3.233 \(\int \frac {1}{(a+b x) (c+d x) (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac {1}{2 B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]

[Out]

-1/2/B/(-a*d+b*c)/n/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2

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Rubi [A]  time = 0.12, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6686} \[ -\frac {1}{2 B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3),x]

[Out]

-1/(2*B*(b*c - a*d)*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx &=-\frac {1}{2 B (b c-a d) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.96 \[ -\frac {1}{2 (b B c n-a B d n) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3),x]

[Out]

-1/2*1/((b*B*c*n - a*B*d*n)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)

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fricas [B]  time = 0.74, size = 238, normalized size = 5.29 \[ -\frac {1}{2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right )^{2} + {\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (d x + c\right )^{2} + {\left (B^{3} b c - B^{3} a d\right )} n \log \relax (e)^{2} + 2 \, {\left (A B^{2} b c - A B^{2} a d\right )} n \log \relax (e) + {\left (A^{2} B b c - A^{2} B a d\right )} n + 2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \relax (e) + {\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right ) + {\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \relax (e) + {\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="fricas")

[Out]

-1/2/((B^3*b*c - B^3*a*d)*n^3*log(b*x + a)^2 + (B^3*b*c - B^3*a*d)*n^3*log(d*x + c)^2 + (B^3*b*c - B^3*a*d)*n*
log(e)^2 + 2*(A*B^2*b*c - A*B^2*a*d)*n*log(e) + (A^2*B*b*c - A^2*B*a*d)*n + 2*((B^3*b*c - B^3*a*d)*n^2*log(e)
+ (A*B^2*b*c - A*B^2*a*d)*n^2)*log(b*x + a) - 2*((B^3*b*c - B^3*a*d)*n^3*log(b*x + a) + (B^3*b*c - B^3*a*d)*n^
2*log(e) + (A*B^2*b*c - A*B^2*a*d)*n^2)*log(d*x + c))

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giac [B]  time = 0.24, size = 301, normalized size = 6.69 \[ -\frac {1}{2 \, {\left (B^{3} b c n^{3} \log \left (b x + a\right )^{2} - B^{3} a d n^{3} \log \left (b x + a\right )^{2} - 2 \, B^{3} b c n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + 2 \, B^{3} a d n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + B^{3} b c n^{3} \log \left (d x + c\right )^{2} - B^{3} a d n^{3} \log \left (d x + c\right )^{2} + 2 \, A B^{2} b c n^{2} \log \left (b x + a\right ) + 2 \, B^{3} b c n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} a d n^{2} \log \left (b x + a\right ) - 2 \, B^{3} a d n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} b c n^{2} \log \left (d x + c\right ) - 2 \, B^{3} b c n^{2} \log \left (d x + c\right ) + 2 \, A B^{2} a d n^{2} \log \left (d x + c\right ) + 2 \, B^{3} a d n^{2} \log \left (d x + c\right ) + A^{2} B b c n + 2 \, A B^{2} b c n + B^{3} b c n - A^{2} B a d n - 2 \, A B^{2} a d n - B^{3} a d n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="giac")

[Out]

-1/2/(B^3*b*c*n^3*log(b*x + a)^2 - B^3*a*d*n^3*log(b*x + a)^2 - 2*B^3*b*c*n^3*log(b*x + a)*log(d*x + c) + 2*B^
3*a*d*n^3*log(b*x + a)*log(d*x + c) + B^3*b*c*n^3*log(d*x + c)^2 - B^3*a*d*n^3*log(d*x + c)^2 + 2*A*B^2*b*c*n^
2*log(b*x + a) + 2*B^3*b*c*n^2*log(b*x + a) - 2*A*B^2*a*d*n^2*log(b*x + a) - 2*B^3*a*d*n^2*log(b*x + a) - 2*A*
B^2*b*c*n^2*log(d*x + c) - 2*B^3*b*c*n^2*log(d*x + c) + 2*A*B^2*a*d*n^2*log(d*x + c) + 2*B^3*a*d*n^2*log(d*x +
c) + A^2*B*b*c*n + 2*A*B^2*b*c*n + B^3*b*c*n - A^2*B*a*d*n - 2*A*B^2*a*d*n - B^3*a*d*n)

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maple [C]  time = 0.18, size = 366, normalized size = 8.13 \[ \frac {2}{\left (a d -b c \right ) \left (-i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i \pi B \,\mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+2 B \ln \relax (e )+2 B \ln \left (\left (b x +a \right )^{n}\right )-2 B \ln \left (\left (d x +c \right )^{n}\right )+2 A \right )^{2} B n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(d*x+c)/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)

[Out]

2/B/n/(a*d-b*c)/(2*A+2*B*ln(e)+2*B*ln((b*x+a)^n)-2*B*ln((d*x+c)^n)-I*B*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n)
)*csgn(I*(b*x+a)^n/((d*x+c)^n))+I*B*Pi*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*csgn(I/((d*x+c
)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-I*B*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-I*B*Pi*csgn(I*e)*csgn(I*(b*x+a)^n
/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+I*B*Pi*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I*B*Pi*csgn(I
*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3)^2

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maxima [B]  time = 3.19, size = 220, normalized size = 4.89 \[ -\frac {1}{2 \, {\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (b c n - a d n\right )} B^{3} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + {\left (b c n - a d n\right )} A^{2} B + 2 \, {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} A B^{2} + {\left (b c n \log \relax (e)^{2} - a d n \log \relax (e)^{2}\right )} B^{3} + 2 \, {\left ({\left (b c n - a d n\right )} A B^{2} + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B^{2} + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="maxima")

[Out]

-1/2/((b*c*n - a*d*n)*B^3*log((b*x + a)^n)^2 + (b*c*n - a*d*n)*B^3*log((d*x + c)^n)^2 + (b*c*n - a*d*n)*A^2*B
+ 2*(b*c*n*log(e) - a*d*n*log(e))*A*B^2 + (b*c*n*log(e)^2 - a*d*n*log(e)^2)*B^3 + 2*((b*c*n - a*d*n)*A*B^2 + (
b*c*n*log(e) - a*d*n*log(e))*B^3)*log((b*x + a)^n) - 2*((b*c*n - a*d*n)*B^3*log((b*x + a)^n) + (b*c*n - a*d*n)
*A*B^2 + (b*c*n*log(e) - a*d*n*log(e))*B^3)*log((d*x + c)^n))

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mupad [B]  time = 4.54, size = 72, normalized size = 1.60 \[ \frac {1}{2\,B\,n\,\left (a\,d-b\,c\right )\,\left (A^2+2\,A\,B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )+B^2\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3*(a + b*x)*(c + d*x)),x)

[Out]

1/(2*B*n*(a*d - b*c)*(B^2*log((e*(a + b*x)^n)/(c + d*x)^n)^2 + A^2 + 2*A*B*log((e*(a + b*x)^n)/(c + d*x)^n)))

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

[Out]

Timed out

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