3.61 \(\int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx\)

Optimal. Leaf size=217 \[ d^2 x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{2 d e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+\frac{e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right ),-p;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 n+1} \]

[Out]

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(2 + n^(-1))/2, -p, (4 + n^
(-1))/2, -((c*x^(2*n))/a)])/((1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (d^2*x*(a + c*x^
(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 +
(c*x^(2*n))/a)^p + (2*d*e*x^(1 + n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + n)/
(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + n)*(1 + (c*x^(2*n))/a)^p)

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Rubi [A]  time = 0.216818, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ d^2 x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{2 d e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+\frac{e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right ),-p;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 n+1} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^n)^2*(a + c*x^(2*n))^p,x]

[Out]

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(2 + n^(-1))/2, -p, (4 + n^
(-1))/2, -((c*x^(2*n))/a)])/((1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (d^2*x*(a + c*x^
(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 +
(c*x^(2*n))/a)^p + (2*d*e*x^(1 + n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + n)/
(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + n)*(1 + (c*x^(2*n))/a)^p)

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Rubi in Sympy [A]  time = 29.3167, size = 170, normalized size = 0.78 \[ d^{2} x \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )} + \frac{2 d e x^{n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{n + 1} + \frac{e^{2} x^{2 n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{n + \frac{1}{2}}{n} \\ 2 + \frac{1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{2 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e*x**n)**2*(a+c*x**(2*n))**p,x)

[Out]

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

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Mathematica [A]  time = 0.206929, size = 171, normalized size = 0.79 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (d (2 n+1) \left (d (n+1) \, _2F_1\left (\frac{1}{2 n},-p;1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+2 e x^n \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )+e^2 (n+1) x^{2 n} \, _2F_1\left (1+\frac{1}{2 n},-p;2+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )\right )}{(n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^n)^2*(a + c*x^(2*n))^p,x]

[Out]

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

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Maple [F]  time = 0.154, size = 0, normalized size = 0. \[ \int \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e*x^n)^2*(a+c*x^(2*n))^p,x)

[Out]

int((d+e*x^n)^2*(a+c*x^(2*n))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{n} + d\right )}^{2}{\left (c x^{2 \, n} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p,x, algorithm="maxima")

[Out]

integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}\right )}{\left (c x^{2 \, n} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p,x, algorithm="fricas")

[Out]

integral((e^2*x^(2*n) + 2*d*e*x^n + d^2)*(c*x^(2*n) + a)^p, x)

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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((d+e*x**n)**2*(a+c*x**(2*n))**p,x)

[Out]

Timed out

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GIAC/XCAS [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((e*x^n + d)^2*(c*x^(2*n) + a)^p,x, algorithm="giac")

[Out]

Exception raised: TypeError