3.17 \(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=89 \[ -\frac {i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {B i^2 (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)} \]

[Out]

-1/9*B*i^2*(d*x+c)^3/(-a*d+b*c)/g^4/(b*x+a)^3-1/3*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)/g^4/(b*
x+a)^3

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Rubi [B]  time = 0.49, antiderivative size = 287, normalized size of antiderivative = 3.22, number of steps used = 14, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2528, 2525, 12, 44} \[ -\frac {d^2 i^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^4 (a+b x)}-\frac {d i^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^4 (a+b x)^2}-\frac {i^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^3 g^4 (a+b x)^3}-\frac {B d^3 i^2 \log (a+b x)}{3 b^3 g^4 (b c-a d)}+\frac {B d^3 i^2 \log (c+d x)}{3 b^3 g^4 (b c-a d)}-\frac {B d i^2 (b c-a d)}{3 b^3 g^4 (a+b x)^2}-\frac {B i^2 (b c-a d)^2}{9 b^3 g^4 (a+b x)^3}-\frac {B d^2 i^2}{3 b^3 g^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^4,x]

[Out]

-(B*(b*c - a*d)^2*i^2)/(9*b^3*g^4*(a + b*x)^3) - (B*d*(b*c - a*d)*i^2)/(3*b^3*g^4*(a + b*x)^2) - (B*d^2*i^2)/(
3*b^3*g^4*(a + b*x)) - (B*d^3*i^2*Log[a + b*x])/(3*b^3*(b*c - a*d)*g^4) - ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(3*b^3*g^4*(a + b*x)^3) - (d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g
^4*(a + b*x)^2) - (d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^4*(a + b*x)) + (B*d^3*i^2*Log[c + d*x]
)/(3*b^3*(b*c - a*d)*g^4)

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 {(17 c+17 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^4}+\frac {578 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^3}+\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac {\left (289 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^4}+\frac {(578 d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^4}+\frac {\left (289 (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^2 g^4}\\ &=-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (289 B d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {(289 B d (b c-a d)) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (289 B (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (289 B d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (289 B d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (289 B (b c-a d)^3\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (289 B d^2 (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^3 g^4}+\frac {\left (289 B d (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^3 g^4}+\frac {\left (289 B (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}{3 b^3 g^4}\\ &=-\frac {289 B (b c-a d)^2}{9 b^3 g^4 (a+b x)^3}-\frac {289 B d (b c-a d)}{3 b^3 g^4 (a+b x)^2}-\frac {289 B d^2}{3 b^3 g^4 (a+b x)}-\frac {289 B d^3 \log (a+b x)}{3 b^3 (b c-a d) g^4}-\frac {289 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {289 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {289 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^4 (a+b x)}+\frac {289 B d^3 \log (c+d x)}{3 b^3 (b c-a d) g^4}\\ \end {align*}

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Mathematica [B]  time = 0.30, size = 315, normalized size = 3.54 \[ -\frac {i^2 \left (-3 a^3 A d^3-3 a^3 B d^3 \log (c+d x)-a^3 B d^3-9 a^2 A b d^3 x+3 B (b c-a d) \left (a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-9 a^2 b B d^3 x \log (c+d x)-3 a^2 b B d^3 x-9 a A b^2 d^3 x^2-9 a b^2 B d^3 x^2 \log (c+d x)-3 a b^2 B d^3 x^2+3 B d^3 (a+b x)^3 \log (a+b x)+3 A b^3 c^3+9 A b^3 c^2 d x+9 A b^3 c d^2 x^2+b^3 B c^3+3 b^3 B c^2 d x-3 b^3 B d^3 x^3 \log (c+d x)+3 b^3 B c d^2 x^2\right )}{9 b^3 g^4 (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^4,x]

[Out]

-1/9*(i^2*(3*A*b^3*c^3 + b^3*B*c^3 - 3*a^3*A*d^3 - a^3*B*d^3 + 9*A*b^3*c^2*d*x + 3*b^3*B*c^2*d*x - 9*a^2*A*b*d
^3*x - 3*a^2*b*B*d^3*x + 9*A*b^3*c*d^2*x^2 + 3*b^3*B*c*d^2*x^2 - 9*a*A*b^2*d^3*x^2 - 3*a*b^2*B*d^3*x^2 + 3*B*d
^3*(a + b*x)^3*Log[a + b*x] + 3*B*(b*c - a*d)*(a^2*d^2 + a*b*d*(c + 3*d*x) + b^2*(c^2 + 3*c*d*x + 3*d^2*x^2))*
Log[(e*(a + b*x))/(c + d*x)] - 3*a^3*B*d^3*Log[c + d*x] - 9*a^2*b*B*d^3*x*Log[c + d*x] - 9*a*b^2*B*d^3*x^2*Log
[c + d*x] - 3*b^3*B*d^3*x^3*Log[c + d*x]))/(b^3*(b*c - a*d)*g^4*(a + b*x)^3)

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fricas [B]  time = 0.74, size = 271, normalized size = 3.04 \[ -\frac {3 \, {\left ({\left (3 \, A + B\right )} b^{3} c d^{2} - {\left (3 \, A + B\right )} a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left ({\left (3 \, A + B\right )} b^{3} c^{2} d - {\left (3 \, A + B\right )} a^{2} b d^{3}\right )} i^{2} x + {\left ({\left (3 \, A + B\right )} b^{3} c^{3} - {\left (3 \, A + B\right )} a^{3} d^{3}\right )} i^{2} + 3 \, {\left (B b^{3} d^{3} i^{2} x^{3} + 3 \, B b^{3} c d^{2} i^{2} x^{2} + 3 \, B b^{3} c^{2} d i^{2} x + B b^{3} c^{3} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{9 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.30, size = 114, normalized size = 1.28 \[ \frac {{\left (3 \, B e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 3 \, A e^{4} + B e^{4}\right )} {\left (d x + c\right )}^{3} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{9 \, {\left (b x e + a e\right )}^{3} g^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

1/9*(3*B*e^4*log((b*x*e + a*e)/(d*x + c)) + 3*A*e^4 + B*e^4)*(d*x + c)^3*(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)^3*g^4)

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maple [B]  time = 0.05, size = 406, normalized size = 4.56 \[ \frac {B a d \,e^{3} i^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {B b c \,e^{3} i^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {A a d \,e^{3} i^{2}}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {A b c \,e^{3} i^{2}}{3 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {B a d \,e^{3} i^{2}}{9 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {B b c \,e^{3} i^{2}}{9 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^2*(B*ln((b*x+a)/(d*x+c)*e)+A)/(b*g*x+a*g)^4,x)

[Out]

1/3*d*e^3*i^2/(a*d-b*c)^2/g^4*A/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a-1/3*e^3*i^2/(a*d-b*c)^2/g^4*A/(1/(
d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*b*c+1/3*d*e^3*i^2/(a*d-b*c)^2/g^4*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d
*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-1/3*e^3*i^2/(a*d-b*c)^2/g^4*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b*c+1/9*d*e^3*i^2/(a*d-b*c)^2/g^4*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
3*a-1/9*e^3*i^2/(a*d-b*c)^2/g^4*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*b*c

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maxima [B]  time = 1.63, size = 1515, normalized size = 17.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/18*B*d^2*i^2*(6*(3*b^2*x^2 + 3*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^4*x^3 + 3*a*b^5*g^4
*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) + (11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d
+ a^2*b^2*d^2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*
g^4*x^3 + 3*(a*b^7*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*
g^4*x + (a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a
)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*lo
g(d*x + c)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4)) - 1/18*B*c*d*i^2*(6*(3*b*x + a)*lo
g(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) + (5*a*b^2*
c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2*b*d^2)*x)
/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2
*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*(3*b*c*d^
2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3
)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/18*B*c^2*i^2*((6*b^2*d^2*x
^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^
4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^
4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^4*x^3 + 3
*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2
- a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/3*(
3*b*x + a)*A*c*d*i^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*(3*b^2*x^2 + 3*a*b*
x + a^2)*A*d^2*i^2/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/3*A*c^2*i^2/(b^4*g^4*x^
3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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mupad [B]  time = 6.19, size = 423, normalized size = 4.75 \[ -\frac {x^2\,\left (3\,A\,b^2\,d^2\,i^2+B\,b^2\,d^2\,i^2\right )+x\,\left (3\,A\,a\,b\,d^2\,i^2+B\,a\,b\,d^2\,i^2+3\,A\,b^2\,c\,d\,i^2+B\,b^2\,c\,d\,i^2\right )+A\,a^2\,d^2\,i^2+A\,b^2\,c^2\,i^2+\frac {B\,a^2\,d^2\,i^2}{3}+\frac {B\,b^2\,c^2\,i^2}{3}+A\,a\,b\,c\,d\,i^2+\frac {B\,a\,b\,c\,d\,i^2}{3}}{3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^4\,g^4}+\frac {B\,c\,d\,i^2}{3\,b^3\,g^4}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^4\,g^4}+\frac {B\,c\,d\,i^2}{3\,b^3\,g^4}\right )+\frac {2\,B\,a\,d^2\,i^2}{3\,b^3\,g^4}+\frac {2\,B\,c\,d\,i^2}{3\,b^2\,g^4}\right )+\frac {B\,c^2\,i^2}{3\,b^2\,g^4}+\frac {B\,d^2\,i^2\,x^2}{b^2\,g^4}\right )}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B\,d^3\,i^2\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^4,x)

[Out]

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

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sympy [B]  time = 26.25, size = 614, normalized size = 6.90 \[ - \frac {B d^{3} i^{2} \log {\left (x + \frac {- \frac {B a^{2} d^{5} i^{2}}{a d - b c} + \frac {2 B a b c d^{4} i^{2}}{a d - b c} + B a d^{4} i^{2} - \frac {B b^{2} c^{2} d^{3} i^{2}}{a d - b c} + B b c d^{3} i^{2}}{2 B b d^{4} i^{2}} \right )}}{3 b^{3} g^{4} \left (a d - b c\right )} + \frac {B d^{3} i^{2} \log {\left (x + \frac {\frac {B a^{2} d^{5} i^{2}}{a d - b c} - \frac {2 B a b c d^{4} i^{2}}{a d - b c} + B a d^{4} i^{2} + \frac {B b^{2} c^{2} d^{3} i^{2}}{a d - b c} + B b c d^{3} i^{2}}{2 B b d^{4} i^{2}} \right )}}{3 b^{3} g^{4} \left (a d - b c\right )} + \frac {- 3 A a^{2} d^{2} i^{2} - 3 A a b c d i^{2} - 3 A b^{2} c^{2} i^{2} - B a^{2} d^{2} i^{2} - B a b c d i^{2} - B b^{2} c^{2} i^{2} + x^{2} \left (- 9 A b^{2} d^{2} i^{2} - 3 B b^{2} d^{2} i^{2}\right ) + x \left (- 9 A a b d^{2} i^{2} - 9 A b^{2} c d i^{2} - 3 B a b d^{2} i^{2} - 3 B b^{2} c d i^{2}\right )}{9 a^{3} b^{3} g^{4} + 27 a^{2} b^{4} g^{4} x + 27 a b^{5} g^{4} x^{2} + 9 b^{6} g^{4} x^{3}} + \frac {\left (- B a^{2} d^{2} i^{2} - B a b c d i^{2} - 3 B a b d^{2} i^{2} x - B b^{2} c^{2} i^{2} - 3 B b^{2} c d i^{2} x - 3 B b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{3 a^{3} b^{3} g^{4} + 9 a^{2} b^{4} g^{4} x + 9 a b^{5} g^{4} x^{2} + 3 b^{6} g^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4,x)

[Out]

-B*d**3*i**2*log(x + (-B*a**2*d**5*i**2/(a*d - b*c) + 2*B*a*b*c*d**4*i**2/(a*d - b*c) + B*a*d**4*i**2 - B*b**2
*c**2*d**3*i**2/(a*d - b*c) + B*b*c*d**3*i**2)/(2*B*b*d**4*i**2))/(3*b**3*g**4*(a*d - b*c)) + B*d**3*i**2*log(
x + (B*a**2*d**5*i**2/(a*d - b*c) - 2*B*a*b*c*d**4*i**2/(a*d - b*c) + B*a*d**4*i**2 + B*b**2*c**2*d**3*i**2/(a
*d - b*c) + B*b*c*d**3*i**2)/(2*B*b*d**4*i**2))/(3*b**3*g**4*(a*d - b*c)) + (-3*A*a**2*d**2*i**2 - 3*A*a*b*c*d
*i**2 - 3*A*b**2*c**2*i**2 - B*a**2*d**2*i**2 - B*a*b*c*d*i**2 - B*b**2*c**2*i**2 + x**2*(-9*A*b**2*d**2*i**2
- 3*B*b**2*d**2*i**2) + x*(-9*A*a*b*d**2*i**2 - 9*A*b**2*c*d*i**2 - 3*B*a*b*d**2*i**2 - 3*B*b**2*c*d*i**2))/(9
*a**3*b**3*g**4 + 27*a**2*b**4*g**4*x + 27*a*b**5*g**4*x**2 + 9*b**6*g**4*x**3) + (-B*a**2*d**2*i**2 - B*a*b*c
*d*i**2 - 3*B*a*b*d**2*i**2*x - B*b**2*c**2*i**2 - 3*B*b**2*c*d*i**2*x - 3*B*b**2*d**2*i**2*x**2)*log(e*(a + b
*x)/(c + d*x))/(3*a**3*b**3*g**4 + 9*a**2*b**4*g**4*x + 9*a*b**5*g**4*x**2 + 3*b**6*g**4*x**3)

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