3.30 \(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^7} \, dx\)

Optimal. Leaf size=281 \[ -\frac {b^2 i^3 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 g^7 (a+b x)^6 (b c-a d)^3}-\frac {d^2 i^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g^7 (a+b x)^4 (b c-a d)^3}+\frac {2 b d i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 g^7 (a+b x)^5 (b c-a d)^3}-\frac {b^2 B i^3 (c+d x)^6}{36 g^7 (a+b x)^6 (b c-a d)^3}-\frac {B d^2 i^3 (c+d x)^4}{16 g^7 (a+b x)^4 (b c-a d)^3}+\frac {2 b B d i^3 (c+d x)^5}{25 g^7 (a+b x)^5 (b c-a d)^3} \]

[Out]

-1/16*B*d^2*i^3*(d*x+c)^4/(-a*d+b*c)^3/g^7/(b*x+a)^4+2/25*b*B*d*i^3*(d*x+c)^5/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/36*
b^2*B*i^3*(d*x+c)^6/(-a*d+b*c)^3/g^7/(b*x+a)^6-1/4*d^2*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/
g^7/(b*x+a)^4+2/5*b*d*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/6*b^2*i^3*(d*x+c)
^6*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^7/(b*x+a)^6

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Rubi [A]  time = 0.98, antiderivative size = 445, normalized size of antiderivative = 1.58, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2528, 2525, 12, 44} \[ -\frac {d^3 i^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^4 g^7 (a+b x)^3}-\frac {3 d^2 i^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b^4 g^7 (a+b x)^4}-\frac {3 d i^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b^4 g^7 (a+b x)^5}-\frac {i^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 b^4 g^7 (a+b x)^6}-\frac {B d^5 i^3}{60 b^4 g^7 (a+b x) (b c-a d)^2}+\frac {B d^4 i^3}{120 b^4 g^7 (a+b x)^2 (b c-a d)}-\frac {19 B d^2 i^3 (b c-a d)}{240 b^4 g^7 (a+b x)^4}-\frac {B d^6 i^3 \log (a+b x)}{60 b^4 g^7 (b c-a d)^3}+\frac {B d^6 i^3 \log (c+d x)}{60 b^4 g^7 (b c-a d)^3}-\frac {13 B d i^3 (b c-a d)^2}{150 b^4 g^7 (a+b x)^5}-\frac {B i^3 (b c-a d)^3}{36 b^4 g^7 (a+b x)^6}-\frac {B d^3 i^3}{180 b^4 g^7 (a+b x)^3} \]

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)^7,x]

[Out]

-(B*(b*c - a*d)^3*i^3)/(36*b^4*g^7*(a + b*x)^6) - (13*B*d*(b*c - a*d)^2*i^3)/(150*b^4*g^7*(a + b*x)^5) - (19*B
*d^2*(b*c - a*d)*i^3)/(240*b^4*g^7*(a + b*x)^4) - (B*d^3*i^3)/(180*b^4*g^7*(a + b*x)^3) + (B*d^4*i^3)/(120*b^4
*(b*c - a*d)*g^7*(a + b*x)^2) - (B*d^5*i^3)/(60*b^4*(b*c - a*d)^2*g^7*(a + b*x)) - (B*d^6*i^3*Log[a + b*x])/(6
0*b^4*(b*c - a*d)^3*g^7) - ((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^4*g^7*(a + b*x)^6) -
(3*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^4*g^7*(a + b*x)^5) - (3*d^2*(b*c - a*d)*i^3*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^4*g^7*(a + b*x)^4) - (d^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/(3*b^4*g^7*(a + b*x)^3) + (B*d^6*i^3*Log[c + d*x])/(60*b^4*(b*c - a*d)^3*g^7)

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])

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 \frac {(30 c+30 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx &=\int \left (\frac {27000 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^7}+\frac {81000 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^6}+\frac {81000 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^5}+\frac {27000 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^7 (a+b x)^4}\right ) \, dx\\ &=\frac {\left (27000 d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^7}+\frac {\left (81000 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)^5} \, dx}{b^3 g^7}+\frac {\left (81000 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)^6} \, dx}{b^3 g^7}+\frac {\left (27000 (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^7} \, dx}{b^3 g^7}\\ &=-\frac {4500 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac {16200 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac {20250 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac {9000 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac {\left (9000 B d^3\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^7}+\frac {\left (20250 B d^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^7}+\frac {\left (16200 B d (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^6 (c+d x)} \, dx}{b^4 g^7}+\frac {\left (4500 B (b c-a d)^3\right ) \int \frac {b c-a d}{(a+b x)^7 (c+d x)} \, dx}{b^4 g^7}\\ &=-\frac {4500 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac {16200 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac {20250 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac {9000 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac {\left (9000 B d^3 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^7}+\frac {\left (20250 B d^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{b^4 g^7}+\frac {\left (16200 B d (b c-a d)^3\right ) \int \frac {1}{(a+b x)^6 (c+d x)} \, dx}{b^4 g^7}+\frac {\left (4500 B (b c-a d)^4\right ) \int \frac {1}{(a+b x)^7 (c+d x)} \, dx}{b^4 g^7}\\ &=-\frac {4500 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac {16200 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac {20250 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac {9000 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac {\left (9000 B d^3 (b c-a d)\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^7}+\frac {\left (20250 B d^2 (b c-a d)^2\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^7}+\frac {\left (16200 B d (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^6}-\frac {b d}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5}{(b c-a d)^6 (a+b x)}+\frac {d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{b^4 g^7}+\frac {\left (4500 B (b c-a d)^4\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^7}-\frac {b d}{(b c-a d)^2 (a+b x)^6}+\frac {b d^2}{(b c-a d)^3 (a+b x)^5}-\frac {b d^3}{(b c-a d)^4 (a+b x)^4}+\frac {b d^4}{(b c-a d)^5 (a+b x)^3}-\frac {b d^5}{(b c-a d)^6 (a+b x)^2}+\frac {b d^6}{(b c-a d)^7 (a+b x)}-\frac {d^7}{(b c-a d)^7 (c+d x)}\right ) \, dx}{b^4 g^7}\\ &=-\frac {750 B (b c-a d)^3}{b^4 g^7 (a+b x)^6}-\frac {2340 B d (b c-a d)^2}{b^4 g^7 (a+b x)^5}-\frac {4275 B d^2 (b c-a d)}{2 b^4 g^7 (a+b x)^4}-\frac {150 B d^3}{b^4 g^7 (a+b x)^3}+\frac {225 B d^4}{b^4 (b c-a d) g^7 (a+b x)^2}-\frac {450 B d^5}{b^4 (b c-a d)^2 g^7 (a+b x)}-\frac {450 B d^6 \log (a+b x)}{b^4 (b c-a d)^3 g^7}-\frac {4500 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^6}-\frac {16200 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^5}-\frac {20250 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^4}-\frac {9000 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^7 (a+b x)^3}+\frac {450 B d^6 \log (c+d x)}{b^4 (b c-a d)^3 g^7}\\ \end {align*}

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Mathematica [B]  time = 1.05, size = 642, normalized size = 2.28 \[ \frac {i^3 \left (1200 d^3 (a+b x)^3 (a d-b c)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2700 d^2 (a+b x)^2 (b c-a d)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2160 d (a+b x) (a d-b c)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-600 (b c-a d)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2100 B d^6 (a+b x)^6 \log (c+d x)+2160 a B d^6 (a+b x)^5 \log (c+d x)+2160 b B d^6 x (a+b x)^5 \log (c+d x)+2100 B d^5 (a+b x)^5 (b c-a d)+2160 a B d^5 (a+b x)^4 (a d-b c)-2160 b B d^5 x (a+b x)^4 (b c-a d)-1050 B d^4 (a+b x)^4 (b c-a d)^2+1080 a B d^4 (a+b x)^3 (b c-a d)^2+1080 b B d^4 x (a+b x)^3 (b c-a d)^2+700 B d^3 (a+b x)^3 (b c-a d)^3+720 a B d^3 (a+b x)^2 (a d-b c)^3-720 b B d^3 x (a+b x)^2 (b c-a d)^3-825 B d^2 (a+b x)^2 (b c-a d)^4+540 a B d^2 (a+b x) (b c-a d)^4+540 b B d^2 x (a+b x) (b c-a d)^4-432 b B d x (b c-a d)^5+120 B d (a+b x) (b c-a d)^5+432 a B d (a d-b c)^5-100 B (b c-a d)^6+2100 B d^6 (a+b x)^6 \log (a+b x)-2160 a B d^6 (a+b x)^5 \log (a+b x)-2160 b B d^6 x (a+b x)^5 \log (a+b x)\right )}{3600 b^4 g^7 (a+b x)^6 (b c-a d)^3} \]

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)^7,x]

[Out]

(i^3*(-100*B*(b*c - a*d)^6 + 432*a*B*d*(-(b*c) + a*d)^5 - 432*b*B*d*(b*c - a*d)^5*x + 540*a*B*d^2*(b*c - a*d)^
4*(a + b*x) + 120*B*d*(b*c - a*d)^5*(a + b*x) + 540*b*B*d^2*(b*c - a*d)^4*x*(a + b*x) - 825*B*d^2*(b*c - a*d)^
4*(a + b*x)^2 + 720*a*B*d^3*(-(b*c) + a*d)^3*(a + b*x)^2 - 720*b*B*d^3*(b*c - a*d)^3*x*(a + b*x)^2 + 1080*a*B*
d^4*(b*c - a*d)^2*(a + b*x)^3 + 700*B*d^3*(b*c - a*d)^3*(a + b*x)^3 + 1080*b*B*d^4*(b*c - a*d)^2*x*(a + b*x)^3
- 1050*B*d^4*(b*c - a*d)^2*(a + b*x)^4 + 2160*a*B*d^5*(-(b*c) + a*d)*(a + b*x)^4 - 2160*b*B*d^5*(b*c - a*d)*x
*(a + b*x)^4 + 2100*B*d^5*(b*c - a*d)*(a + b*x)^5 - 2160*a*B*d^6*(a + b*x)^5*Log[a + b*x] - 2160*b*B*d^6*x*(a
+ b*x)^5*Log[a + b*x] + 2100*B*d^6*(a + b*x)^6*Log[a + b*x] - 600*(b*c - a*d)^6*(A + B*Log[(e*(a + b*x))/(c +
d*x)]) + 2160*d*(-(b*c) + a*d)^5*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2700*d^2*(b*c - a*d)^4*(a +
b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 1200*d^3*(-(b*c) + a*d)^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(
c + d*x)]) + 2160*a*B*d^6*(a + b*x)^5*Log[c + d*x] + 2160*b*B*d^6*x*(a + b*x)^5*Log[c + d*x] - 2100*B*d^6*(a +
b*x)^6*Log[c + d*x]))/(3600*b^4*(b*c - a*d)^3*g^7*(a + b*x)^6)

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fricas [B]  time = 0.94, size = 991, normalized size = 3.53 \[ -\frac {60 \, {\left (B b^{6} c d^{5} - B a b^{5} d^{6}\right )} i^{3} x^{5} - 30 \, {\left (B b^{6} c^{2} d^{4} - 12 \, B a b^{5} c d^{5} + 11 \, B a^{2} b^{4} d^{6}\right )} i^{3} x^{4} + 20 \, {\left ({\left (60 \, A + B\right )} b^{6} c^{3} d^{3} - 9 \, {\left (20 \, A + B\right )} a b^{5} c^{2} d^{4} + 45 \, {\left (4 \, A + B\right )} a^{2} b^{4} c d^{5} - {\left (60 \, A + 37 \, B\right )} a^{3} b^{3} d^{6}\right )} i^{3} x^{3} + 15 \, {\left ({\left (180 \, A + 19 \, B\right )} b^{6} c^{4} d^{2} - 24 \, {\left (20 \, A + 3 \, B\right )} a b^{5} c^{3} d^{3} + 90 \, {\left (4 \, A + B\right )} a^{2} b^{4} c^{2} d^{4} - {\left (60 \, A + 37 \, B\right )} a^{4} b^{2} d^{6}\right )} i^{3} x^{2} + 6 \, {\left (4 \, {\left (90 \, A + 13 \, B\right )} b^{6} c^{5} d - 15 \, {\left (60 \, A + 11 \, B\right )} a b^{5} c^{4} d^{2} + 150 \, {\left (4 \, A + B\right )} a^{2} b^{4} c^{3} d^{3} - {\left (60 \, A + 37 \, B\right )} a^{5} b d^{6}\right )} i^{3} x + {\left (100 \, {\left (6 \, A + B\right )} b^{6} c^{6} - 288 \, {\left (5 \, A + B\right )} a b^{5} c^{5} d + 225 \, {\left (4 \, A + B\right )} a^{2} b^{4} c^{4} d^{2} - {\left (60 \, A + 37 \, B\right )} a^{6} d^{6}\right )} i^{3} + 60 \, {\left (B b^{6} d^{6} i^{3} x^{6} + 6 \, B a b^{5} d^{6} i^{3} x^{5} + 15 \, B a^{2} b^{4} d^{6} i^{3} x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} - 3 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} i^{3} x^{3} + 15 \, {\left (3 \, B b^{6} c^{4} d^{2} - 8 \, B a b^{5} c^{3} d^{3} + 6 \, B a^{2} b^{4} c^{2} d^{4}\right )} i^{3} x^{2} + 6 \, {\left (6 \, B b^{6} c^{5} d - 15 \, B a b^{5} c^{4} d^{2} + 10 \, B a^{2} b^{4} c^{3} d^{3}\right )} i^{3} x + {\left (10 \, B b^{6} c^{6} - 24 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2}\right )} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3600 \, {\left ({\left (b^{13} c^{3} - 3 \, a b^{12} c^{2} d + 3 \, a^{2} b^{11} c d^{2} - a^{3} b^{10} d^{3}\right )} g^{7} x^{6} + 6 \, {\left (a b^{12} c^{3} - 3 \, a^{2} b^{11} c^{2} d + 3 \, a^{3} b^{10} c d^{2} - a^{4} b^{9} d^{3}\right )} g^{7} x^{5} + 15 \, {\left (a^{2} b^{11} c^{3} - 3 \, a^{3} b^{10} c^{2} d + 3 \, a^{4} b^{9} c d^{2} - a^{5} b^{8} d^{3}\right )} g^{7} x^{4} + 20 \, {\left (a^{3} b^{10} c^{3} - 3 \, a^{4} b^{9} c^{2} d + 3 \, a^{5} b^{8} c d^{2} - a^{6} b^{7} d^{3}\right )} g^{7} x^{3} + 15 \, {\left (a^{4} b^{9} c^{3} - 3 \, a^{5} b^{8} c^{2} d + 3 \, a^{6} b^{7} c d^{2} - a^{7} b^{6} d^{3}\right )} g^{7} x^{2} + 6 \, {\left (a^{5} b^{8} c^{3} - 3 \, a^{6} b^{7} c^{2} d + 3 \, a^{7} b^{6} c d^{2} - a^{8} b^{5} d^{3}\right )} g^{7} x + {\left (a^{6} b^{7} c^{3} - 3 \, a^{7} b^{6} c^{2} d + 3 \, a^{8} b^{5} c d^{2} - a^{9} b^{4} d^{3}\right )} g^{7}\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

-1/3600*(60*(B*b^6*c*d^5 - B*a*b^5*d^6)*i^3*x^5 - 30*(B*b^6*c^2*d^4 - 12*B*a*b^5*c*d^5 + 11*B*a^2*b^4*d^6)*i^3
*x^4 + 20*((60*A + B)*b^6*c^3*d^3 - 9*(20*A + B)*a*b^5*c^2*d^4 + 45*(4*A + B)*a^2*b^4*c*d^5 - (60*A + 37*B)*a^
3*b^3*d^6)*i^3*x^3 + 15*((180*A + 19*B)*b^6*c^4*d^2 - 24*(20*A + 3*B)*a*b^5*c^3*d^3 + 90*(4*A + B)*a^2*b^4*c^2
*d^4 - (60*A + 37*B)*a^4*b^2*d^6)*i^3*x^2 + 6*(4*(90*A + 13*B)*b^6*c^5*d - 15*(60*A + 11*B)*a*b^5*c^4*d^2 + 15
0*(4*A + B)*a^2*b^4*c^3*d^3 - (60*A + 37*B)*a^5*b*d^6)*i^3*x + (100*(6*A + B)*b^6*c^6 - 288*(5*A + B)*a*b^5*c^
5*d + 225*(4*A + B)*a^2*b^4*c^4*d^2 - (60*A + 37*B)*a^6*d^6)*i^3 + 60*(B*b^6*d^6*i^3*x^6 + 6*B*a*b^5*d^6*i^3*x
^5 + 15*B*a^2*b^4*d^6*i^3*x^4 + 20*(B*b^6*c^3*d^3 - 3*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*i^3*x^3 + 15*(3*B*b
^6*c^4*d^2 - 8*B*a*b^5*c^3*d^3 + 6*B*a^2*b^4*c^2*d^4)*i^3*x^2 + 6*(6*B*b^6*c^5*d - 15*B*a*b^5*c^4*d^2 + 10*B*a
^2*b^4*c^3*d^3)*i^3*x + (10*B*b^6*c^6 - 24*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2)*i^3)*log((b*e*x + a*e)/(d*x +
c)))/((b^13*c^3 - 3*a*b^12*c^2*d + 3*a^2*b^11*c*d^2 - a^3*b^10*d^3)*g^7*x^6 + 6*(a*b^12*c^3 - 3*a^2*b^11*c^2*
d + 3*a^3*b^10*c*d^2 - a^4*b^9*d^3)*g^7*x^5 + 15*(a^2*b^11*c^3 - 3*a^3*b^10*c^2*d + 3*a^4*b^9*c*d^2 - a^5*b^8*
d^3)*g^7*x^4 + 20*(a^3*b^10*c^3 - 3*a^4*b^9*c^2*d + 3*a^5*b^8*c*d^2 - a^6*b^7*d^3)*g^7*x^3 + 15*(a^4*b^9*c^3 -
3*a^5*b^8*c^2*d + 3*a^6*b^7*c*d^2 - a^7*b^6*d^3)*g^7*x^2 + 6*(a^5*b^8*c^3 - 3*a^6*b^7*c^2*d + 3*a^7*b^6*c*d^2
- a^8*b^5*d^3)*g^7*x + (a^6*b^7*c^3 - 3*a^7*b^6*c^2*d + 3*a^8*b^5*c*d^2 - a^9*b^4*d^3)*g^7)

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giac [A]  time = 4.08, size = 391, normalized size = 1.39 \[ \frac {{\left (600 \, B b^{2} i e^{7} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {1440 \, {\left (b x e + a e\right )} B b d i e^{6} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {900 \, {\left (b x e + a e\right )}^{2} B d^{2} i e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + 600 \, A b^{2} i e^{7} + 100 \, B b^{2} i e^{7} - \frac {1440 \, {\left (b x e + a e\right )} A b d i e^{6}}{d x + c} - \frac {288 \, {\left (b x e + a e\right )} B b d i e^{6}}{d x + c} + \frac {900 \, {\left (b x e + a e\right )}^{2} A d^{2} i e^{5}}{{\left (d x + c\right )}^{2}} + \frac {225 \, {\left (b x e + a e\right )}^{2} B d^{2} i e^{5}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{3600 \, {\left (\frac {{\left (b x e + a e\right )}^{6} b^{2} c^{2} g^{7}}{{\left (d x + c\right )}^{6}} - \frac {2 \, {\left (b x e + a e\right )}^{6} a b c d g^{7}}{{\left (d x + c\right )}^{6}} + \frac {{\left (b x e + a e\right )}^{6} a^{2} d^{2} g^{7}}{{\left (d x + c\right )}^{6}}\right )}} \]

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)^7,x, algorithm="giac")

[Out]

1/3600*(600*B*b^2*i*e^7*log((b*x*e + a*e)/(d*x + c)) - 1440*(b*x*e + a*e)*B*b*d*i*e^6*log((b*x*e + a*e)/(d*x +
c))/(d*x + c) + 900*(b*x*e + a*e)^2*B*d^2*i*e^5*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 600*A*b^2*i*e^7 +
100*B*b^2*i*e^7 - 1440*(b*x*e + a*e)*A*b*d*i*e^6/(d*x + c) - 288*(b*x*e + a*e)*B*b*d*i*e^6/(d*x + c) + 900*(b*
x*e + a*e)^2*A*d^2*i*e^5/(d*x + c)^2 + 225*(b*x*e + a*e)^2*B*d^2*i*e^5/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c
- a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/((b*x*e + a*e)^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*x*e + a*e)^6*a*b
*c*d*g^7/(d*x + c)^6 + (b*x*e + a*e)^6*a^2*d^2*g^7/(d*x + c)^6)

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maple [B]  time = 0.05, size = 1262, normalized size = 4.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^3*(B*ln((b*x+a)/(d*x+c)*e)+A)/(b*g*x+a*g)^7,x)

[Out]

1/4*d^3*e^4*i^3/(a*d-b*c)^4/g^7*A/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*a-1/4*d^2*e^4*i^3/(a*d-b*c)^4/g^7*
A/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*b*c-2/5*d^2*e^5*i^3/(a*d-b*c)^4/g^7*A*b/(1/(d*x+c)*a*e-1/(d*x+c)*b
*c/d*e+b/d*e)^5*a+2/5*d*e^5*i^3/(a*d-b*c)^4/g^7*A*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*c+1/6*d*e^6*i^
3/(a*d-b*c)^4/g^7*A*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^6*a-1/6*e^6*i^3/(a*d-b*c)^4/g^7*A*b^3/(1/(d*x+
c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^6*c+1/4*d^3*e^4*i^3/(a*d-b*c)^4/g^7*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-1/4*d^2*e^4*i^3/(a*d-b*c)^4/g^7*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b*c+1/16*d^3*e^4*i^3/(a*d-b*c)^4/g^7*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*
e)^4*a-1/16*d^2*e^4*i^3/(a*d-b*c)^4/g^7*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*b*c-2/5*d^2*e^5*i^3/(a*d-b
*c)^4/g^7*B*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a+2/5*d*e^5*i^3/(a*d-b
*c)^4/g^7*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c-2/25*d^2*e^5*i^3/(
a*d-b*c)^4/g^7*B*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*a+2/25*d*e^5*i^3/(a*d-b*c)^4/g^7*B*b^2/(1/(d*x+c)
*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*c+1/6*d*e^6*i^3/(a*d-b*c)^4/g^7*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
6*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-1/6*e^6*i^3/(a*d-b*c)^4/g^7*B*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
6*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c+1/36*d*e^6*i^3/(a*d-b*c)^4/g^7*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*
e)^6*a-1/36*e^6*i^3/(a*d-b*c)^4/g^7*B*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^6*c

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maxima [B]  time = 5.15, size = 5524, normalized size = 19.66 \[ \text {result too large to display} \]

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)^7,x, algorithm="maxima")

[Out]

-1/3600*B*d^3*i^3*(60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^10
*g^7*x^6 + 6*a*b^9*g^7*x^5 + 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x +
a^6*b^4*g^7) + (57*a^3*b^5*c^5 - 405*a^4*b^4*c^4*d + 1470*a^5*b^3*c^3*d^2 - 730*a^6*b^2*c^2*d^3 + 245*a^7*b*c*
d^4 - 37*a^8*d^5 + 60*(20*b^8*c^3*d^2 - 15*a*b^7*c^2*d^3 + 6*a^2*b^6*c*d^4 - a^3*b^5*d^5)*x^5 - 30*(20*b^8*c^4
*d - 235*a*b^7*c^3*d^2 + 171*a^2*b^6*c^2*d^3 - 67*a^3*b^5*c*d^4 + 11*a^4*b^4*d^5)*x^4 + 20*(20*b^8*c^5 - 175*a
*b^7*c^4*d + 866*a^2*b^6*c^3*d^2 - 604*a^3*b^5*c^2*d^3 + 230*a^4*b^4*c*d^4 - 37*a^5*b^3*d^5)*x^3 + 15*(35*a*b^
7*c^5 - 271*a^2*b^6*c^4*d + 1128*a^3*b^5*c^3*d^2 - 700*a^4*b^4*c^2*d^3 + 245*a^5*b^3*c*d^4 - 37*a^6*b^2*d^5)*x
^2 + 6*(47*a^2*b^6*c^5 - 345*a^3*b^5*c^4*d + 1320*a^4*b^4*c^3*d^2 - 730*a^5*b^3*c^2*d^3 + 245*a^6*b^2*c*d^4 -
37*a^7*b*d^5)*x)/((b^15*c^5 - 5*a*b^14*c^4*d + 10*a^2*b^13*c^3*d^2 - 10*a^3*b^12*c^2*d^3 + 5*a^4*b^11*c*d^4 -
a^5*b^10*d^5)*g^7*x^6 + 6*(a*b^14*c^5 - 5*a^2*b^13*c^4*d + 10*a^3*b^12*c^3*d^2 - 10*a^4*b^11*c^2*d^3 + 5*a^5*b
^10*c*d^4 - a^6*b^9*d^5)*g^7*x^5 + 15*(a^2*b^13*c^5 - 5*a^3*b^12*c^4*d + 10*a^4*b^11*c^3*d^2 - 10*a^5*b^10*c^2
*d^3 + 5*a^6*b^9*c*d^4 - a^7*b^8*d^5)*g^7*x^4 + 20*(a^3*b^12*c^5 - 5*a^4*b^11*c^4*d + 10*a^5*b^10*c^3*d^2 - 10
*a^6*b^9*c^2*d^3 + 5*a^7*b^8*c*d^4 - a^8*b^7*d^5)*g^7*x^3 + 15*(a^4*b^11*c^5 - 5*a^5*b^10*c^4*d + 10*a^6*b^9*c
^3*d^2 - 10*a^7*b^8*c^2*d^3 + 5*a^8*b^7*c*d^4 - a^9*b^6*d^5)*g^7*x^2 + 6*(a^5*b^10*c^5 - 5*a^6*b^9*c^4*d + 10*
a^7*b^8*c^3*d^2 - 10*a^8*b^7*c^2*d^3 + 5*a^9*b^6*c*d^4 - a^10*b^5*d^5)*g^7*x + (a^6*b^9*c^5 - 5*a^7*b^8*c^4*d
+ 10*a^8*b^7*c^3*d^2 - 10*a^9*b^6*c^2*d^3 + 5*a^10*b^5*c*d^4 - a^11*b^4*d^5)*g^7) + 60*(20*b^3*c^3*d^3 - 15*a*
b^2*c^2*d^4 + 6*a^2*b*c*d^5 - a^3*d^6)*log(b*x + a)/((b^10*c^6 - 6*a*b^9*c^5*d + 15*a^2*b^8*c^4*d^2 - 20*a^3*b
^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7) - 60*(20*b^3*c^3*d^3 - 15*a*b^2*c^2*d^4
+ 6*a^2*b*c*d^5 - a^3*d^6)*log(d*x + c)/((b^10*c^6 - 6*a*b^9*c^5*d + 15*a^2*b^8*c^4*d^2 - 20*a^3*b^7*c^3*d^3 +
15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7)) - 1/1200*B*c*d^2*i^3*(60*(15*b^2*x^2 + 6*a*b*x + a^
2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^9*g^7*x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x
^3 + 15*a^4*b^5*g^7*x^2 + 6*a^5*b^4*g^7*x + a^6*b^3*g^7) + (37*a^2*b^5*c^5 - 245*a^3*b^4*c^4*d + 730*a^4*b^3*c
^3*d^2 - 1470*a^5*b^2*c^2*d^3 + 405*a^6*b*c*d^4 - 57*a^7*d^5 - 60*(15*b^7*c^2*d^3 - 6*a*b^6*c*d^4 + a^2*b^5*d^
5)*x^5 + 30*(15*b^7*c^3*d^2 - 171*a*b^6*c^2*d^3 + 67*a^2*b^5*c*d^4 - 11*a^3*b^4*d^5)*x^4 - 20*(15*b^7*c^4*d -
126*a*b^6*c^3*d^2 + 604*a^2*b^5*c^2*d^3 - 230*a^3*b^4*c*d^4 + 37*a^4*b^3*d^5)*x^3 + 15*(15*b^7*c^5 - 111*a*b^6
*c^4*d + 388*a^2*b^5*c^3*d^2 - 1000*a^3*b^4*c^2*d^3 + 365*a^4*b^3*c*d^4 - 57*a^5*b^2*d^5)*x^2 + 6*(27*a*b^6*c^
5 - 185*a^2*b^5*c^4*d + 580*a^3*b^4*c^3*d^2 - 1270*a^4*b^3*c^2*d^3 + 405*a^5*b^2*c*d^4 - 57*a^6*b*d^5)*x)/((b^
14*c^5 - 5*a*b^13*c^4*d + 10*a^2*b^12*c^3*d^2 - 10*a^3*b^11*c^2*d^3 + 5*a^4*b^10*c*d^4 - a^5*b^9*d^5)*g^7*x^6
+ 6*(a*b^13*c^5 - 5*a^2*b^12*c^4*d + 10*a^3*b^11*c^3*d^2 - 10*a^4*b^10*c^2*d^3 + 5*a^5*b^9*c*d^4 - a^6*b^8*d^5
)*g^7*x^5 + 15*(a^2*b^12*c^5 - 5*a^3*b^11*c^4*d + 10*a^4*b^10*c^3*d^2 - 10*a^5*b^9*c^2*d^3 + 5*a^6*b^8*c*d^4 -
a^7*b^7*d^5)*g^7*x^4 + 20*(a^3*b^11*c^5 - 5*a^4*b^10*c^4*d + 10*a^5*b^9*c^3*d^2 - 10*a^6*b^8*c^2*d^3 + 5*a^7*
b^7*c*d^4 - a^8*b^6*d^5)*g^7*x^3 + 15*(a^4*b^10*c^5 - 5*a^5*b^9*c^4*d + 10*a^6*b^8*c^3*d^2 - 10*a^7*b^7*c^2*d^
3 + 5*a^8*b^6*c*d^4 - a^9*b^5*d^5)*g^7*x^2 + 6*(a^5*b^9*c^5 - 5*a^6*b^8*c^4*d + 10*a^7*b^7*c^3*d^2 - 10*a^8*b^
6*c^2*d^3 + 5*a^9*b^5*c*d^4 - a^10*b^4*d^5)*g^7*x + (a^6*b^8*c^5 - 5*a^7*b^7*c^4*d + 10*a^8*b^6*c^3*d^2 - 10*a
^9*b^5*c^2*d^3 + 5*a^10*b^4*c*d^4 - a^11*b^3*d^5)*g^7) - 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*log(b*x +
a)/((b^9*c^6 - 6*a*b^8*c^5*d + 15*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*b^4*c*d^5
+ a^6*b^3*d^6)*g^7) + 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*log(d*x + c)/((b^9*c^6 - 6*a*b^8*c^5*d + 15
*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*b^4*c*d^5 + a^6*b^3*d^6)*g^7)) - 1/600*B*c^
2*d*i^3*(60*(6*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x
^4 + 20*a^3*b^5*g^7*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) + (22*a*b^5*c^5 - 140*a^2*b^4*c^
4*d + 385*a^3*b^3*c^3*d^2 - 615*a^4*b^2*c^2*d^3 + 735*a^5*b*c*d^4 - 87*a^6*d^5 + 60*(6*b^6*c*d^4 - a*b^5*d^5)*
x^5 - 30*(6*b^6*c^2*d^3 - 67*a*b^5*c*d^4 + 11*a^2*b^4*d^5)*x^4 + 20*(6*b^6*c^3*d^2 - 49*a*b^5*c^2*d^3 + 230*a^
2*b^4*c*d^4 - 37*a^3*b^3*d^5)*x^3 - 15*(6*b^6*c^4*d - 43*a*b^5*c^3*d^2 + 145*a^2*b^4*c^2*d^3 - 365*a^3*b^3*c*d
^4 + 57*a^4*b^2*d^5)*x^2 + 6*(12*b^6*c^5 - 80*a*b^5*c^4*d + 235*a^2*b^4*c^3*d^2 - 415*a^3*b^3*c^2*d^3 + 585*a^
4*b^2*c*d^4 - 87*a^5*b*d^5)*x)/((b^13*c^5 - 5*a*b^12*c^4*d + 10*a^2*b^11*c^3*d^2 - 10*a^3*b^10*c^2*d^3 + 5*a^4
*b^9*c*d^4 - a^5*b^8*d^5)*g^7*x^6 + 6*(a*b^12*c^5 - 5*a^2*b^11*c^4*d + 10*a^3*b^10*c^3*d^2 - 10*a^4*b^9*c^2*d^
3 + 5*a^5*b^8*c*d^4 - a^6*b^7*d^5)*g^7*x^5 + 15*(a^2*b^11*c^5 - 5*a^3*b^10*c^4*d + 10*a^4*b^9*c^3*d^2 - 10*a^5
*b^8*c^2*d^3 + 5*a^6*b^7*c*d^4 - a^7*b^6*d^5)*g^7*x^4 + 20*(a^3*b^10*c^5 - 5*a^4*b^9*c^4*d + 10*a^5*b^8*c^3*d^
2 - 10*a^6*b^7*c^2*d^3 + 5*a^7*b^6*c*d^4 - a^8*b^5*d^5)*g^7*x^3 + 15*(a^4*b^9*c^5 - 5*a^5*b^8*c^4*d + 10*a^6*b
^7*c^3*d^2 - 10*a^7*b^6*c^2*d^3 + 5*a^8*b^5*c*d^4 - a^9*b^4*d^5)*g^7*x^2 + 6*(a^5*b^8*c^5 - 5*a^6*b^7*c^4*d +
10*a^7*b^6*c^3*d^2 - 10*a^8*b^5*c^2*d^3 + 5*a^9*b^4*c*d^4 - a^10*b^3*d^5)*g^7*x + (a^6*b^7*c^5 - 5*a^7*b^6*c^4
*d + 10*a^8*b^5*c^3*d^2 - 10*a^9*b^4*c^2*d^3 + 5*a^10*b^3*c*d^4 - a^11*b^2*d^5)*g^7) + 60*(6*b*c*d^5 - a*d^6)*
log(b*x + a)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*
b^3*c*d^5 + a^6*b^2*d^6)*g^7) - 60*(6*b*c*d^5 - a*d^6)*log(d*x + c)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4
*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*g^7)) + 1/360*B*c^3*i^3*((60*b
^5*d^5*x^5 - 10*b^5*c^5 + 62*a*b^4*c^4*d - 163*a^2*b^3*c^3*d^2 + 237*a^3*b^2*c^2*d^3 - 213*a^4*b*c*d^4 + 147*a
^5*d^5 - 30*(b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 8*a*b^4*c*d^4 + 37*a^2*b^3*d^5)*x^3 - 15*(b^5*c
^3*d^2 - 7*a*b^4*c^2*d^3 + 23*a^2*b^3*c*d^4 - 57*a^3*b^2*d^5)*x^2 + 6*(2*b^5*c^4*d - 13*a*b^4*c^3*d^2 + 37*a^2
*b^3*c^2*d^3 - 63*a^3*b^2*c*d^4 + 87*a^4*b*d^5)*x)/((b^12*c^5 - 5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2 - 10*a^3*
b^9*c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)*g^7*x^6 + 6*(a*b^11*c^5 - 5*a^2*b^10*c^4*d + 10*a^3*b^9*c^3*d^2 -
10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^6*b^6*d^5)*g^7*x^5 + 15*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a^4*b^8
*c^3*d^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^7*b^5*d^5)*g^7*x^4 + 20*(a^3*b^9*c^5 - 5*a^4*b^8*c^4*d + 1
0*a^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + 5*a^7*b^5*c*d^4 - a^8*b^4*d^5)*g^7*x^3 + 15*(a^4*b^8*c^5 - 5*a^5*b^7*
c^4*d + 10*a^6*b^6*c^3*d^2 - 10*a^7*b^5*c^2*d^3 + 5*a^8*b^4*c*d^4 - a^9*b^3*d^5)*g^7*x^2 + 6*(a^5*b^7*c^5 - 5*
a^6*b^6*c^4*d + 10*a^7*b^5*c^3*d^2 - 10*a^8*b^4*c^2*d^3 + 5*a^9*b^3*c*d^4 - a^10*b^2*d^5)*g^7*x + (a^6*b^6*c^5
- 5*a^7*b^5*c^4*d + 10*a^8*b^4*c^3*d^2 - 10*a^9*b^3*c^2*d^3 + 5*a^10*b^2*c*d^4 - a^11*b*d^5)*g^7) - 60*log(b*
e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 + 15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a
^4*b^3*g^7*x^2 + 6*a^5*b^2*g^7*x + a^6*b*g^7) + 60*d^6*log(b*x + a)/((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4
*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*g^7) - 60*d^6*log(d*x + c)/((b^7
*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*
d^6)*g^7)) - 1/10*(6*b*x + a)*A*c^2*d*i^3/(b^8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x^4 + 20*a^3*b^5*g^7
*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) - 1/20*(15*b^2*x^2 + 6*a*b*x + a^2)*A*c*d^2*i^3/(b^
9*g^7*x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x^3 + 15*a^4*b^5*g^7*x^2 + 6*a^5*b^4*g^7*x +
a^6*b^3*g^7) - 1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*A*d^3*i^3/(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5 +
15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^6*b^4*g^7) - 1/6*A*c^3*i^3
/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 + 15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a^4*b^3*g^7*x^2 + 6*a^5*b^2*g^7
*x + a^6*b*g^7)

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mupad [B]  time = 9.73, size = 1396, normalized size = 4.97 \[ \frac {B\,d^6\,i^3\,\mathrm {atanh}\left (\frac {60\,a^3\,b^4\,d^3\,g^7-60\,a^2\,b^5\,c\,d^2\,g^7-60\,a\,b^6\,c^2\,d\,g^7+60\,b^7\,c^3\,g^7}{60\,b^4\,g^7\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{30\,b^4\,g^7\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x^2\,\left (b\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{60\,b^5\,g^7}+\frac {B\,c\,d^2\,i^3}{20\,b^4\,g^7}\right )+\frac {B\,a\,d^3\,i^3}{15\,b^4\,g^7}+\frac {B\,c\,d^2\,i^3}{5\,b^3\,g^7}\right )+\frac {B\,a\,d^3\,i^3}{6\,b^3\,g^7}+\frac {B\,c\,d^2\,i^3}{2\,b^2\,g^7}\right )+x\,\left (b\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{60\,b^5\,g^7}+\frac {B\,c\,d^2\,i^3}{20\,b^4\,g^7}\right )+\frac {B\,c^2\,d\,i^3}{10\,b^3\,g^7}\right )+a\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{60\,b^5\,g^7}+\frac {B\,c\,d^2\,i^3}{20\,b^4\,g^7}\right )+\frac {B\,a\,d^3\,i^3}{15\,b^4\,g^7}+\frac {B\,c\,d^2\,i^3}{5\,b^3\,g^7}\right )+\frac {B\,c^2\,d\,i^3}{2\,b^2\,g^7}\right )+a\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{60\,b^5\,g^7}+\frac {B\,c\,d^2\,i^3}{20\,b^4\,g^7}\right )+\frac {B\,c^2\,d\,i^3}{10\,b^3\,g^7}\right )+\frac {B\,c^3\,i^3}{6\,b^2\,g^7}+\frac {B\,d^3\,i^3\,x^3}{3\,b^2\,g^7}\right )}{6\,a^5\,x+\frac {a^6}{b}+b^5\,x^6+15\,a^4\,b\,x^2+6\,a\,b^4\,x^5+20\,a^3\,b^2\,x^3+15\,a^2\,b^3\,x^4}-\frac {\frac {60\,A\,a^5\,d^5\,i^3+600\,A\,b^5\,c^5\,i^3+37\,B\,a^5\,d^5\,i^3+100\,B\,b^5\,c^5\,i^3+60\,A\,a^2\,b^3\,c^3\,d^2\,i^3+60\,A\,a^3\,b^2\,c^2\,d^3\,i^3+37\,B\,a^2\,b^3\,c^3\,d^2\,i^3+37\,B\,a^3\,b^2\,c^2\,d^3\,i^3-840\,A\,a\,b^4\,c^4\,d\,i^3+60\,A\,a^4\,b\,c\,d^4\,i^3-188\,B\,a\,b^4\,c^4\,d\,i^3+37\,B\,a^4\,b\,c\,d^4\,i^3}{60\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (60\,A\,a^3\,b^2\,d^5\,i^3+37\,B\,a^3\,b^2\,d^5\,i^3+180\,A\,b^5\,c^3\,d^2\,i^3+19\,B\,b^5\,c^3\,d^2\,i^3-300\,A\,a\,b^4\,c^2\,d^3\,i^3+60\,A\,a^2\,b^3\,c\,d^4\,i^3-53\,B\,a\,b^4\,c^2\,d^3\,i^3+37\,B\,a^2\,b^3\,c\,d^4\,i^3\right )}{4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (60\,A\,a^4\,b\,d^5\,i^3+37\,B\,a^4\,b\,d^5\,i^3+360\,A\,b^5\,c^4\,d\,i^3+52\,B\,b^5\,c^4\,d\,i^3-540\,A\,a\,b^4\,c^3\,d^2\,i^3+60\,A\,a^3\,b^2\,c\,d^4\,i^3-113\,B\,a\,b^4\,c^3\,d^2\,i^3+37\,B\,a^3\,b^2\,c\,d^4\,i^3+60\,A\,a^2\,b^3\,c^2\,d^3\,i^3+37\,B\,a^2\,b^3\,c^2\,d^3\,i^3\right )}{10\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^3\,\left (60\,A\,a^2\,b^3\,d^5\,i^3+37\,B\,a^2\,b^3\,d^5\,i^3+60\,A\,b^5\,c^2\,d^3\,i^3+B\,b^5\,c^2\,d^3\,i^3-120\,A\,a\,b^4\,c\,d^4\,i^3-8\,B\,a\,b^4\,c\,d^4\,i^3\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^4\,\left (11\,B\,a\,b^4\,d^4\,i^3-B\,b^5\,c\,d^3\,i^3\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^5\,d^5\,i^3\,x^5}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{60\,a^6\,b^4\,g^7+360\,a^5\,b^5\,g^7\,x+900\,a^4\,b^6\,g^7\,x^2+1200\,a^3\,b^7\,g^7\,x^3+900\,a^2\,b^8\,g^7\,x^4+360\,a\,b^9\,g^7\,x^5+60\,b^{10}\,g^7\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^7,x)

[Out]

(B*d^6*i^3*atanh((60*b^7*c^3*g^7 + 60*a^3*b^4*d^3*g^7 - 60*a*b^6*c^2*d*g^7 - 60*a^2*b^5*c*d^2*g^7)/(60*b^4*g^7
*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(30*b^4*g^7*(a*d - b*c)^3) - (log(
(e*(a + b*x))/(c + d*x))*(x^2*(b*(b*((B*a*d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^7)) + (B*a*d^3*i^3)/
(15*b^4*g^7) + (B*c*d^2*i^3)/(5*b^3*g^7)) + (B*a*d^3*i^3)/(6*b^3*g^7) + (B*c*d^2*i^3)/(2*b^2*g^7)) + x*(b*(a*(
(B*a*d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^7)) + (B*c^2*d*i^3)/(10*b^3*g^7)) + a*(b*((B*a*d^3*i^3)/(
60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^7)) + (B*a*d^3*i^3)/(15*b^4*g^7) + (B*c*d^2*i^3)/(5*b^3*g^7)) + (B*c^2*d
*i^3)/(2*b^2*g^7)) + a*(a*((B*a*d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^7)) + (B*c^2*d*i^3)/(10*b^3*g^
7)) + (B*c^3*i^3)/(6*b^2*g^7) + (B*d^3*i^3*x^3)/(3*b^2*g^7)))/(6*a^5*x + a^6/b + b^5*x^6 + 15*a^4*b*x^2 + 6*a*
b^4*x^5 + 20*a^3*b^2*x^3 + 15*a^2*b^3*x^4) - ((60*A*a^5*d^5*i^3 + 600*A*b^5*c^5*i^3 + 37*B*a^5*d^5*i^3 + 100*B
*b^5*c^5*i^3 + 60*A*a^2*b^3*c^3*d^2*i^3 + 60*A*a^3*b^2*c^2*d^3*i^3 + 37*B*a^2*b^3*c^3*d^2*i^3 + 37*B*a^3*b^2*c
^2*d^3*i^3 - 840*A*a*b^4*c^4*d*i^3 + 60*A*a^4*b*c*d^4*i^3 - 188*B*a*b^4*c^4*d*i^3 + 37*B*a^4*b*c*d^4*i^3)/(60*
(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x^2*(60*A*a^3*b^2*d^5*i^3 + 37*B*a^3*b^2*d^5*i^3 + 180*A*b^5*c^3*d^2*i^3 +
19*B*b^5*c^3*d^2*i^3 - 300*A*a*b^4*c^2*d^3*i^3 + 60*A*a^2*b^3*c*d^4*i^3 - 53*B*a*b^4*c^2*d^3*i^3 + 37*B*a^2*b
^3*c*d^4*i^3))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(60*A*a^4*b*d^5*i^3 + 37*B*a^4*b*d^5*i^3 + 360*A*b^5*c
^4*d*i^3 + 52*B*b^5*c^4*d*i^3 - 540*A*a*b^4*c^3*d^2*i^3 + 60*A*a^3*b^2*c*d^4*i^3 - 113*B*a*b^4*c^3*d^2*i^3 + 3
7*B*a^3*b^2*c*d^4*i^3 + 60*A*a^2*b^3*c^2*d^3*i^3 + 37*B*a^2*b^3*c^2*d^3*i^3))/(10*(a^2*d^2 + b^2*c^2 - 2*a*b*c
*d)) + (x^3*(60*A*a^2*b^3*d^5*i^3 + 37*B*a^2*b^3*d^5*i^3 + 60*A*b^5*c^2*d^3*i^3 + B*b^5*c^2*d^3*i^3 - 120*A*a*
b^4*c*d^4*i^3 - 8*B*a*b^4*c*d^4*i^3))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*x^4*(11*B*a*b^4*d^4*i^3 - B*b^5
*c*d^3*i^3))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B*b^5*d^5*i^3*x^5)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(60*a^
6*b^4*g^7 + 60*b^10*g^7*x^6 + 360*a^5*b^5*g^7*x + 360*a*b^9*g^7*x^5 + 900*a^4*b^6*g^7*x^2 + 1200*a^3*b^7*g^7*x
^3 + 900*a^2*b^8*g^7*x^4)

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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((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**7,x)

[Out]

Timed out

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