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

Optimal. Leaf size=352 \[ \frac {b^2 g^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}+\frac {g^2 i^2 (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3}-\frac {b g^2 i^2 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3}-\frac {B g^2 i^2 n (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}-\frac {B g^2 i^2 n (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac {B g^2 i^2 n x (b c-a d)^4}{30 b^2 d^2}-\frac {B g^2 i^2 n (c+d x)^2 (b c-a d)^3}{60 b d^3}+\frac {B g^2 i^2 n (c+d x)^3 (b c-a d)^2}{10 d^3}-\frac {b B g^2 i^2 n (c+d x)^4 (b c-a d)}{20 d^3} \]

[Out]

-1/30*B*(-a*d+b*c)^4*g^2*i^2*n*x/b^2/d^2-1/60*B*(-a*d+b*c)^3*g^2*i^2*n*(d*x+c)^2/b/d^3+1/10*B*(-a*d+b*c)^2*g^2
*i^2*n*(d*x+c)^3/d^3-1/20*b*B*(-a*d+b*c)*g^2*i^2*n*(d*x+c)^4/d^3+1/3*(-a*d+b*c)^2*g^2*i^2*(d*x+c)^3*(A+B*ln(e*
((b*x+a)/(d*x+c))^n))/d^3-1/2*b*(-a*d+b*c)*g^2*i^2*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/5*b^2*g^2*i
^2*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/30*B*(-a*d+b*c)^5*g^2*i^2*n*ln((b*x+a)/(d*x+c))/b^3/d^3-1/3
0*B*(-a*d+b*c)^5*g^2*i^2*n*ln(d*x+c)/b^3/d^3

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Rubi [A]  time = 0.54, antiderivative size = 310, normalized size of antiderivative = 0.88, number of steps used = 14, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2528, 2525, 12, 43} \[ \frac {d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3}+\frac {g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3}+\frac {d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3}+\frac {B g^2 i^2 n x (b c-a d)^4}{30 b^2 d^2}-\frac {B g^2 i^2 n (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac {B g^2 i^2 n (a+b x)^2 (b c-a d)^3}{60 b^3 d}-\frac {B g^2 i^2 n (a+b x)^3 (b c-a d)^2}{10 b^3}-\frac {B d g^2 i^2 n (a+b x)^4 (b c-a d)}{20 b^3} \]

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))^n]),x]

[Out]

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

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])

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 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]

Rubi steps

\begin {align*} \int (118 c+118 d x)^2 (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 (118 c+118 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}-\frac {b (b c-a d) g^2 (118 c+118 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{59 d^2}+\frac {b^2 g^2 (118 c+118 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{13924 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (118 c+118 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{13924 d^2}-\frac {\left (b (b c-a d) g^2\right ) \int (118 c+118 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{59 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (118 c+118 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^2}\\ &=\frac {13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac {13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {\left (b^2 B g^2 n\right ) \int \frac {22877577568 (b c-a d) (c+d x)^4}{a+b x} \, dx}{8215160 d^3}+\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {193877776 (b c-a d) (c+d x)^3}{a+b x} \, dx}{27848 d^3}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {1643032 (b c-a d) (c+d x)^2}{a+b x} \, dx}{354 d^3}\\ &=\frac {13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac {13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {\left (13924 b^2 B (b c-a d) g^2 n\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^3}+\frac {\left (6962 b B (b c-a d)^2 g^2 n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^3}-\frac {\left (13924 B (b c-a d)^3 g^2 n\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 d^3}\\ &=\frac {13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac {13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac {\left (13924 b^2 B (b c-a d) g^2 n\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 (6962 b B (b c-a d)^2 g^2 n\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}{d^3}-\frac {\left (13924 B (b c-a d)^3 g^2 n\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 {6962 B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac {3481 B (b c-a d)^3 g^2 n (c+d x)^2}{15 b d^3}+\frac {6962 B (b c-a d)^2 g^2 n (c+d x)^3}{5 d^3}-\frac {3481 b B (b c-a d) g^2 n (c+d x)^4}{5 d^3}-\frac {6962 B (b c-a d)^5 g^2 n \log (a+b x)}{15 b^3 d^3}+\frac {13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac {13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 374, normalized size = 1.06 \[ \frac {g^2 i^2 \left (12 d^5 (a+b x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+30 d^4 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+20 d^3 (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+10 B n (b c-a d)^3 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )-5 B n (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+B n (b c-a d) \left (4 d^3 (a+b x)^3 (b c-a d)-6 d^2 (a+b x)^2 (b c-a d)^2+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )\right )}{60 b^3 d^3} \]

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))^n]),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))^n]) + 30*d^4*(b*c - a*d)*(a + b*
x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 12*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 10*B
*(b*c - a*d)^3*n*(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*n*
(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)*n*(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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fricas [B]  time = 1.16, size = 774, normalized size = 2.20 \[ \frac {12 \, A b^{5} d^{5} g^{2} i^{2} x^{5} + 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} n \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} n \log \left (d x + c\right ) - 3 \, {\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{2} i^{2} n - 10 \, {\left (A b^{5} c d^{4} + A a b^{4} d^{5}\right )} g^{2} i^{2}\right )} x^{4} - 2 \, {\left (3 \, {\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} n - 10 \, {\left (A b^{5} c^{2} d^{3} + 4 \, A a b^{4} c d^{4} + A a^{2} b^{3} d^{5}\right )} g^{2} i^{2}\right )} x^{3} - {\left ({\left (B b^{5} c^{3} d^{2} + 15 \, B a b^{4} c^{2} d^{3} - 15 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{2} i^{2} n - 60 \, {\left (A a b^{4} c^{2} d^{3} + A a^{2} b^{3} c d^{4}\right )} g^{2} i^{2}\right )} x^{2} + 2 \, {\left (30 \, A a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} + {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{2} i^{2} n\right )} x + 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 \relax (e) + 2 \, {\left (6 \, B b^{5} d^{5} g^{2} i^{2} n x^{5} + 30 \, B a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} n x + 15 \, {\left (B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g^{2} i^{2} n 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} n 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} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{60 \, b^{3} d^{3}} \]

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))^n)),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 7.34, size = 2995, normalized size = 8.51 \[ \text {result too large to display} \]

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))^n)),x, algorithm="giac")

[Out]

-1/60*(2*(B*b^8*c^6*g^2*n - 6*B*a*b^7*c^5*d*g^2*n - 5*(b*x + a)*B*b^7*c^6*d*g^2*n/(d*x + c) + 15*B*a^2*b^6*c^4
*d^2*g^2*n + 30*(b*x + a)*B*a*b^6*c^5*d^2*g^2*n/(d*x + c) + 10*(b*x + a)^2*B*b^6*c^6*d^2*g^2*n/(d*x + c)^2 - 2
0*B*a^3*b^5*c^3*d^3*g^2*n - 75*(b*x + a)*B*a^2*b^5*c^4*d^3*g^2*n/(d*x + c) - 60*(b*x + a)^2*B*a*b^5*c^5*d^3*g^
2*n/(d*x + c)^2 + 15*B*a^4*b^4*c^2*d^4*g^2*n + 100*(b*x + a)*B*a^3*b^4*c^3*d^4*g^2*n/(d*x + c) + 150*(b*x + a)
^2*B*a^2*b^4*c^4*d^4*g^2*n/(d*x + c)^2 - 6*B*a^5*b^3*c*d^5*g^2*n - 75*(b*x + a)*B*a^4*b^3*c^2*d^5*g^2*n/(d*x +
c) - 200*(b*x + a)^2*B*a^3*b^3*c^3*d^5*g^2*n/(d*x + c)^2 + B*a^6*b^2*d^6*g^2*n + 30*(b*x + a)*B*a^5*b^2*c*d^6
*g^2*n/(d*x + c) + 150*(b*x + a)^2*B*a^4*b^2*c^2*d^6*g^2*n/(d*x + c)^2 - 5*(b*x + a)*B*a^6*b*d^7*g^2*n/(d*x +
c) - 60*(b*x + a)^2*B*a^5*b*c*d^7*g^2*n/(d*x + c)^2 + 10*(b*x + a)^2*B*a^6*d^8*g^2*n/(d*x + c)^2)*log((b*x + a
)/(d*x + c))/(b^5*d^3 - 5*(b*x + a)*b^4*d^4/(d*x + c) + 10*(b*x + a)^2*b^3*d^5/(d*x + c)^2 - 10*(b*x + a)^3*b^
2*d^6/(d*x + c)^3 + 5*(b*x + a)^4*b*d^7/(d*x + c)^4 - (b*x + a)^5*d^8/(d*x + c)^5) + (2*(b*x + a)*B*b^9*c^6*d*
g^2*n/(d*x + c) - 12*(b*x + a)*B*a*b^8*c^5*d^2*g^2*n/(d*x + c) - 9*(b*x + a)^2*B*b^8*c^6*d^2*g^2*n/(d*x + c)^2
+ 30*(b*x + a)*B*a^2*b^7*c^4*d^3*g^2*n/(d*x + c) + 54*(b*x + a)^2*B*a*b^7*c^5*d^3*g^2*n/(d*x + c)^2 + 9*(b*x
+ a)^3*B*b^7*c^6*d^3*g^2*n/(d*x + c)^3 - 40*(b*x + a)*B*a^3*b^6*c^3*d^4*g^2*n/(d*x + c) - 135*(b*x + a)^2*B*a^
2*b^6*c^4*d^4*g^2*n/(d*x + c)^2 - 54*(b*x + a)^3*B*a*b^6*c^5*d^4*g^2*n/(d*x + c)^3 - 2*(b*x + a)^4*B*b^6*c^6*d
^4*g^2*n/(d*x + c)^4 + 30*(b*x + a)*B*a^4*b^5*c^2*d^5*g^2*n/(d*x + c) + 180*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g^2*
n/(d*x + c)^2 + 135*(b*x + a)^3*B*a^2*b^5*c^4*d^5*g^2*n/(d*x + c)^3 + 12*(b*x + a)^4*B*a*b^5*c^5*d^5*g^2*n/(d*
x + c)^4 - 12*(b*x + a)*B*a^5*b^4*c*d^6*g^2*n/(d*x + c) - 135*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g^2*n/(d*x + c)^2
- 180*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g^2*n/(d*x + c)^3 - 30*(b*x + a)^4*B*a^2*b^4*c^4*d^6*g^2*n/(d*x + c)^4 + 2
*(b*x + a)*B*a^6*b^3*d^7*g^2*n/(d*x + c) + 54*(b*x + a)^2*B*a^5*b^3*c*d^7*g^2*n/(d*x + c)^2 + 135*(b*x + a)^3*
B*a^4*b^3*c^2*d^7*g^2*n/(d*x + c)^3 + 40*(b*x + a)^4*B*a^3*b^3*c^3*d^7*g^2*n/(d*x + c)^4 - 9*(b*x + a)^2*B*a^6
*b^2*d^8*g^2*n/(d*x + c)^2 - 54*(b*x + a)^3*B*a^5*b^2*c*d^8*g^2*n/(d*x + c)^3 - 30*(b*x + a)^4*B*a^4*b^2*c^2*d
^8*g^2*n/(d*x + c)^4 + 9*(b*x + a)^3*B*a^6*b*d^9*g^2*n/(d*x + c)^3 + 12*(b*x + a)^4*B*a^5*b*c*d^9*g^2*n/(d*x +
c)^4 - 2*(b*x + a)^4*B*a^6*d^10*g^2*n/(d*x + c)^4 + 2*A*b^10*c^6*g^2 + 2*B*b^10*c^6*g^2 - 12*A*a*b^9*c^5*d*g^
2 - 12*B*a*b^9*c^5*d*g^2 - 10*(b*x + a)*A*b^9*c^6*d*g^2/(d*x + c) - 10*(b*x + a)*B*b^9*c^6*d*g^2/(d*x + c) + 3
0*A*a^2*b^8*c^4*d^2*g^2 + 30*B*a^2*b^8*c^4*d^2*g^2 + 60*(b*x + a)*A*a*b^8*c^5*d^2*g^2/(d*x + c) + 60*(b*x + a)
*B*a*b^8*c^5*d^2*g^2/(d*x + c) + 20*(b*x + a)^2*A*b^8*c^6*d^2*g^2/(d*x + c)^2 + 20*(b*x + a)^2*B*b^8*c^6*d^2*g
^2/(d*x + c)^2 - 40*A*a^3*b^7*c^3*d^3*g^2 - 40*B*a^3*b^7*c^3*d^3*g^2 - 150*(b*x + a)*A*a^2*b^7*c^4*d^3*g^2/(d*
x + c) - 150*(b*x + a)*B*a^2*b^7*c^4*d^3*g^2/(d*x + c) - 120*(b*x + a)^2*A*a*b^7*c^5*d^3*g^2/(d*x + c)^2 - 120
*(b*x + a)^2*B*a*b^7*c^5*d^3*g^2/(d*x + c)^2 + 30*A*a^4*b^6*c^2*d^4*g^2 + 30*B*a^4*b^6*c^2*d^4*g^2 + 200*(b*x
+ a)*A*a^3*b^6*c^3*d^4*g^2/(d*x + c) + 200*(b*x + a)*B*a^3*b^6*c^3*d^4*g^2/(d*x + c) + 300*(b*x + a)^2*A*a^2*b
^6*c^4*d^4*g^2/(d*x + c)^2 + 300*(b*x + a)^2*B*a^2*b^6*c^4*d^4*g^2/(d*x + c)^2 - 12*A*a^5*b^5*c*d^5*g^2 - 12*B
*a^5*b^5*c*d^5*g^2 - 150*(b*x + a)*A*a^4*b^5*c^2*d^5*g^2/(d*x + c) - 150*(b*x + a)*B*a^4*b^5*c^2*d^5*g^2/(d*x
+ c) - 400*(b*x + a)^2*A*a^3*b^5*c^3*d^5*g^2/(d*x + c)^2 - 400*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g^2/(d*x + c)^2 +
2*A*a^6*b^4*d^6*g^2 + 2*B*a^6*b^4*d^6*g^2 + 60*(b*x + a)*A*a^5*b^4*c*d^6*g^2/(d*x + c) + 60*(b*x + a)*B*a^5*b
^4*c*d^6*g^2/(d*x + c) + 300*(b*x + a)^2*A*a^4*b^4*c^2*d^6*g^2/(d*x + c)^2 + 300*(b*x + a)^2*B*a^4*b^4*c^2*d^6
*g^2/(d*x + c)^2 - 10*(b*x + a)*A*a^6*b^3*d^7*g^2/(d*x + c) - 10*(b*x + a)*B*a^6*b^3*d^7*g^2/(d*x + c) - 120*(
b*x + a)^2*A*a^5*b^3*c*d^7*g^2/(d*x + c)^2 - 120*(b*x + a)^2*B*a^5*b^3*c*d^7*g^2/(d*x + c)^2 + 20*(b*x + a)^2*
A*a^6*b^2*d^8*g^2/(d*x + c)^2 + 20*(b*x + a)^2*B*a^6*b^2*d^8*g^2/(d*x + c)^2)/(b^7*d^3 - 5*(b*x + a)*b^6*d^4/(
d*x + c) + 10*(b*x + a)^2*b^5*d^5/(d*x + c)^2 - 10*(b*x + a)^3*b^4*d^6/(d*x + c)^3 + 5*(b*x + a)^4*b^3*d^7/(d*
x + c)^4 - (b*x + a)^5*b^2*d^8/(d*x + c)^5) + 2*(B*b^6*c^6*g^2*n - 6*B*a*b^5*c^5*d*g^2*n + 15*B*a^2*b^4*c^4*d^
2*g^2*n - 20*B*a^3*b^3*c^3*d^3*g^2*n + 15*B*a^4*b^2*c^2*d^4*g^2*n - 6*B*a^5*b*c*d^5*g^2*n + B*a^6*d^6*g^2*n)*l
og(b - (b*x + a)*d/(d*x + c))/(b^3*d^3) - 2*(B*b^6*c^6*g^2*n - 6*B*a*b^5*c^5*d*g^2*n + 15*B*a^2*b^4*c^4*d^2*g^
2*n - 20*B*a^3*b^3*c^3*d^3*g^2*n + 15*B*a^4*b^2*c^2*d^4*g^2*n - 6*B*a^5*b*c*d^5*g^2*n + B*a^6*d^6*g^2*n)*log((
b*x + a)/(d*x + c))/(b^3*d^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.43, size = 1336, normalized size = 3.80 \[ \text {result too large to display} \]

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))^n)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.14, size = 1328, normalized size = 3.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*(((30*a*d + 30*b*c)*((((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b*d*g^2*i^2*(30*a*d
+ 30*b*c))/30)*(30*a*d + 30*b*c))/(30*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n +
18*A*a*b*c*d))/2 + A*a*b*c*d*g^2*i^2))/(60*b*d) - (a*c*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d*n - B*b*c*n
))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(2*b*d) + (g^2*i^2*(3*A*a^3*d^3 + 3*A*b^3*c^3 + B*a^3*d^3*n - B*
b^3*c^3*n + 27*A*a*b^2*c^2*d + 27*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(6*b*d)) + log(e*((a
+ b*x)/(c + d*x))^n)*((B*g^2*i^2*x^3*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d))/3 + B*a^2*c^2*g^2*i^2*x + (B*b^2*d^2*g^
2*i^2*x^5)/5 + B*a*c*g^2*i^2*x^2*(a*d + b*c) + (B*b*d*g^2*i^2*x^4*(a*d + b*c))/2) - x^3*((((b*d*g^2*i^2*(15*A*
a*d + 15*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30)*(30*a*d + 30*b*c))/(90*b*d) - (
g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 18*A*a*b*c*d))/6 + (A*a*b*c*d*g^2*i^2)/3) + x
*((a*c*((((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30)*(3
0*a*d + 30*b*c))/(30*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 18*A*a*b*c*d))/2
+ A*a*b*c*d*g^2*i^2))/(b*d) - ((30*a*d + 30*b*c)*(((30*a*d + 30*b*c)*((((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B
*a*d*n - B*b*c*n))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30)*(30*a*d + 30*b*c))/(30*b*d) - (g^2*i^2*(6*A*a^2*d
^2 + 6*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 18*A*a*b*c*d))/2 + A*a*b*c*d*g^2*i^2))/(30*b*d) - (a*c*((b*d*g^
2*i^2*(15*A*a*d + 15*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(b*d) + (g^2*i^2*(
3*A*a^3*d^3 + 3*A*b^3*c^3 + B*a^3*d^3*n - B*b^3*c^3*n + 27*A*a*b^2*c^2*d + 27*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*
n + 3*B*a^2*b*c*d^2*n))/(3*b*d)))/(30*b*d) + (a*c*g^2*i^2*(3*A*a^2*d^2 + 3*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2
*n + 9*A*a*b*c*d))/(b*d)) + x^4*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d*n - B*b*c*n))/20 - (A*b*d*g^2*i^2*(
30*a*d + 30*b*c))/120) + (log(a + b*x)*(B*a^5*d^2*g^2*i^2*n + 10*B*a^3*b^2*c^2*g^2*i^2*n - 5*B*a^4*b*c*d*g^2*i
^2*n))/(30*b^3) - (log(c + d*x)*(B*b^2*c^5*g^2*i^2*n + 10*B*a^2*c^3*d^2*g^2*i^2*n - 5*B*a*b*c^4*d*g^2*i^2*n))/
(30*d^3) + (A*b^2*d^2*g^2*i^2*x^5)/5

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**n)),x)

[Out]

Timed out

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