3.315 \(\int (a+b x^n)^p (c+d x^n) \, dx\)

Optimal. Leaf size=98 \[ \frac{d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)}-\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (a d-b c (n p+n+1)) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b (n p+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0473692, antiderivative size = 89, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {388, 246, 245} \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c-\frac{a d}{b n p+b n+b}\right ) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+\frac{d x \left (a+b x^n\right )^{p+1}}{b (n p+n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n),x]

[Out]

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

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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])

Rubi steps

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

Mathematica [A]  time = 0.0429777, size = 94, normalized size = 0.96 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left ((b c (n p+n+1)-a d) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+d \left (a+b x^n\right ) \left (\frac{b x^n}{a}+1\right )^p\right )}{b (n p+n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.401, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.2984, size = 87, normalized size = 0.89 \begin{align*} \frac{a^{p} c x \Gamma \left (\frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, - p \\ 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac{1}{n}\right )} + \frac{a^{p} d x x^{n} \Gamma \left (1 + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, 1 + \frac{1}{n} \\ 2 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError