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

Optimal. Leaf size=28 \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

[Out]

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

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Rubi [A]  time = 0.150258, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021 \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.3048, size = 20, normalized size = 0.71 \[ x \left (a + b x^{n}\right )^{- \frac{1}{n}} \left (c + d x^{n}\right )^{- \frac{1}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**n)**(-1/n)*(c + d*x**n)**(-1/n)

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Mathematica [A]  time = 0.202057, size = 28, normalized size = 1. \[ x \left (a+b x^n\right )^{-1/n} \left (c+d x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.393, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{{\frac{-1-n}{n}}} \left ( c+d{x}^{n} \right ) ^{{\frac{-1-n}{n}}} \left ( ac-bd{x}^{2\,n} \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 2.75057, size = 41, normalized size = 1.46 \[ x e^{\left (-\frac{\log \left (b x^{n} + a\right )}{n} - \frac{\log \left (d x^{n} + c\right )}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241394, size = 82, normalized size = 2.93 \[ \frac{b d x x^{2 \, n} + a c x +{\left (b c + a d\right )} x x^{n}}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}}{\left (d x^{n} + c\right )}^{\frac{n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.256566, size = 354, normalized size = 12.64 \[ b d x e^{\left (2 \, n{\rm ln}\left (x\right ) - \frac{n{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) +{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right )}{n} - \frac{n{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) +{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right )}{n}\right )} + b c x e^{\left (n{\rm ln}\left (x\right ) - \frac{n{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) +{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right )}{n} - \frac{n{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) +{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right )}{n}\right )} + a d x e^{\left (n{\rm ln}\left (x\right ) - \frac{n{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) +{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right )}{n} - \frac{n{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) +{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right )}{n}\right )} + a c x e^{\left (-\frac{n{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) +{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right )}{n} - \frac{n{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) +{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right )}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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