3.125 \(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=189 \[ -\frac {b i^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^2}+\frac {d i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^2}-\frac {b B i^2 n (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^2}+\frac {B d i^2 n (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.60, antiderivative size = 340, normalized size of antiderivative = 1.80, 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, 44} \[ -\frac {d^2 i^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^5 (a+b x)^2}-\frac {2 d i^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^5 (a+b x)^3}-\frac {i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^3 g^5 (a+b x)^4}+\frac {B d^3 i^2 n}{12 b^3 g^5 (a+b x) (b c-a d)}+\frac {B d^4 i^2 n \log (a+b x)}{12 b^3 g^5 (b c-a d)^2}-\frac {B d^4 i^2 n \log (c+d x)}{12 b^3 g^5 (b c-a d)^2}-\frac {5 B d i^2 n (b c-a d)}{36 b^3 g^5 (a+b x)^3}-\frac {B i^2 n (b c-a d)^2}{16 b^3 g^5 (a+b x)^4}-\frac {B d^2 i^2 n}{24 b^3 g^5 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^5,x]

[Out]

-(B*(b*c - a*d)^2*i^2*n)/(16*b^3*g^5*(a + b*x)^4) - (5*B*d*(b*c - a*d)*i^2*n)/(36*b^3*g^5*(a + b*x)^3) - (B*d^
2*i^2*n)/(24*b^3*g^5*(a + b*x)^2) + (B*d^3*i^2*n)/(12*b^3*(b*c - a*d)*g^5*(a + b*x)) + (B*d^4*i^2*n*Log[a + b*
x])/(12*b^3*(b*c - a*d)^2*g^5) - ((b*c - a*d)^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^3*g^5*(a + b*
x)^4) - (2*d*(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*g^5*(a + b*x)^3) - (d^2*i^2*(A + B
*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^3*g^5*(a + b*x)^2) - (B*d^4*i^2*n*Log[c + d*x])/(12*b^3*(b*c - a*d)^2*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 {(125 c+125 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac {15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^5 (a+b x)^5}+\frac {31250 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^5 (a+b x)^4}+\frac {15625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac {\left (15625 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {(31250 d (b c-a d)) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {\left (15625 (b c-a d)^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac {15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac {31250 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac {15625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}+\frac {\left (15625 B d^2 n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^5}+\frac {(31250 B d (b c-a d) n) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (15625 B (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^5}\\ &=-\frac {15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac {31250 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac {15625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}+\frac {\left (15625 B d^2 (b c-a d) n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^5}+\frac {\left (31250 B d (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (15625 B (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^5}\\ &=-\frac {15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac {31250 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac {15625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}+\frac {\left (15625 B d^2 (b c-a d) n\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}{2 b^3 g^5}+\frac {\left (31250 B d (b c-a d)^2 n\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^5}+\frac {\left (15625 B (b c-a d)^3 n\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}{4 b^3 g^5}\\ &=-\frac {15625 B (b c-a d)^2 n}{16 b^3 g^5 (a+b x)^4}-\frac {78125 B d (b c-a d) n}{36 b^3 g^5 (a+b x)^3}-\frac {15625 B d^2 n}{24 b^3 g^5 (a+b x)^2}+\frac {15625 B d^3 n}{12 b^3 (b c-a d) g^5 (a+b x)}+\frac {15625 B d^4 n \log (a+b x)}{12 b^3 (b c-a d)^2 g^5}-\frac {15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac {31250 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac {15625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}-\frac {15625 B d^4 n \log (c+d x)}{12 b^3 (b c-a d)^2 g^5}\\ \end {align*}

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Mathematica [B]  time = 0.42, size = 474, normalized size = 2.51 \[ -\frac {d^2 i^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^5 (a+b x)^2}-\frac {2 d i^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^5 (a+b x)^3}-\frac {i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^3 g^5 (a+b x)^4}-\frac {B d^2 i^2 n \left (-\frac {2 d^2 \log (a+b x)}{(b c-a d)^2}+\frac {2 d^2 \log (c+d x)}{(b c-a d)^2}-\frac {2 d}{(a+b x) (b c-a d)}+\frac {1}{(a+b x)^2}\right )}{4 b^3 g^5}-\frac {B d i^2 n \left (\frac {6 d^3 \log (a+b x)}{(b c-a d)^2}-\frac {6 d^3 \log (c+d x)}{(b c-a d)^2}+\frac {6 d^2}{(a+b x) (b c-a d)}+\frac {2 (b c-a d)}{(a+b x)^3}-\frac {3 d}{(a+b x)^2}\right )}{9 b^3 g^5}-\frac {B i^2 n \left (-\frac {12 d^4 \log (a+b x)}{(b c-a d)^2}+\frac {12 d^4 \log (c+d x)}{(b c-a d)^2}-\frac {12 d^3}{(a+b x) (b c-a d)}-\frac {4 d (b c-a d)}{(a+b x)^3}+\frac {3 (b c-a d)^2}{(a+b x)^4}+\frac {6 d^2}{(a+b x)^2}\right )}{48 b^3 g^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^5,x]

[Out]

-1/4*((b*c - a*d)^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^5*(a + b*x)^4) - (2*d*(b*c - a*d)*i^2*(
A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*g^5*(a + b*x)^3) - (d^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/(2*b^3*g^5*(a + b*x)^2) - (B*d^2*i^2*n*((a + b*x)^(-2) - (2*d)/((b*c - a*d)*(a + b*x)) - (2*d^2*Log[a + b*
x])/(b*c - a*d)^2 + (2*d^2*Log[c + d*x])/(b*c - a*d)^2))/(4*b^3*g^5) - (B*d*i^2*n*((2*(b*c - a*d))/(a + b*x)^3
- (3*d)/(a + b*x)^2 + (6*d^2)/((b*c - a*d)*(a + b*x)) + (6*d^3*Log[a + b*x])/(b*c - a*d)^2 - (6*d^3*Log[c + d
*x])/(b*c - a*d)^2))/(9*b^3*g^5) - (B*i^2*n*((3*(b*c - a*d)^2)/(a + b*x)^4 - (4*d*(b*c - a*d))/(a + b*x)^3 + (
6*d^2)/(a + b*x)^2 - (12*d^3)/((b*c - a*d)*(a + b*x)) - (12*d^4*Log[a + b*x])/(b*c - a*d)^2 + (12*d^4*Log[c +
d*x])/(b*c - a*d)^2))/(48*b^3*g^5)

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fricas [B]  time = 0.87, size = 710, normalized size = 3.76 \[ \frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i^{2} n x^{3} - {\left (9 \, B b^{4} c^{4} - 16 \, B a b^{3} c^{3} d + 7 \, B a^{4} d^{4}\right )} i^{2} n - 12 \, {\left (3 \, A b^{4} c^{4} - 4 \, A a b^{3} c^{3} d + A a^{4} d^{4}\right )} i^{2} - 6 \, {\left ({\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i^{2} n + 12 \, {\left (A b^{4} c^{2} d^{2} - 2 \, A a b^{3} c d^{3} + A a^{2} b^{2} d^{4}\right )} i^{2}\right )} x^{2} - 4 \, {\left ({\left (5 \, B b^{4} c^{3} d - 12 \, B a b^{3} c^{2} d^{2} + 7 \, B a^{3} b d^{4}\right )} i^{2} n + 12 \, {\left (2 \, A b^{4} c^{3} d - 3 \, A a b^{3} c^{2} d^{2} + A a^{3} b d^{4}\right )} i^{2}\right )} x - 12 \, {\left (6 \, {\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} i^{2} x^{2} + 4 \, {\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + B a^{3} b d^{4}\right )} i^{2} x + {\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + B a^{4} d^{4}\right )} i^{2}\right )} \log \relax (e) + 12 \, {\left (B b^{4} d^{4} i^{2} n x^{4} + 4 \, B a b^{3} d^{4} i^{2} n x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3}\right )} i^{2} n x^{2} - 4 \, {\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2}\right )} i^{2} n x - {\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\right )}} \]

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))^n))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i^2*n*x^3 - (9*B*b^4*c^4 - 16*B*a*b^3*c^3*d + 7*B*a^4*d^4)*i^2*n - 12*(3
*A*b^4*c^4 - 4*A*a*b^3*c^3*d + A*a^4*d^4)*i^2 - 6*((B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^2*d^4)*i^2*n +
12*(A*b^4*c^2*d^2 - 2*A*a*b^3*c*d^3 + A*a^2*b^2*d^4)*i^2)*x^2 - 4*((5*B*b^4*c^3*d - 12*B*a*b^3*c^2*d^2 + 7*B*
a^3*b*d^4)*i^2*n + 12*(2*A*b^4*c^3*d - 3*A*a*b^3*c^2*d^2 + A*a^3*b*d^4)*i^2)*x - 12*(6*(B*b^4*c^2*d^2 - 2*B*a*
b^3*c*d^3 + B*a^2*b^2*d^4)*i^2*x^2 + 4*(2*B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2 + B*a^3*b*d^4)*i^2*x + (3*B*b^4*c^4
- 4*B*a*b^3*c^3*d + B*a^4*d^4)*i^2)*log(e) + 12*(B*b^4*d^4*i^2*n*x^4 + 4*B*a*b^3*d^4*i^2*n*x^3 - 6*(B*b^4*c^2*
d^2 - 2*B*a*b^3*c*d^3)*i^2*n*x^2 - 4*(2*B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2)*i^2*n*x - (3*B*b^4*c^4 - 4*B*a*b^3*c^
3*d)*i^2*n)*log((b*x + a)/(d*x + c)))/((b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^
7*c*d + a^3*b^6*d^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)*g^5*x^2 + 4*(a^3*b^6*c^2 - 2*a^4*
b^5*c*d + a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*g^5)

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giac [A]  time = 78.92, size = 222, normalized size = 1.17 \[ \frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b n - \frac {4 \, {\left (b x + a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {9 \, B b n - \frac {16 \, {\left (b x + a\right )} B d n}{d x + c} + 36 \, A b + 36 \, B b - \frac {48 \, {\left (b x + a\right )} A d}{d x + c} - \frac {48 \, {\left (b x + a\right )} B d}{d x + c}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

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))^n))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

1/144*(12*(3*B*b*n - 4*(b*x + a)*B*d*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x + a)^4*b*c*g^5/(d*x + c)^4 -
(b*x + a)^4*a*d*g^5/(d*x + c)^4) + (9*B*b*n - 16*(b*x + a)*B*d*n/(d*x + c) + 36*A*b + 36*B*b - 48*(b*x + a)*A*
d/(d*x + c) - 48*(b*x + a)*B*d/(d*x + c))/((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4))
*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^2*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)/(b*g*x+a*g)^5,x)

[Out]

int((d*i*x+c*i)^2*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)/(b*g*x+a*g)^5,x)

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maxima [B]  time = 2.86, size = 2247, normalized size = 11.89 \[ \text {result too large to display} \]

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))^n))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

1/48*B*c^2*i^2*n*((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*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)) - 1/144*B*d^2*i^2*n*((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)) - 1/72*B*c*d*i^2*n*((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
/6*(4*b*x + a)*B*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(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) - 1/12*(6*b^2*x^2 + 4*a*b*x + a^2)*B*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d
*x + c))^n)/(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) - 1/6*(4*b*x +
a)*A*c*d*i^2/(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) - 1/12*(6*b^
2*x^2 + 4*a*b*x + a^2)*A*d^2*i^2/(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*B*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(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) - 1/4*A*c^2*i^2/(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 = 6.07, size = 652, normalized size = 3.45 \[ -\frac {\frac {12\,A\,a^3\,d^3\,i^2-36\,A\,b^3\,c^3\,i^2+7\,B\,a^3\,d^3\,i^2\,n-9\,B\,b^3\,c^3\,i^2\,n+12\,A\,a\,b^2\,c^2\,d\,i^2+12\,A\,a^2\,b\,c\,d^2\,i^2+7\,B\,a\,b^2\,c^2\,d\,i^2\,n+7\,B\,a^2\,b\,c\,d^2\,i^2\,n}{12\,\left (a\,d-b\,c\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i^2-24\,A\,b^3\,c^2\,d\,i^2+12\,A\,a\,b^2\,c\,d^2\,i^2+7\,B\,a^2\,b\,d^3\,i^2\,n-5\,B\,b^3\,c^2\,d\,i^2\,n+7\,B\,a\,b^2\,c\,d^2\,i^2\,n\right )}{3\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (12\,A\,a\,b^2\,d^3\,i^2-12\,A\,b^3\,c\,d^2\,i^2+7\,B\,a\,b^2\,d^3\,i^2\,n-B\,b^3\,c\,d^2\,i^2\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^3\,d^3\,i^2\,n\,x^3}{a\,d-b\,c}}{12\,a^4\,b^3\,g^5+48\,a^3\,b^4\,g^5\,x+72\,a^2\,b^5\,g^5\,x^2+48\,a\,b^6\,g^5\,x^3+12\,b^7\,g^5\,x^4}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{12\,b^3}+\frac {B\,c\,d\,i^2}{6\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{12\,b^3}+\frac {B\,c\,d\,i^2}{6\,b^2}\right )+\frac {B\,a\,d^2\,i^2}{4\,b^2}+\frac {B\,c\,d\,i^2}{2\,b}\right )+\frac {B\,c^2\,i^2}{4\,b}+\frac {B\,d^2\,i^2\,x^2}{2\,b}\right )}{a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4}-\frac {B\,d^4\,i^2\,n\,\mathrm {atanh}\left (\frac {12\,b^5\,c^2\,g^5-12\,a^2\,b^3\,d^2\,g^5}{12\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{6\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2} \]

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))^n)))/(a*g + b*g*x)^5,x)

[Out]

- ((12*A*a^3*d^3*i^2 - 36*A*b^3*c^3*i^2 + 7*B*a^3*d^3*i^2*n - 9*B*b^3*c^3*i^2*n + 12*A*a*b^2*c^2*d*i^2 + 12*A*
a^2*b*c*d^2*i^2 + 7*B*a*b^2*c^2*d*i^2*n + 7*B*a^2*b*c*d^2*i^2*n)/(12*(a*d - b*c)) + (x*(12*A*a^2*b*d^3*i^2 - 2
4*A*b^3*c^2*d*i^2 + 12*A*a*b^2*c*d^2*i^2 + 7*B*a^2*b*d^3*i^2*n - 5*B*b^3*c^2*d*i^2*n + 7*B*a*b^2*c*d^2*i^2*n))
/(3*(a*d - b*c)) + (x^2*(12*A*a*b^2*d^3*i^2 - 12*A*b^3*c*d^2*i^2 + 7*B*a*b^2*d^3*i^2*n - B*b^3*c*d^2*i^2*n))/(
2*(a*d - b*c)) + (B*b^3*d^3*i^2*n*x^3)/(a*d - b*c))/(12*a^4*b^3*g^5 + 12*b^7*g^5*x^4 + 48*a^3*b^4*g^5*x + 48*a
*b^6*g^5*x^3 + 72*a^2*b^5*g^5*x^2) - (log(e*((a + b*x)/(c + d*x))^n)*(a*((B*a*d^2*i^2)/(12*b^3) + (B*c*d*i^2)/
(6*b^2)) + x*(b*((B*a*d^2*i^2)/(12*b^3) + (B*c*d*i^2)/(6*b^2)) + (B*a*d^2*i^2)/(4*b^2) + (B*c*d*i^2)/(2*b)) +
(B*c^2*i^2)/(4*b) + (B*d^2*i^2*x^2)/(2*b)))/(a^4*g^5 + b^4*g^5*x^4 + 4*a*b^3*g^5*x^3 + 6*a^2*b^2*g^5*x^2 + 4*a
^3*b*g^5*x) - (B*d^4*i^2*n*atanh((12*b^5*c^2*g^5 - 12*a^2*b^3*d^2*g^5)/(12*b^3*g^5*(a*d - b*c)^2) - (2*b*d*x)/
(a*d - b*c)))/(6*b^3*g^5*(a*d - b*c)^2)

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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)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**5,x)

[Out]

Timed out

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