3.232 \(\int \frac {1}{(a+b x) (c+d x) (A+B \log (e (a+b x)^n (c+d x)^{-n}))^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac {1}{B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 43, 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} \[ -\frac {1}{B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )} \]

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])^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.95 \[ -\frac {1}{(b B c n-a B d n) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )} \]

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])^2),x]

[Out]

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

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fricas [A]  time = 0.61, size = 86, normalized size = 2.00 \[ -\frac {1}{{\left (B^{2} b c - B^{2} a d\right )} n^{2} \log \left (b x + a\right ) - {\left (B^{2} b c - B^{2} a d\right )} n^{2} \log \left (d x + c\right ) + {\left (B^{2} b c - B^{2} a d\right )} n \log \relax (e) + {\left (A B b c - A B a d\right )} n} \]

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)))^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.18, size = 95, normalized size = 2.21 \[ -\frac {1}{B^{2} b c n^{2} \log \left (b x + a\right ) - B^{2} a d n^{2} \log \left (b x + a\right ) - B^{2} b c n^{2} \log \left (d x + c\right ) + B^{2} a d n^{2} \log \left (d x + c\right ) + A B b c n + B^{2} b c n - A B a d n - B^{2} a d n} \]

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)))^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.18, size = 366, normalized size = 8.51 \[ \frac {2}{\left (a d -b c \right ) \left (-i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i \pi B \,\mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i \pi B \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+2 B \ln \relax (e )+2 B \ln \left (\left (b x +a \right )^{n}\right )-2 B \ln \left (\left (d x +c \right )^{n}\right )+2 A \right ) B n} \]

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)))^2,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)

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maxima [A]  time = 2.47, size = 81, normalized size = 1.88 \[ -\frac {1}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B + {\left (b c n \log \relax (e) - a d n \log \relax (e)\right )} B^{2}} \]

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)))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.49, size = 42, normalized size = 0.98 \[ \frac {1}{B\,n\,\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )\,\left (a\,d-b\,c\right )} \]

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))^2*(a + b*x)*(c + d*x)),x)

[Out]

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

[Out]

Timed out

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