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

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 = 0.100881, 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, Rules used = {1437, 246, 245, 365, 364} \[ d^2 x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{2 d e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+\frac{e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right ),-p;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 n+1} \]

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)

Rule 1437

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)
^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ((
IntegersQ[p, q] &&  !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] &&  !IntegerQ[n]))

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx &=\int \left (d^2 \left (a+c x^{2 n}\right )^p+2 d e x^n \left (a+c x^{2 n}\right )^p+e^2 x^{2 n} \left (a+c x^{2 n}\right )^p\right ) \, dx\\ &=d^2 \int \left (a+c x^{2 n}\right )^p \, dx+(2 d e) \int x^n \left (a+c x^{2 n}\right )^p \, dx+e^2 \int x^{2 n} \left (a+c x^{2 n}\right )^p \, dx\\ &=\left (d^2 \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int \left (1+\frac{c x^{2 n}}{a}\right )^p \, dx+\left (2 d e \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int x^n \left (1+\frac{c x^{2 n}}{a}\right )^p \, dx+\left (e^2 \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int x^{2 n} \left (1+\frac{c x^{2 n}}{a}\right )^p \, dx\\ &=\frac{e^2 x^{1+2 n} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\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 )}{1+2 n}+d^2 x \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\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^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac{1+n}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.108526, size = 171, normalized size = 0.79 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (d (2 n+1) \left (d (n+1) \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+2 e x^n \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )+e^2 (n+1) x^{2 n} \, _2F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right ),-p;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )}{(n+1) (2 n+1)} \]

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[(2 + n^(-1))/2, -p, (4 + n^(-1))/2, -((c*x^(2*n))/
a)] + d*(1 + 2*n)*(d*(1 + n)*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + 2*e*x^n*Hyperg
eometric2F1[(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 = 0.094, size = 0, normalized size = 0. \begin{align*} \int \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{p}\, dx \end{align*}

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. \begin{align*} \int{\left (e x^{n} + d\right )}^{2}{\left (c x^{2 \, n} + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^n)^2*(a+c*x^(2*n))^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. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^n)^2*(a+c*x^(2*n))^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. \begin{align*} \text{Timed out} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^n)^2*(a+c*x^(2*n))^p,x, algorithm="giac")

[Out]

Exception raised: TypeError