3.2916 \(\int (d x)^m \left (a+b (c x)^n\right )^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.116388, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(d x)^{m+1} \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b (c x)^n}{a}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.137, size = 65, normalized size = 0.89 \[ \frac{\left (c x\right )^{- m} \left (c x\right )^{m + 1} \left (d x\right )^{m} \left (1 + \frac{b \left (c x\right )^{n}}{a}\right )^{- p} \left (a + b \left (c x\right )^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b \left (c x\right )^{n}}{a}} \right )}}{c \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.122609, size = 70, normalized size = 0.96 \[ \frac{x (d x)^m \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+1}{n}+1;-\frac{b (c x)^n}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.467, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(a+b*(c*x)^n)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m*(a + b*(c*x)**n)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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