3.128 \(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\)

Optimal. Leaf size=387 \[ \frac {b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3}+\frac {g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3}-\frac {2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log (c+d x)}{60 b^4 d^3}-\frac {B g^2 i^3 n x (b c-a d)^5}{60 b^3 d^2}-\frac {B g^2 i^3 n (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac {B g^2 i^3 n (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac {7 B g^2 i^3 n (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac {b B g^2 i^3 n (c+d x)^5 (b c-a d)}{30 d^3} \]

[Out]

-1/60*B*(-a*d+b*c)^5*g^2*i^3*n*x/b^3/d^2-1/120*B*(-a*d+b*c)^4*g^2*i^3*n*(d*x+c)^2/b^2/d^3-1/180*B*(-a*d+b*c)^3
*g^2*i^3*n*(d*x+c)^3/b/d^3+7/120*B*(-a*d+b*c)^2*g^2*i^3*n*(d*x+c)^4/d^3-1/30*b*B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^
5/d^3+1/4*(-a*d+b*c)^2*g^2*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-2/5*b*(-a*d+b*c)*g^2*i^3*(d*x+c)^
5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/6*b^2*g^2*i^3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/60*B*(-a
*d+b*c)^6*g^2*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^3-1/60*B*(-a*d+b*c)^6*g^2*i^3*n*ln(d*x+c)/b^4/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.70, antiderivative size = 345, normalized size of antiderivative = 0.89, 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} \[ \frac {b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3}+\frac {g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3}-\frac {2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}-\frac {B g^2 i^3 n x (b c-a d)^5}{60 b^3 d^2}-\frac {B g^2 i^3 n (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log (a+b x)}{60 b^4 d^3}-\frac {B g^2 i^3 n (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac {7 B g^2 i^3 n (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac {b B g^2 i^3 n (c+d x)^5 (b c-a d)}{30 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

-(B*(b*c - a*d)^5*g^2*i^3*n*x)/(60*b^3*d^2) - (B*(b*c - a*d)^4*g^2*i^3*n*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*c
- a*d)^3*g^2*i^3*n*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*n*(c + d*x)^4)/(120*d^3) - (b*B*(b*c
- a*d)*g^2*i^3*n*(c + d*x)^5)/(30*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[a + b*x])/(60*b^4*d^3) + ((b*c - a*d)^
2*g^2*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(
A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^3) + (b^2*g^2*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/(6*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 (128 c+128 d x)^3 (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 (128 c+128 d x)^3 \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 (128 c+128 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 d^2}+\frac {b^2 g^2 (128 c+128 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16384 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (128 c+128 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{16384 d^2}-\frac {\left (b (b c-a d) g^2\right ) \int (128 c+128 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{64 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (128 c+128 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^2}\\ &=\frac {524288 (b c-a d)^2 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 {4194304 b (b c-a d) 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 {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {\left (b^2 B g^2 n\right ) \int \frac {4398046511104 (b c-a d) (c+d x)^5}{a+b x} \, dx}{12582912 d^3}+\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {34359738368 (b c-a d) (c+d x)^4}{a+b x} \, dx}{40960 d^3}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {268435456 (b c-a d) (c+d x)^3}{a+b x} \, dx}{512 d^3}\\ &=\frac {524288 (b c-a d)^2 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 {4194304 b (b c-a d) 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 {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {\left (1048576 b^2 B (b c-a d) g^2 n\right ) \int \frac {(c+d x)^5}{a+b x} \, dx}{3 d^3}+\frac {\left (4194304 b B (b c-a d)^2 g^2 n\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^3}-\frac {\left (524288 B (b c-a d)^3 g^2 n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^3}\\ &=\frac {524288 (b c-a d)^2 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 {4194304 b (b c-a d) 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 {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {\left (1048576 b^2 B (b c-a d) g^2 n\right ) \int \left (\frac {d (b c-a d)^4}{b^5}+\frac {(b c-a d)^5}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x)}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2}{b^3}+\frac {d (b c-a d) (c+d x)^3}{b^2}+\frac {d (c+d x)^4}{b}\right ) \, dx}{3 d^3}+\frac {\left (4194304 b B (b c-a d)^2 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 (524288 B (b c-a d)^3 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 {524288 B (b c-a d)^5 g^2 n x}{15 b^3 d^2}-\frac {262144 B (b c-a d)^4 g^2 n (c+d x)^2}{15 b^2 d^3}-\frac {524288 B (b c-a d)^3 g^2 n (c+d x)^3}{45 b d^3}+\frac {1835008 B (b c-a d)^2 g^2 n (c+d x)^4}{15 d^3}-\frac {1048576 b B (b c-a d) g^2 n (c+d x)^5}{15 d^3}-\frac {524288 B (b c-a d)^6 g^2 n \log (a+b x)}{15 b^4 d^3}+\frac {524288 (b c-a d)^2 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 {4194304 b (b c-a d) 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 {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 441, normalized size = 1.14 \[ \frac {g^2 i^3 \left (60 b^6 (c+d x)^6 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-144 b^5 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+90 b^4 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-15 B n (b c-a d)^3 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )+12 B n (b c-a d)^2 \left (4 b^3 (c+d x)^3 (b c-a d)+6 b^2 (c+d x)^2 (b c-a d)^2+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )-B n (b c-a d) \left (15 b^4 (c+d x)^4 (b c-a d)+20 b^3 (c+d x)^3 (b c-a d)^2+30 b^2 (c+d x)^2 (b c-a d)^3+60 b d x (b c-a d)^4+60 (b c-a d)^5 \log (a+b x)+12 b^5 (c+d x)^5\right )\right )}{360 b^4 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

(g^2*i^3*(-15*B*(b*c - a*d)^3*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6
*(b*c - a*d)^3*Log[a + b*x]) + 12*B*(b*c - a*d)^2*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2
+ 4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]) - B*(b*c - a*d)*n*(60*b*d
*(b*c - a*d)^4*x + 30*b^2*(b*c - a*d)^3*(c + d*x)^2 + 20*b^3*(b*c - a*d)^2*(c + d*x)^3 + 15*b^4*(b*c - a*d)*(c
+ d*x)^4 + 12*b^5*(c + d*x)^5 + 60*(b*c - a*d)^5*Log[a + b*x]) + 90*b^4*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]) - 144*b^5*(b*c - a*d)*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 60*b^6*
(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(360*b^4*d^3)

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fricas [B]  time = 1.50, size = 1075, normalized size = 2.78 \[ \frac {60 \, A b^{6} d^{6} g^{2} i^{3} x^{6} + 6 \, {\left (20 \, B a^{3} b^{3} c^{3} d^{3} - 15 \, B a^{4} b^{2} c^{2} d^{4} + 6 \, B a^{5} b c d^{5} - B a^{6} d^{6}\right )} g^{2} i^{3} n \log \left (b x + a\right ) - 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2}\right )} g^{2} i^{3} n \log \left (d x + c\right ) - 12 \, {\left ({\left (B b^{6} c d^{5} - B a b^{5} d^{6}\right )} g^{2} i^{3} n - 6 \, {\left (3 \, A b^{6} c d^{5} + 2 \, A a b^{5} d^{6}\right )} g^{2} i^{3}\right )} x^{5} - 3 \, {\left ({\left (13 \, B b^{6} c^{2} d^{4} - 6 \, B a b^{5} c d^{5} - 7 \, B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} n - 30 \, {\left (3 \, A b^{6} c^{2} d^{4} + 6 \, A a b^{5} c d^{5} + A a^{2} b^{4} d^{6}\right )} g^{2} i^{3}\right )} x^{4} - 2 \, {\left ({\left (19 \, B b^{6} c^{3} d^{3} + 21 \, B a b^{5} c^{2} d^{4} - 39 \, B a^{2} b^{4} c d^{5} - B a^{3} b^{3} d^{6}\right )} g^{2} i^{3} n - 60 \, {\left (A b^{6} c^{3} d^{3} + 6 \, A a b^{5} c^{2} d^{4} + 3 \, A a^{2} b^{4} c d^{5}\right )} g^{2} i^{3}\right )} x^{3} - 3 \, {\left ({\left (B b^{6} c^{4} d^{2} + 34 \, B a b^{5} c^{3} d^{3} - 30 \, B a^{2} b^{4} c^{2} d^{4} - 6 \, B a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{2} i^{3} n - 60 \, {\left (2 \, A a b^{5} c^{3} d^{3} + 3 \, A a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3}\right )} x^{2} + 6 \, {\left (60 \, A a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} + {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} - 5 \, B a^{2} b^{4} c^{3} d^{3} + 15 \, B a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{2} i^{3} n\right )} x + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} x^{2}\right )} \log \relax (e) + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} n x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} n x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} n x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} n x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} n x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{360 \, b^{4} 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)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

1/360*(60*A*b^6*d^6*g^2*i^3*x^6 + 6*(20*B*a^3*b^3*c^3*d^3 - 15*B*a^4*b^2*c^2*d^4 + 6*B*a^5*b*c*d^5 - B*a^6*d^6
)*g^2*i^3*n*log(b*x + a) - 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2)*g^2*i^3*n*log(d*x + c) - 12*
((B*b^6*c*d^5 - B*a*b^5*d^6)*g^2*i^3*n - 6*(3*A*b^6*c*d^5 + 2*A*a*b^5*d^6)*g^2*i^3)*x^5 - 3*((13*B*b^6*c^2*d^4
- 6*B*a*b^5*c*d^5 - 7*B*a^2*b^4*d^6)*g^2*i^3*n - 30*(3*A*b^6*c^2*d^4 + 6*A*a*b^5*c*d^5 + A*a^2*b^4*d^6)*g^2*i
^3)*x^4 - 2*((19*B*b^6*c^3*d^3 + 21*B*a*b^5*c^2*d^4 - 39*B*a^2*b^4*c*d^5 - B*a^3*b^3*d^6)*g^2*i^3*n - 60*(A*b^
6*c^3*d^3 + 6*A*a*b^5*c^2*d^4 + 3*A*a^2*b^4*c*d^5)*g^2*i^3)*x^3 - 3*((B*b^6*c^4*d^2 + 34*B*a*b^5*c^3*d^3 - 30*
B*a^2*b^4*c^2*d^4 - 6*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^2*i^3*n - 60*(2*A*a*b^5*c^3*d^3 + 3*A*a^2*b^4*c^2*d^4
)*g^2*i^3)*x^2 + 6*(60*A*a^2*b^4*c^3*d^3*g^2*i^3 + (B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 - 5*B*a^2*b^4*c^3*d^3 + 15
*B*a^3*b^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^2*i^3*n)*x + 6*(10*B*b^6*d^6*g^2*i^3*x^6 + 60*B*a^2*b^
4*c^3*d^3*g^2*i^3*x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*x^5 + 15*(3*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 +
B*a^2*b^4*d^6)*g^2*i^3*x^4 + 20*(B*b^6*c^3*d^3 + 6*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*g^2*i^3*x^3 + 30*(2*B
*a*b^5*c^3*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*x^2)*log(e) + 6*(10*B*b^6*d^6*g^2*i^3*n*x^6 + 60*B*a^2*b^4*c^3*d
^3*g^2*i^3*n*x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*n*x^5 + 15*(3*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + B*
a^2*b^4*d^6)*g^2*i^3*n*x^4 + 20*(B*b^6*c^3*d^3 + 6*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*g^2*i^3*n*x^3 + 30*(2*
B*a*b^5*c^3*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*n*x^2)*log((b*x + a)/(d*x + c)))/(b^4*d^3)

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giac [B]  time = 10.15, size = 3980, normalized size = 10.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

-1/360*(6*(B*b^9*c^7*g^2*i*n - 7*B*a*b^8*c^6*d*g^2*i*n - 6*(b*x + a)*B*b^8*c^7*d*g^2*i*n/(d*x + c) + 21*B*a^2*
b^7*c^5*d^2*g^2*i*n + 42*(b*x + a)*B*a*b^7*c^6*d^2*g^2*i*n/(d*x + c) + 15*(b*x + a)^2*B*b^7*c^7*d^2*g^2*i*n/(d
*x + c)^2 - 35*B*a^3*b^6*c^4*d^3*g^2*i*n - 126*(b*x + a)*B*a^2*b^6*c^5*d^3*g^2*i*n/(d*x + c) - 105*(b*x + a)^2
*B*a*b^6*c^6*d^3*g^2*i*n/(d*x + c)^2 + 35*B*a^4*b^5*c^3*d^4*g^2*i*n + 210*(b*x + a)*B*a^3*b^5*c^4*d^4*g^2*i*n/
(d*x + c) + 315*(b*x + a)^2*B*a^2*b^5*c^5*d^4*g^2*i*n/(d*x + c)^2 - 21*B*a^5*b^4*c^2*d^5*g^2*i*n - 210*(b*x +
a)*B*a^4*b^4*c^3*d^5*g^2*i*n/(d*x + c) - 525*(b*x + a)^2*B*a^3*b^4*c^4*d^5*g^2*i*n/(d*x + c)^2 + 7*B*a^6*b^3*c
*d^6*g^2*i*n + 126*(b*x + a)*B*a^5*b^3*c^2*d^6*g^2*i*n/(d*x + c) + 525*(b*x + a)^2*B*a^4*b^3*c^3*d^6*g^2*i*n/(
d*x + c)^2 - B*a^7*b^2*d^7*g^2*i*n - 42*(b*x + a)*B*a^6*b^2*c*d^7*g^2*i*n/(d*x + c) - 315*(b*x + a)^2*B*a^5*b^
2*c^2*d^7*g^2*i*n/(d*x + c)^2 + 6*(b*x + a)*B*a^7*b*d^8*g^2*i*n/(d*x + c) + 105*(b*x + a)^2*B*a^6*b*c*d^8*g^2*
i*n/(d*x + c)^2 - 15*(b*x + a)^2*B*a^7*d^9*g^2*i*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/(b^6*d^3 - 6*(b*x + a
)*b^5*d^4/(d*x + c) + 15*(b*x + a)^2*b^4*d^5/(d*x + c)^2 - 20*(b*x + a)^3*b^3*d^6/(d*x + c)^3 + 15*(b*x + a)^4
*b^2*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b*d^8/(d*x + c)^5 + (b*x + a)^6*d^9/(d*x + c)^6) - (2*B*b^12*c^7*g^2*i*n
- 14*B*a*b^11*c^6*d*g^2*i*n - 18*(b*x + a)*B*b^11*c^7*d*g^2*i*n/(d*x + c) + 42*B*a^2*b^10*c^5*d^2*g^2*i*n + 12
6*(b*x + a)*B*a*b^10*c^6*d^2*g^2*i*n/(d*x + c) + 63*(b*x + a)^2*B*b^10*c^7*d^2*g^2*i*n/(d*x + c)^2 - 70*B*a^3*
b^9*c^4*d^3*g^2*i*n - 378*(b*x + a)*B*a^2*b^9*c^5*d^3*g^2*i*n/(d*x + c) - 441*(b*x + a)^2*B*a*b^9*c^6*d^3*g^2*
i*n/(d*x + c)^2 - 74*(b*x + a)^3*B*b^9*c^7*d^3*g^2*i*n/(d*x + c)^3 + 70*B*a^4*b^8*c^3*d^4*g^2*i*n + 630*(b*x +
a)*B*a^3*b^8*c^4*d^4*g^2*i*n/(d*x + c) + 1323*(b*x + a)^2*B*a^2*b^8*c^5*d^4*g^2*i*n/(d*x + c)^2 + 518*(b*x +
a)^3*B*a*b^8*c^6*d^4*g^2*i*n/(d*x + c)^3 + 33*(b*x + a)^4*B*b^8*c^7*d^4*g^2*i*n/(d*x + c)^4 - 42*B*a^5*b^7*c^2
*d^5*g^2*i*n - 630*(b*x + a)*B*a^4*b^7*c^3*d^5*g^2*i*n/(d*x + c) - 2205*(b*x + a)^2*B*a^3*b^7*c^4*d^5*g^2*i*n/
(d*x + c)^2 - 1554*(b*x + a)^3*B*a^2*b^7*c^5*d^5*g^2*i*n/(d*x + c)^3 - 231*(b*x + a)^4*B*a*b^7*c^6*d^5*g^2*i*n
/(d*x + c)^4 - 6*(b*x + a)^5*B*b^7*c^7*d^5*g^2*i*n/(d*x + c)^5 + 14*B*a^6*b^6*c*d^6*g^2*i*n + 378*(b*x + a)*B*
a^5*b^6*c^2*d^6*g^2*i*n/(d*x + c) + 2205*(b*x + a)^2*B*a^4*b^6*c^3*d^6*g^2*i*n/(d*x + c)^2 + 2590*(b*x + a)^3*
B*a^3*b^6*c^4*d^6*g^2*i*n/(d*x + c)^3 + 693*(b*x + a)^4*B*a^2*b^6*c^5*d^6*g^2*i*n/(d*x + c)^4 + 42*(b*x + a)^5
*B*a*b^6*c^6*d^6*g^2*i*n/(d*x + c)^5 - 2*B*a^7*b^5*d^7*g^2*i*n - 126*(b*x + a)*B*a^6*b^5*c*d^7*g^2*i*n/(d*x +
c) - 1323*(b*x + a)^2*B*a^5*b^5*c^2*d^7*g^2*i*n/(d*x + c)^2 - 2590*(b*x + a)^3*B*a^4*b^5*c^3*d^7*g^2*i*n/(d*x
+ c)^3 - 1155*(b*x + a)^4*B*a^3*b^5*c^4*d^7*g^2*i*n/(d*x + c)^4 - 126*(b*x + a)^5*B*a^2*b^5*c^5*d^7*g^2*i*n/(d
*x + c)^5 + 18*(b*x + a)*B*a^7*b^4*d^8*g^2*i*n/(d*x + c) + 441*(b*x + a)^2*B*a^6*b^4*c*d^8*g^2*i*n/(d*x + c)^2
+ 1554*(b*x + a)^3*B*a^5*b^4*c^2*d^8*g^2*i*n/(d*x + c)^3 + 1155*(b*x + a)^4*B*a^4*b^4*c^3*d^8*g^2*i*n/(d*x +
c)^4 + 210*(b*x + a)^5*B*a^3*b^4*c^4*d^8*g^2*i*n/(d*x + c)^5 - 63*(b*x + a)^2*B*a^7*b^3*d^9*g^2*i*n/(d*x + c)^
2 - 518*(b*x + a)^3*B*a^6*b^3*c*d^9*g^2*i*n/(d*x + c)^3 - 693*(b*x + a)^4*B*a^5*b^3*c^2*d^9*g^2*i*n/(d*x + c)^
4 - 210*(b*x + a)^5*B*a^4*b^3*c^3*d^9*g^2*i*n/(d*x + c)^5 + 74*(b*x + a)^3*B*a^7*b^2*d^10*g^2*i*n/(d*x + c)^3
+ 231*(b*x + a)^4*B*a^6*b^2*c*d^10*g^2*i*n/(d*x + c)^4 + 126*(b*x + a)^5*B*a^5*b^2*c^2*d^10*g^2*i*n/(d*x + c)^
5 - 33*(b*x + a)^4*B*a^7*b*d^11*g^2*i*n/(d*x + c)^4 - 42*(b*x + a)^5*B*a^6*b*c*d^11*g^2*i*n/(d*x + c)^5 + 6*(b
*x + a)^5*B*a^7*d^12*g^2*i*n/(d*x + c)^5 - 6*A*b^12*c^7*g^2*i - 6*B*b^12*c^7*g^2*i + 42*A*a*b^11*c^6*d*g^2*i +
42*B*a*b^11*c^6*d*g^2*i + 36*(b*x + a)*A*b^11*c^7*d*g^2*i/(d*x + c) + 36*(b*x + a)*B*b^11*c^7*d*g^2*i/(d*x +
c) - 126*A*a^2*b^10*c^5*d^2*g^2*i - 126*B*a^2*b^10*c^5*d^2*g^2*i - 252*(b*x + a)*A*a*b^10*c^6*d^2*g^2*i/(d*x +
c) - 252*(b*x + a)*B*a*b^10*c^6*d^2*g^2*i/(d*x + c) - 90*(b*x + a)^2*A*b^10*c^7*d^2*g^2*i/(d*x + c)^2 - 90*(b
*x + a)^2*B*b^10*c^7*d^2*g^2*i/(d*x + c)^2 + 210*A*a^3*b^9*c^4*d^3*g^2*i + 210*B*a^3*b^9*c^4*d^3*g^2*i + 756*(
b*x + a)*A*a^2*b^9*c^5*d^3*g^2*i/(d*x + c) + 756*(b*x + a)*B*a^2*b^9*c^5*d^3*g^2*i/(d*x + c) + 630*(b*x + a)^2
*A*a*b^9*c^6*d^3*g^2*i/(d*x + c)^2 + 630*(b*x + a)^2*B*a*b^9*c^6*d^3*g^2*i/(d*x + c)^2 - 210*A*a^4*b^8*c^3*d^4
*g^2*i - 210*B*a^4*b^8*c^3*d^4*g^2*i - 1260*(b*x + a)*A*a^3*b^8*c^4*d^4*g^2*i/(d*x + c) - 1260*(b*x + a)*B*a^3
*b^8*c^4*d^4*g^2*i/(d*x + c) - 1890*(b*x + a)^2*A*a^2*b^8*c^5*d^4*g^2*i/(d*x + c)^2 - 1890*(b*x + a)^2*B*a^2*b
^8*c^5*d^4*g^2*i/(d*x + c)^2 + 126*A*a^5*b^7*c^2*d^5*g^2*i + 126*B*a^5*b^7*c^2*d^5*g^2*i + 1260*(b*x + a)*A*a^
4*b^7*c^3*d^5*g^2*i/(d*x + c) + 1260*(b*x + a)*B*a^4*b^7*c^3*d^5*g^2*i/(d*x + c) + 3150*(b*x + a)^2*A*a^3*b^7*
c^4*d^5*g^2*i/(d*x + c)^2 + 3150*(b*x + a)^2*B*a^3*b^7*c^4*d^5*g^2*i/(d*x + c)^2 - 42*A*a^6*b^6*c*d^6*g^2*i -
42*B*a^6*b^6*c*d^6*g^2*i - 756*(b*x + a)*A*a^5*b^6*c^2*d^6*g^2*i/(d*x + c) - 756*(b*x + a)*B*a^5*b^6*c^2*d^6*g
^2*i/(d*x + c) - 3150*(b*x + a)^2*A*a^4*b^6*c^3*d^6*g^2*i/(d*x + c)^2 - 3150*(b*x + a)^2*B*a^4*b^6*c^3*d^6*g^2
*i/(d*x + c)^2 + 6*A*a^7*b^5*d^7*g^2*i + 6*B*a^7*b^5*d^7*g^2*i + 252*(b*x + a)*A*a^6*b^5*c*d^7*g^2*i/(d*x + c)
+ 252*(b*x + a)*B*a^6*b^5*c*d^7*g^2*i/(d*x + c) + 1890*(b*x + a)^2*A*a^5*b^5*c^2*d^7*g^2*i/(d*x + c)^2 + 1890
*(b*x + a)^2*B*a^5*b^5*c^2*d^7*g^2*i/(d*x + c)^2 - 36*(b*x + a)*A*a^7*b^4*d^8*g^2*i/(d*x + c) - 36*(b*x + a)*B
*a^7*b^4*d^8*g^2*i/(d*x + c) - 630*(b*x + a)^2*A*a^6*b^4*c*d^8*g^2*i/(d*x + c)^2 - 630*(b*x + a)^2*B*a^6*b^4*c
*d^8*g^2*i/(d*x + c)^2 + 90*(b*x + a)^2*A*a^7*b^3*d^9*g^2*i/(d*x + c)^2 + 90*(b*x + a)^2*B*a^7*b^3*d^9*g^2*i/(
d*x + c)^2)/(b^9*d^3 - 6*(b*x + a)*b^8*d^4/(d*x + c) + 15*(b*x + a)^2*b^7*d^5/(d*x + c)^2 - 20*(b*x + a)^3*b^6
*d^6/(d*x + c)^3 + 15*(b*x + a)^4*b^5*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b^4*d^8/(d*x + c)^5 + (b*x + a)^6*b^3*d^
9/(d*x + c)^6) + 6*(B*b^7*c^7*g^2*i*n - 7*B*a*b^6*c^6*d*g^2*i*n + 21*B*a^2*b^5*c^5*d^2*g^2*i*n - 35*B*a^3*b^4*
c^4*d^3*g^2*i*n + 35*B*a^4*b^3*c^3*d^4*g^2*i*n - 21*B*a^5*b^2*c^2*d^5*g^2*i*n + 7*B*a^6*b*c*d^6*g^2*i*n - B*a^
7*d^7*g^2*i*n)*log(-b + (b*x + a)*d/(d*x + c))/(b^4*d^3) - 6*(B*b^7*c^7*g^2*i*n - 7*B*a*b^6*c^6*d*g^2*i*n + 21
*B*a^2*b^5*c^5*d^2*g^2*i*n - 35*B*a^3*b^4*c^4*d^3*g^2*i*n + 35*B*a^4*b^3*c^3*d^4*g^2*i*n - 21*B*a^5*b^2*c^2*d^
5*g^2*i*n + 7*B*a^6*b*c*d^6*g^2*i*n - B*a^7*d^7*g^2*i*n)*log((b*x + a)/(d*x + c))/(b^4*d^3))*(b*c/(b*c - a*d)^
2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^2*(d*i*x+c*i)^3*(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)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

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maxima [B]  time = 1.68, size = 1978, normalized size = 5.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/6*B*b^2*d^3*g^2*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A*b^2*d^3*g^2*i^3*x^6 + 3/5*B*b^2*c*d^2
*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/5*B*a*b*d^3*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + 3/5*A*b^2*c*d^2*g^2*i^3*x^5 + 2/5*A*a*b*d^3*g^2*i^3*x^5 + 3/4*B*b^2*c^2*d*g^2*i^3*x^4*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n) + 3/2*B*a*b*c*d^2*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a^2*d^3
*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A*b^2*c^2*d*g^2*i^3*x^4 + 3/2*A*a*b*c*d^2*g^2*i^3*x^
4 + 1/4*A*a^2*d^3*g^2*i^3*x^4 + 1/3*B*b^2*c^3*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*B*a*b*c^2
*d*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a^2*c*d^2*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + 1/3*A*b^2*c^3*g^2*i^3*x^3 + 2*A*a*b*c^2*d*g^2*i^3*x^3 + A*a^2*c*d^2*g^2*i^3*x^3 + B*a*b*c^3*g^2*i^3
*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a^2*c^2*d*g^2*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^
n) + A*a*b*c^3*g^2*i^3*x^2 + 3/2*A*a^2*c^2*d*g^2*i^3*x^2 - 1/360*B*b^2*d^3*g^2*i^3*n*(60*a^6*log(b*x + a)/b^6
- 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3
*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/20*B*b^2*c
*d^2*g^2*i^3*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)) + 1/30*B*a*b*d
^3*g^2*i^3*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)) - 1/8*B*b^2*c^2*
d*g^2*i^3*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)) - 1/4*B*a*b*c*d^2*g^2*i^3*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)) - 1/24*B*a^2*d^3*g^2*i^3*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^3*g^2*i^3*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*d*g^2*i^3*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)) + 1/2*B*a^2*c*d^2*g^2*i^3*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^3*g^2*i^3*n*(a^2*log(b*x + a)/b^2 - c
^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - 3/2*B*a^2*c^2*d*g^2*i^3*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^3*g^2*i^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + B*a^2*c^3*g^2*i^3*x
*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^2*c^3*g^2*i^3*x

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mupad [B]  time = 6.26, size = 2547, normalized size = 6.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)

[Out]

x^2*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))
/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 6
0*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(2*b*d) - ((60*a*d + 60*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16
*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b
*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^
2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a
^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(60*b*d) - (a*c*((b*d^2*g^2*
i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(b*d)))/(120*b*d)
+ (c*g^2*i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n + 36*A*a*b^2*c^2*d + 54*A*a^2*b*c*d^2 -
5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(6*b*d)) + x^3*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n -
3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(12*b) + ((60*a*
d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c
))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n +
60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(180*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c
+ B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(3*b*d)) - x^4*((((b*d^2*g^2*i^3*(18*A*a*d
+ 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(240*b*d) - (d
*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/20 + (A*a
*b*c*d^2*g^2*i^3)/4) + x^5*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/30 - (A*b*d^2*g^2*i^3*(6
0*a*d + 60*b*c))/300) - x*(((60*a*d + 60*b*c)*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n
))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b
^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(b*d) - ((60*a
*d + 60*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2
*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24
*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*
i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^
2*g^2*i^3))/(60*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(6
0*a*d + 60*b*c))/60))/(b*d)))/(60*b*d) + (c*g^2*i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n
+ 36*A*a*b^2*c^2*d + 54*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(3*b*d)))/(60*b*d) + (a*c*((g^
2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*
b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n
- B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2
+ 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(60*b
*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/
60))/(b*d)))/(b*d) - (a*c^2*g^2*i^3*(12*A*a^2*d^2 + 6*A*b^2*c^2 + 3*B*a^2*d^2*n - 2*B*b^2*c^2*n + 24*A*a*b*c*d
- B*a*b*c*d*n))/(2*b*d)) + log(e*((a + b*x)/(c + d*x))^n)*(B*a^2*c^3*g^2*i^3*x + (B*c*g^2*i^3*x^3*(3*a^2*d^2
+ b^2*c^2 + 6*a*b*c*d))/3 + (B*d*g^2*i^3*x^4*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d))/4 + (B*b^2*d^3*g^2*i^3*x^6)/6
+ (B*a*c^2*g^2*i^3*x^2*(3*a*d + 2*b*c))/2 + (B*b*d^2*g^2*i^3*x^5*(2*a*d + 3*b*c))/5) - (log(a + b*x)*(B*a^6*d^
3*g^2*i^3*n - 20*B*a^3*b^3*c^3*g^2*i^3*n + 15*B*a^4*b^2*c^2*d*g^2*i^3*n - 6*B*a^5*b*c*d^2*g^2*i^3*n))/(60*b^4)
- (log(c + d*x)*(B*b^2*c^6*g^2*i^3*n + 15*B*a^2*c^4*d^2*g^2*i^3*n - 6*B*a*b*c^5*d*g^2*i^3*n))/(60*d^3) + (A*b
^2*d^3*g^2*i^3*x^6)/6

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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)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

Timed out

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