3.874 \(\int \frac{x^{-1+2 n}}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx\)

Optimal. Leaf size=75 \[ \frac{a}{b n (b c-a d) \left (a+b x^n\right )}+\frac{c \log \left (a+b x^n\right )}{n (b c-a d)^2}-\frac{c \log \left (c+d x^n\right )}{n (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.18436, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a}{b n (b c-a d) \left (a+b x^n\right )}+\frac{c \log \left (a+b x^n\right )}{n (b c-a d)^2}-\frac{c \log \left (c+d x^n\right )}{n (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)/((a + b*x^n)^2*(c + d*x^n)),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.6123, size = 58, normalized size = 0.77 \[ - \frac{a}{b n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{c \log{\left (a + b x^{n} \right )}}{n \left (a d - b c\right )^{2}} - \frac{c \log{\left (c + d x^{n} \right )}}{n \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)/(a+b*x**n)**2/(c+d*x**n),x)

[Out]

-a/(b*n*(a + b*x**n)*(a*d - b*c)) + c*log(a + b*x**n)/(n*(a*d - b*c)**2) - c*log
(c + d*x**n)/(n*(a*d - b*c)**2)

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Mathematica [A]  time = 0.121836, size = 75, normalized size = 1. \[ \frac{a}{b n (b c-a d) \left (a+b x^n\right )}+\frac{c \log \left (a+b x^n\right )}{n (b c-a d)^2}-\frac{c \log \left (c+d x^n\right )}{n (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)/((a + b*x^n)^2*(c + d*x^n)),x]

[Out]

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

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Maple [A]  time = 0.048, size = 109, normalized size = 1.5 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{ \left ( ad-bc \right ) n \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{c\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(a+b*x^n)^2/(c+d*x^n),x)

[Out]

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

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Maxima [A]  time = 1.47418, size = 163, normalized size = 2.17 \[ \frac{c \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} c^{2} n - 2 \, a b c d n + a^{2} d^{2} n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{b^{2} c^{2} n - 2 \, a b c d n + a^{2} d^{2} n} + \frac{a}{a b^{2} c n - a^{2} b d n +{\left (b^{3} c n - a b^{2} d n\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.234974, size = 162, normalized size = 2.16 \[ \frac{a b c - a^{2} d +{\left (b^{2} c x^{n} + a b c\right )} \log \left (b x^{n} + a\right ) -{\left (b^{2} c x^{n} + a b c\right )} \log \left (d x^{n} + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} n x^{n} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="fricas")

[Out]

(a*b*c - a^2*d + (b^2*c*x^n + a*b*c)*log(b*x^n + a) - (b^2*c*x^n + a*b*c)*log(d*
x^n + c))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*n*x^n + (a*b^3*c^2 - 2*a^2*b^2*
c*d + a^3*b*d^2)*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)/(a+b*x**n)**2/(c+d*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)), x)