3.2793 \(\int (c x)^{-1-3 n-n p} \left (a+b x^n\right )^p \, dx\)

Optimal. Leaf size=127 \[ -\frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+3}}{a^3 c n (p+1) (p+2) (p+3)}+\frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac{(c x)^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

[Out]

-((a + b*x^n)^(1 + p)/(a*c*n*(1 + p)*(c*x)^(n*(3 + p)))) + (2*(a + b*x^n)^(2 + p
))/(a^2*c*n*(1 + p)*(2 + p)*(c*x)^(n*(3 + p))) - (2*(a + b*x^n)^(3 + p))/(a^3*c*
n*(1 + p)*(2 + p)*(3 + p)*(c*x)^(n*(3 + p)))

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Rubi [A]  time = 0.195138, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+3}}{a^3 c n (p+1) (p+2) (p+3)}+\frac{2 (c x)^{-n (p+3)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac{(c x)^{-n (p+3)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x^n)^(1 + p)/(a*c*n*(1 + p)*(c*x)^(n*(3 + p)))) + (2*(a + b*x^n)^(2 + p
))/(a^2*c*n*(1 + p)*(2 + p)*(c*x)^(n*(3 + p))) - (2*(a + b*x^n)^(3 + p))/(a^3*c*
n*(1 + p)*(2 + p)*(3 + p)*(c*x)^(n*(3 + p)))

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Rubi in Sympy [A]  time = 9.42377, size = 53, normalized size = 0.42 \[ - \frac{\left (c x\right )^{- n \left (p + 3\right )} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p - 3 \\ - p - 2 \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c n \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c*x)**(-n*(p + 3))*(1 + b*x**n/a)**(-p)*(a + b*x**n)**p*hyper((-p, -p - 3), (-
p - 2,), -b*x**n/a)/(c*n*(p + 3))

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Mathematica [C]  time = 0.0886295, size = 69, normalized size = 0.54 \[ -\frac{x (c x)^{-n (p+3)-1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p-3,-p;-p-2;-\frac{b x^n}{a}\right )}{n (p+3)} \]

Antiderivative was successfully verified.

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

[Out]

-((x*(c*x)^(-1 - n*(3 + p))*(a + b*x^n)^p*Hypergeometric2F1[-3 - p, -p, -2 - p,
-((b*x^n)/a)])/(n*(3 + p)*(1 + (b*x^n)/a)^p))

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{-np-3\,n-1} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x)^(-n*p-3*n-1)*(a+b*x^n)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237157, size = 290, normalized size = 2.28 \[ \frac{{\left (2 \, a b^{2} p x x^{2 \, n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} - 2 \, b^{3} x x^{3 \, n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} -{\left (a^{2} b p^{2} + a^{2} b p\right )} x x^{n} e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )} -{\left (a^{3} p^{2} + 3 \, a^{3} p + 2 \, a^{3}\right )} x e^{\left (-{\left (n p + 3 \, n + 1\right )} \log \left (c\right ) -{\left (n p + 3 \, n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}}{a^{3} n p^{3} + 6 \, a^{3} n p^{2} + 11 \, a^{3} n p + 6 \, a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a*b^2*p*x*x^(2*n)*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)) - 2*b^
3*x*x^(3*n)*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)) - (a^2*b*p^2 +
a^2*b*p)*x*x^n*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)) - (a^3*p^2 +
3*a^3*p + 2*a^3)*x*e^(-(n*p + 3*n + 1)*log(c) - (n*p + 3*n + 1)*log(x)))*(b*x^n
+ a)^p/(a^3*n*p^3 + 6*a^3*n*p^2 + 11*a^3*n*p + 6*a^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((c*x)**(-n*p-3*n-1)*(a+b*x**n)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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