3.1.62 \(\int (d+e x^n) (a+c x^{2 n})^p \, dx\) [62]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {1447, 252, 251, 372, 371} \begin {gather*} 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} \end {gather*}

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 251

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 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 1447

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]

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*}

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Mathematica [A]
time = 0.07, size = 110, normalized size = 0.81 \begin {gather*} \frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (d (1+n) \, _2F_1\left (\frac {1}{2 n},-p;1+\frac {1}{2 n};-\frac {c x^{2 n}}{a}\right )+e x^n \, _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 )\right )}{1+n} \end {gather*}

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, 1 + 1/(2*n), -((c*x^(2*n))/a)] + e*x^n*Hypergeo
metric2F1[(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 = 0.05, size = 0, normalized size = 0.00 \[\int \left (d +e \,x^{n}\right ) \left (a +c \,x^{2 n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^n)*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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((x^n*e + d)*(c*x^(2*n) + a)^p, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^n*e + d)*(c*x^(2*n) + a)^p, x)

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Sympy [C] Result contains complex when optimal does not.
time = 129.60, size = 114, normalized size = 0.84 \begin {gather*} \frac {a^{p} d x \Gamma \left (\frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2 n}, - p \\ 1 + \frac {1}{2 n} \end {matrix}\middle | {\frac {c x^{2 n} e^{i \pi }}{a}} \right )}}{2 n \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{p} e x x^{n} \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} + \frac {1}{2 n} \\ \frac {3}{2} + \frac {1}{2 n} \end {matrix}\middle | {\frac {c x^{2 n} e^{i \pi }}{a}} \right )}}{2 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^n*e + d)*(c*x^(2*n) + a)^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+c\,x^{2\,n}\right )}^p\,\left (d+e\,x^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^(2*n))^p*(d + e*x^n),x)

[Out]

int((a + c*x^(2*n))^p*(d + e*x^n), x)

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