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

Optimal. Leaf size=120 \[ -\frac{a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}+\frac{a (2 b c-a d)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac{c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac{c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

[Out]

-a^2/(2*b^2*(b*c - a*d)*n*(a + b*x^n)^2) + (a*(2*b*c - a*d))/(b^2*(b*c - a*d)^2*
n*(a + b*x^n)) + (c^2*Log[a + b*x^n])/((b*c - a*d)^3*n) - (c^2*Log[c + d*x^n])/(
(b*c - a*d)^3*n)

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Rubi [A]  time = 0.296202, antiderivative size = 120, 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 \[ -\frac{a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}+\frac{a (2 b c-a d)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac{c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac{c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(2*b^2*(b*c - a*d)*n*(a + b*x^n)^2) + (a*(2*b*c - a*d))/(b^2*(b*c - a*d)^2*
n*(a + b*x^n)) + (c^2*Log[a + b*x^n])/((b*c - a*d)^3*n) - (c^2*Log[c + d*x^n])/(
(b*c - a*d)^3*n)

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Rubi in Sympy [A]  time = 43.96, size = 99, normalized size = 0.82 \[ \frac{a^{2}}{2 b^{2} n \left (a + b x^{n}\right )^{2} \left (a d - b c\right )} - \frac{a \left (a d - 2 b c\right )}{b^{2} n \left (a + b x^{n}\right ) \left (a d - b c\right )^{2}} - \frac{c^{2} \log{\left (a + b x^{n} \right )}}{n \left (a d - b c\right )^{3}} + \frac{c^{2} \log{\left (c + d x^{n} \right )}}{n \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.225513, size = 120, normalized size = 1. \[ -\frac{a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}-\frac{a (a d-2 b c)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac{c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac{c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(2*b^2*(b*c - a*d)*n*(a + b*x^n)^2) - (a*(-2*b*c + a*d))/(b^2*(b*c - a*d)^2
*n*(a + b*x^n)) + (c^2*Log[a + b*x^n])/((b*c - a*d)^3*n) - (c^2*Log[c + d*x^n])/
((b*c - a*d)^3*n)

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Maple [A]  time = 0.086, size = 214, normalized size = 1.8 \[{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{ \left ( -ad+2\,bc \right ) a{{\rm e}^{n\ln \left ( x \right ) }}}{nb \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }}+{\frac{{a}^{2} \left ( -ad+3\,bc \right ) }{ \left ( 2\,{a}^{2}{d}^{2}-4\,cabd+2\,{b}^{2}{c}^{2} \right ){b}^{2}n}} \right ) }+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) }}-{\frac{{c}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.40517, size = 354, normalized size = 2.95 \[ \frac{c^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac{c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac{3 \, a^{2} b c - a^{3} d + 2 \,{\left (2 \, a b^{2} c - a^{2} b d\right )} x^{n}}{2 \,{\left (a^{2} b^{4} c^{2} n - 2 \, a^{3} b^{3} c d n + a^{4} b^{2} d^{2} n +{\left (b^{6} c^{2} n - 2 \, a b^{5} c d n + a^{2} b^{4} d^{2} n\right )} x^{2 \, n} + 2 \,{\left (a b^{5} c^{2} n - 2 \, a^{2} b^{4} c d n + a^{3} b^{3} d^{2} n\right )} x^{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^2*log((b*x^n + a)/b)/(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a^3*d^3*
n) - c^2*log((d*x^n + c)/d)/(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a^3
*d^3*n) + 1/2*(3*a^2*b*c - a^3*d + 2*(2*a*b^2*c - a^2*b*d)*x^n)/(a^2*b^4*c^2*n -
2*a^3*b^3*c*d*n + a^4*b^2*d^2*n + (b^6*c^2*n - 2*a*b^5*c*d*n + a^2*b^4*d^2*n)*x
^(2*n) + 2*(a*b^5*c^2*n - 2*a^2*b^4*c*d*n + a^3*b^3*d^2*n)*x^n)

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Fricas [A]  time = 0.242588, size = 406, normalized size = 3.38 \[ \frac{3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \,{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{n} + 2 \,{\left (b^{4} c^{2} x^{2 \, n} + 2 \, a b^{3} c^{2} x^{n} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{n} + a\right ) - 2 \,{\left (b^{4} c^{2} x^{2 \, n} + 2 \, a b^{3} c^{2} x^{n} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{n} + c\right )}{2 \,{\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} n x^{2 \, n} + 2 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} n x^{n} +{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*a^2*b^2*c^2 - 4*a^3*b*c*d + a^4*d^2 + 2*(2*a*b^3*c^2 - 3*a^2*b^2*c*d + a^
3*b*d^2)*x^n + 2*(b^4*c^2*x^(2*n) + 2*a*b^3*c^2*x^n + a^2*b^2*c^2)*log(b*x^n + a
) - 2*(b^4*c^2*x^(2*n) + 2*a*b^3*c^2*x^n + a^2*b^2*c^2)*log(d*x^n + c))/((b^7*c^
3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*n*x^(2*n) + 2*(a*b^6*c^3 - 3*
a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*n*x^n + (a^2*b^5*c^3 - 3*a^3*b^4*
c^2*d + 3*a^4*b^3*c*d^2 - a^5*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)**3/(c+d*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}{\left (d x^{n} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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