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

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

[Out]

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

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Rubi [B]  time = 0.72, antiderivative size = 373, normalized size of antiderivative = 4.19, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2528, 2525, 12, 44} \[ -\frac {d^3 i^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^5 (a+b x)}-\frac {3 d^2 i^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 g^5 (a+b x)^2}-\frac {d i^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^5 (a+b x)^3}-\frac {i^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b^4 g^5 (a+b x)^4}-\frac {3 B d^2 i^3 (b c-a d)}{8 b^4 g^5 (a+b x)^2}-\frac {B d^4 i^3 \log (a+b x)}{4 b^4 g^5 (b c-a d)}+\frac {B d^4 i^3 \log (c+d x)}{4 b^4 g^5 (b c-a d)}-\frac {B d i^3 (b c-a d)^2}{4 b^4 g^5 (a+b x)^3}-\frac {B i^3 (b c-a d)^3}{16 b^4 g^5 (a+b x)^4}-\frac {B d^3 i^3}{4 b^4 g^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B]  time = 0.49, size = 427, normalized size = 4.80 \[ -\frac {i^3 \left (-4 a^4 A d^4-4 a^4 B d^4 \log (c+d x)-a^4 B d^4-16 a^3 A b d^4 x-16 a^3 b B d^4 x \log (c+d x)-4 a^3 b B d^4 x-24 a^2 A b^2 d^4 x^2-24 a^2 b^2 B d^4 x^2 \log (c+d x)-6 a^2 b^2 B d^4 x^2+4 B \left (-a^4 d^4-4 a^3 b d^4 x-6 a^2 b^2 d^4 x^2-4 a b^3 d^4 x^3+b^4 c \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-16 a A b^3 d^4 x^3-16 a b^3 B d^4 x^3 \log (c+d x)-4 a b^3 B d^4 x^3+4 B d^4 (a+b x)^4 \log (a+b x)+4 A b^4 c^4+16 A b^4 c^3 d x+24 A b^4 c^2 d^2 x^2+16 A b^4 c d^3 x^3+b^4 B c^4+4 b^4 B c^3 d x+6 b^4 B c^2 d^2 x^2-4 b^4 B d^4 x^4 \log (c+d x)+4 b^4 B c d^3 x^3\right )}{16 b^4 g^5 (a+b x)^4 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(i^3*(4*A*b^4*c^4 + b^4*B*c^4 - 4*a^4*A*d^4 - a^4*B*d^4 + 16*A*b^4*c^3*d*x + 4*b^4*B*c^3*d*x - 16*a^3*A*
b*d^4*x - 4*a^3*b*B*d^4*x + 24*A*b^4*c^2*d^2*x^2 + 6*b^4*B*c^2*d^2*x^2 - 24*a^2*A*b^2*d^4*x^2 - 6*a^2*b^2*B*d^
4*x^2 + 16*A*b^4*c*d^3*x^3 + 4*b^4*B*c*d^3*x^3 - 16*a*A*b^3*d^4*x^3 - 4*a*b^3*B*d^4*x^3 + 4*B*d^4*(a + b*x)^4*
Log[a + b*x] + 4*B*(-(a^4*d^4) - 4*a^3*b*d^4*x - 6*a^2*b^2*d^4*x^2 - 4*a*b^3*d^4*x^3 + b^4*c*(c^3 + 4*c^2*d*x
+ 6*c*d^2*x^2 + 4*d^3*x^3))*Log[(e*(a + b*x))/(c + d*x)] - 4*a^4*B*d^4*Log[c + d*x] - 16*a^3*b*B*d^4*x*Log[c +
d*x] - 24*a^2*b^2*B*d^4*x^2*Log[c + d*x] - 16*a*b^3*B*d^4*x^3*Log[c + d*x] - 4*b^4*B*d^4*x^4*Log[c + d*x]))/(
b^4*(b*c - a*d)*g^5*(a + b*x)^4)

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fricas [B]  time = 0.86, size = 355, normalized size = 3.99 \[ -\frac {4 \, {\left ({\left (4 \, A + B\right )} b^{4} c d^{3} - {\left (4 \, A + B\right )} a b^{3} d^{4}\right )} i^{3} x^{3} + 6 \, {\left ({\left (4 \, A + B\right )} b^{4} c^{2} d^{2} - {\left (4 \, A + B\right )} a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 4 \, {\left ({\left (4 \, A + B\right )} b^{4} c^{3} d - {\left (4 \, A + B\right )} a^{3} b d^{4}\right )} i^{3} x + {\left ({\left (4 \, A + B\right )} b^{4} c^{4} - {\left (4 \, A + B\right )} a^{4} d^{4}\right )} i^{3} + 4 \, {\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x + B b^{4} c^{4} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{16 \, {\left ({\left (b^{9} c - a b^{8} d\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c - a^{2} b^{7} d\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c - a^{3} b^{6} d\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c - a^{4} b^{5} d\right )} g^{5} x + {\left (a^{4} b^{5} c - a^{5} b^{4} d\right )} g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 3.27, size = 117, normalized size = 1.31 \[ \frac {{\left (4 \, B i e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) + 4 \, A i e^{5} + B i e^{5}\right )} {\left (d x + c\right )}^{4} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{16 \, {\left (b x e + a e\right )}^{4} g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 406, normalized size = 4.56 \[ \frac {B a d \,e^{4} i^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{4} g^{5}}-\frac {B b c \,e^{4} i^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{4} g^{5}}+\frac {A a d \,e^{4} i^{3}}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{4} g^{5}}-\frac {A b c \,e^{4} i^{3}}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{4} g^{5}}+\frac {B a d \,e^{4} i^{3}}{16 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{4} g^{5}}-\frac {B b c \,e^{4} i^{3}}{16 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{4} g^{5}} \]

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)^5,x)

[Out]

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

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maxima [B]  time = 2.71, size = 3107, normalized size = 34.91 \[ \text {result too large to display} \]

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)^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.16, size = 780, normalized size = 8.76 \[ -\frac {x^3\,\left (4\,A\,b^3\,d^3\,i^3+B\,b^3\,d^3\,i^3\right )+x^2\,\left (6\,A\,a\,b^2\,d^3\,i^3+\frac {3\,B\,a\,b^2\,d^3\,i^3}{2}+6\,A\,b^3\,c\,d^2\,i^3+\frac {3\,B\,b^3\,c\,d^2\,i^3}{2}\right )+x\,\left (4\,A\,a^2\,b\,d^3\,i^3+B\,a^2\,b\,d^3\,i^3+4\,A\,b^3\,c^2\,d\,i^3+B\,b^3\,c^2\,d\,i^3+4\,A\,a\,b^2\,c\,d^2\,i^3+B\,a\,b^2\,c\,d^2\,i^3\right )+A\,a^3\,d^3\,i^3+A\,b^3\,c^3\,i^3+\frac {B\,a^3\,d^3\,i^3}{4}+\frac {B\,b^3\,c^3\,i^3}{4}+A\,a\,b^2\,c^2\,d\,i^3+A\,a^2\,b\,c\,d^2\,i^3+\frac {B\,a\,b^2\,c^2\,d\,i^3}{4}+\frac {B\,a^2\,b\,c\,d^2\,i^3}{4}}{4\,a^4\,b^4\,g^5+16\,a^3\,b^5\,g^5\,x+24\,a^2\,b^6\,g^5\,x^2+16\,a\,b^7\,g^5\,x^3+4\,b^8\,g^5\,x^4}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x^2\,\left (b\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,a\,d^3\,i^3}{2\,b^4\,g^5}+\frac {B\,c\,d^2\,i^3}{2\,b^3\,g^5}\right )+\frac {3\,B\,a\,d^3\,i^3}{4\,b^3\,g^5}+\frac {3\,B\,c\,d^2\,i^3}{4\,b^2\,g^5}\right )+x\,\left (b\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,c^2\,d\,i^3}{4\,b^3\,g^5}\right )+a\,\left (b\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,a\,d^3\,i^3}{2\,b^4\,g^5}+\frac {B\,c\,d^2\,i^3}{2\,b^3\,g^5}\right )+\frac {3\,B\,c^2\,d\,i^3}{4\,b^2\,g^5}\right )+a\,\left (a\,\left (\frac {B\,a\,d^3\,i^3}{4\,b^5\,g^5}+\frac {B\,c\,d^2\,i^3}{4\,b^4\,g^5}\right )+\frac {B\,c^2\,d\,i^3}{4\,b^3\,g^5}\right )+\frac {B\,c^3\,i^3}{4\,b^2\,g^5}+\frac {B\,d^3\,i^3\,x^3}{b^2\,g^5}\right )}{4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3}-\frac {B\,d^4\,i^3\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^4\,g^5\,\left (a\,d-b\,c\right )} \]

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)^5,x)

[Out]

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

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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)**5,x)

[Out]

Timed out

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