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

Optimal. Leaf size=135 \[ d x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1} \]

[Out]

(d*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
+ (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 = 0.0604615, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1433, 246, 245, 365, 364} \[ d x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^n)*(a + c*x^(2*n))^p,x]

[Out]

(d*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
+ (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 1433

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)*(a +
c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*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 ) \left (a+c x^{2 n}\right )^p \, dx &=\int \left (d \left (a+c x^{2 n}\right )^p+e x^n \left (a+c x^{2 n}\right )^p\right ) \, dx\\ &=d \int \left (a+c x^{2 n}\right )^p \, dx+e \int x^n \left (a+c x^{2 n}\right )^p \, dx\\ &=\left (d \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 (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\\ &=d 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{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 = 0.0497929, size = 110, normalized size = 0.81 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (d (n+1) \, _2F_1\left (\frac{1}{2 n},-p;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+e x^n \, _2F_1\left (\frac{n+1}{2 n},-p;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + c*x^(2*n))^p*(d*(1 + n)*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + e*x^n*Hyper
geometric2F1[(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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Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int \left ( d+e{x}^{n} \right ) \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)*(a+c*x^(2*n))^p,x)

[Out]

int((d+e*x^n)*(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 )}{\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)*(a+c*x^(2*n))^p,x, algorithm="maxima")

[Out]

integrate((e*x^n + d)*(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 x^{n} + d\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)*(a+c*x^(2*n))^p,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{n} + d\right )}{\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)*(a+c*x^(2*n))^p,x, algorithm="giac")

[Out]

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