3.3041 \(\int x \left (a+b \left (c x^q\right )^n\right )^p \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{2} x^2 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{2}{n q};1+\frac{2}{n q};-\frac{b \left (c x^q\right )^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0671424, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{2} x^2 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{2}{n q};1+\frac{2}{n q};-\frac{b \left (c x^q\right )^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*(c*x^q)^n)^p,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*(c*x**q)**n)**p,x)

[Out]

Integral(x*(a + b*(c*x**q)**n)**p, x)

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Mathematica [A]  time = 0.0895389, size = 73, normalized size = 1. \[ \frac{1}{2} x^2 \left (a+b \left (c x^q\right )^n\right )^p \left (\frac{b \left (c x^q\right )^n}{a}+1\right )^{-p} \, _2F_1\left (-p,\frac{2}{n q};1+\frac{2}{n q};-\frac{b \left (c x^q\right )^n}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.338, size = 0, normalized size = 0. \[ \int x \left ( a+b \left ( c{x}^{q} \right ) ^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*(c*x^q)^n)^p,x)

[Out]

int(x*(a+b*(c*x^q)^n)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^q)^n*b + a)^p*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (\left (c x^{q}\right )^{n} b + a\right )}^{p} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x^q)^n*b + a)^p*x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (a + b \left (c x^{q}\right )^{n}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*(c*x**q)**n)**p,x)

[Out]

Integral(x*(a + b*(c*x**q)**n)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x^q)^n*b + a)^p*x, x)