3.880 \(\int \frac{x^{-1+3 n}}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^2}{b^2 n (b c-a d) \left (a+b x^n\right )}-\frac{a (2 b c-a d) \log \left (a+b x^n\right )}{b^2 n (b c-a d)^2}+\frac{c^2 \log \left (c+d x^n\right )}{d n (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.253573, antiderivative size = 95, 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 \[ -\frac{a^2}{b^2 n (b c-a d) \left (a+b x^n\right )}-\frac{a (2 b c-a d) \log \left (a+b x^n\right )}{b^2 n (b c-a d)^2}+\frac{c^2 \log \left (c+d x^n\right )}{d n (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)/((a + b*x^n)^2*(c + d*x^n)),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.6333, size = 76, normalized size = 0.8 \[ \frac{a^{2}}{b^{2} n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{a \left (a d - 2 b c\right ) \log{\left (a + b x^{n} \right )}}{b^{2} n \left (a d - b c\right )^{2}} + \frac{c^{2} \log{\left (c + d x^{n} \right )}}{d n \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)/(a+b*x**n)**2/(c+d*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.187446, size = 93, normalized size = 0.98 \[ -\frac{a^2}{b^2 n (b c-a d) \left (a+b x^n\right )}+\frac{a (a d-2 b c) \log \left (a+b x^n\right )}{b^2 n (b c-a d)^2}+\frac{c^2 \log \left (c+d x^n\right )}{d n (a d-b c)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)/((a + b*x^n)^2*(c + d*x^n)),x]

[Out]

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

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Maple [A]  time = 0.049, size = 163, normalized size = 1.7 \[{\frac{{a}^{2}}{ \left ( ad-bc \right ){b}^{2}n \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{dn \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }}+{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) d}{{b}^{2} \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) n}}-2\,{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) c}{ \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) bn}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)/(a+b*x^n)^2/(c+d*x^n),x)

[Out]

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

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Maxima [A]  time = 1.41626, size = 198, normalized size = 2.08 \[ \frac{c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{b^{2} c^{2} d n - 2 \, a b c d^{2} n + a^{2} d^{3} n} - \frac{a^{2}}{a b^{3} c n - a^{2} b^{2} d n +{\left (b^{4} c n - a b^{3} d n\right )} x^{n}} - \frac{{\left (2 \, a b c - a^{2} d\right )} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} c^{2} n - 2 \, a b^{3} c d n + a^{2} b^{2} d^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.245683, size = 224, normalized size = 2.36 \[ -\frac{a^{2} b c d - a^{3} d^{2} +{\left (2 \, a^{2} b c d - a^{3} d^{2} +{\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{n}\right )} \log \left (b x^{n} + a\right ) -{\left (b^{3} c^{2} x^{n} + a b^{2} c^{2}\right )} \log \left (d x^{n} + c\right )}{{\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} n x^{n} +{\left (a b^{4} c^{2} d - 2 \, a^{2} b^{3} c d^{2} + a^{3} b^{2} d^{3}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="fricas")

[Out]

-(a^2*b*c*d - a^3*d^2 + (2*a^2*b*c*d - a^3*d^2 + (2*a*b^2*c*d - a^2*b*d^2)*x^n)*
log(b*x^n + a) - (b^3*c^2*x^n + a*b^2*c^2)*log(d*x^n + c))/((b^5*c^2*d - 2*a*b^4
*c*d^2 + a^2*b^3*d^3)*n*x^n + (a*b^4*c^2*d - 2*a^2*b^3*c*d^2 + a^3*b^2*d^3)*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+3*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. \[ \int \frac{x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="giac")

[Out]

integrate(x^(3*n - 1)/((b*x^n + a)^2*(d*x^n + c)), x)