maxima [B]  time = 1.45388, size = 1620, normalized size = 4.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/5*A*b^2*d^2*g^2*i^2*x^5 + 1/2*A*b^2*c*d*g^2*i^2*x^4 + 1/2*A*a*b*d^2*g^2*i^2*x^4 + 1/3*A*b^2*c^2*g^2*i^2*x^3
+ 4/3*A*a*b*c*d*g^2*i^2*x^3 + 1/3*A*a^2*d^2*g^2*i^2*x^3 + A*a*b*c^2*g^2*i^2*x^2 + A*a^2*c*d*g^2*i^2*x^2 + (x*l
og(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a^2*c^2*g^2*i^2 + (x^2*log(b*e*x/
(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a*b*c^2*g^2*
i^2 + 1/6*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^
2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*b^2*c^2*g^2*i^2 + (x^2*log(b*e*x/(d*x + c) + a*e/
(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^2*c*d*g^2*i^2 + 2/3*(2*x^3
*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)
*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*b*c*d*g^2*i^2 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)
) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^
3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^2*c*d*g^2*i^2 + 1/6*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c
)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^
2*d^2))*B*a^2*d^2*g^2*i^2 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*
log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(
b^3*d^3))*B*a*b*d^2*g^2*i^2 + 1/60*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12
*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^
3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*b^2*d^2*g^2*i^2 + A*a^2*c^2*g^2*i^2*x

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Fricas [A]  time = 1.39599, size = 1102, normalized size = 3.27 \begin{align*} \frac{12 \, A b^{5} d^{5} g^{2} i^{2} x^{5} + 3 \,{\left ({\left (10 \, A - B\right )} b^{5} c d^{4} +{\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 2 \,{\left ({\left (10 \, A - 3 \, B\right )} b^{5} c^{2} d^{3} + 40 \, A a b^{4} c d^{4} +{\left (10 \, A + 3 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} -{\left (B b^{5} c^{3} d^{2} - 15 \,{\left (4 \, A - B\right )} a b^{4} c^{2} d^{3} - 15 \,{\left (4 \, A + B\right )} a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{2} i^{2} x^{2} + 2 \,{\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 30 \, A a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{2} i^{2} x + 2 \,{\left (10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{2} i^{2} \log \left (b x + a\right ) - 2 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} g^{2} i^{2} \log \left (d x + c\right ) + 2 \,{\left (6 \, B b^{5} d^{5} g^{2} i^{2} x^{5} + 30 \, B a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} x + 15 \,{\left (B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 10 \,{\left (B b^{5} c^{2} d^{3} + 4 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} + 30 \,{\left (B a b^{4} c^{2} d^{3} + B a^{2} b^{3} c d^{4}\right )} g^{2} i^{2} x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{60 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/60*(12*A*b^5*d^5*g^2*i^2*x^5 + 3*((10*A - B)*b^5*c*d^4 + (10*A + B)*a*b^4*d^5)*g^2*i^2*x^4 + 2*((10*A - 3*B)
*b^5*c^2*d^3 + 40*A*a*b^4*c*d^4 + (10*A + 3*B)*a^2*b^3*d^5)*g^2*i^2*x^3 - (B*b^5*c^3*d^2 - 15*(4*A - B)*a*b^4*
c^2*d^3 - 15*(4*A + B)*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g^2*i^2*x^2 + 2*(B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 30*A*
a^2*b^3*c^2*d^3 + 5*B*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g^2*i^2*x + 2*(10*B*a^3*b^2*c^2*d^3 - 5*B*a^4*b*c*d^4 + B*a
^5*d^5)*g^2*i^2*log(b*x + a) - 2*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2)*g^2*i^2*log(d*x + c) + 2
*(6*B*b^5*d^5*g^2*i^2*x^5 + 30*B*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(B*b^5*c*d^4 + B*a*b^4*d^5)*g^2*i^2*x^4 + 10*(
B*b^5*c^2*d^3 + 4*B*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g^2*i^2*x^3 + 30*(B*a*b^4*c^2*d^3 + B*a^2*b^3*c*d^4)*g^2*i^2*
x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^3*d^3)

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Sympy [B]  time = 9.6747, size = 1292, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**2*(d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**2*d**2*g**2*i**2*x**5/5 + B*a**3*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)*log(x + (B*a**5*c*d**4*
g**2*i**2 - 5*B*a**4*b*c**2*d**3*g**2*i**2 + B*a**4*d**3*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)/b +
20*B*a**3*b**2*c**3*d**2*g**2*i**2 - B*a**3*c*d**2*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2) - 5*B*a**2
*b**3*c**4*d*g**2*i**2 + B*a*b**4*c**5*g**2*i**2)/(B*a**5*d**5*g**2*i**2 - 5*B*a**4*b*c*d**4*g**2*i**2 + 10*B*
a**3*b**2*c**2*d**3*g**2*i**2 + 10*B*a**2*b**3*c**3*d**2*g**2*i**2 - 5*B*a*b**4*c**4*d*g**2*i**2 + B*b**5*c**5
*g**2*i**2))/(30*b**3) - B*c**3*g**2*i**2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)*log(x + (B*a**5*c*d**4*g**2*i
**2 - 5*B*a**4*b*c**2*d**3*g**2*i**2 + 20*B*a**3*b**2*c**3*d**2*g**2*i**2 - 5*B*a**2*b**3*c**4*d*g**2*i**2 + B
*a*b**4*c**5*g**2*i**2 - B*a*b**2*c**3*g**2*i**2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2) + B*b**3*c**4*g**2*i**
2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)/d)/(B*a**5*d**5*g**2*i**2 - 5*B*a**4*b*c*d**4*g**2*i**2 + 10*B*a**3*b
**2*c**2*d**3*g**2*i**2 + 10*B*a**2*b**3*c**3*d**2*g**2*i**2 - 5*B*a*b**4*c**4*d*g**2*i**2 + B*b**5*c**5*g**2*
i**2))/(30*d**3) + x**4*(A*a*b*d**2*g**2*i**2/2 + A*b**2*c*d*g**2*i**2/2 + B*a*b*d**2*g**2*i**2/20 - B*b**2*c*
d*g**2*i**2/20) + x**3*(A*a**2*d**2*g**2*i**2/3 + 4*A*a*b*c*d*g**2*i**2/3 + A*b**2*c**2*g**2*i**2/3 + B*a**2*d
**2*g**2*i**2/10 - B*b**2*c**2*g**2*i**2/10) + (B*a**2*c**2*g**2*i**2*x + B*a**2*c*d*g**2*i**2*x**2 + B*a**2*d
**2*g**2*i**2*x**3/3 + B*a*b*c**2*g**2*i**2*x**2 + 4*B*a*b*c*d*g**2*i**2*x**3/3 + B*a*b*d**2*g**2*i**2*x**4/2
+ B*b**2*c**2*g**2*i**2*x**3/3 + B*b**2*c*d*g**2*i**2*x**4/2 + B*b**2*d**2*g**2*i**2*x**5/5)*log(e*(a + b*x)/(
c + d*x)) + x**2*(60*A*a**2*b*c*d**2*g**2*i**2 + 60*A*a*b**2*c**2*d*g**2*i**2 + B*a**3*d**3*g**2*i**2 + 15*B*a
**2*b*c*d**2*g**2*i**2 - 15*B*a*b**2*c**2*d*g**2*i**2 - B*b**3*c**3*g**2*i**2)/(60*b*d) - x*(-30*A*a**2*b**2*c
**2*d**2*g**2*i**2 + B*a**4*d**4*g**2*i**2 - 5*B*a**3*b*c*d**3*g**2*i**2 + 5*B*a*b**3*c**3*d*g**2*i**2 - B*b**
4*c**4*g**2*i**2)/(30*b**2*d**2)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

Timed out