3.2782 \(\int (c x)^m \left (a+b x^n\right )^p \, dx\)

Optimal. Leaf size=59 \[ \frac{(c x)^{m+1} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,\frac{m+1}{n}+p+1;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0670979, antiderivative size = 67, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{(c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(a + b*x^n)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.06727, size = 49, normalized size = 0.83 \[ \frac{\left (c x\right )^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0587902, size = 64, normalized size = 1.08 \[ \frac{x (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(a+b*x^n)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^p*(c*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^n + a)^p*(c*x)^m, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^p*(c*x)^m, x)