3.314 \(\int (a+b x^n)^p (c+d x^n)^2 \, dx\)

Optimal. Leaf size=202 \[ -\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (b c (n p+n+1) (a d-b c (n (p+2)+1))-a d (a d (n+1)-b c (n (p+3)+1))) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b^2 (n p+n+1) (n (p+2)+1)}-\frac{d x \left (a+b x^n\right )^{p+1} (a d (n+1)-b c (n (p+3)+1))}{b^2 (n p+n+1) (n (p+2)+1)}+\frac{d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b (n p+2 n+1)} \]

[Out]

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

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Rubi [A]  time = 0.26866, antiderivative size = 197, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {416, 388, 246, 245} \[ -\frac{d x \left (a+b x^n\right )^{p+1} (a d (n+1)-b (c n (p+3)+c))}{b^2 (n p+n+1) (n (p+2)+1)}-\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c (a d-b (c n (p+2)+c))-\frac{a d (a d (n+1)-b (c n (p+3)+c))}{b (n p+n+1)}\right ) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b (n (p+2)+1)}+\frac{d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1}}{b (n (p+2)+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p*(c + d*x^n)^2,x]

[Out]

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

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

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

Mathematica [A]  time = 5.1585, size = 140, normalized size = 0.69 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left ((n+1) \left (c^2 (2 n+1) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+d^2 x^{2 n} \, _2F_1\left (2+\frac{1}{n},-p;3+\frac{1}{n};-\frac{b x^n}{a}\right )\right )+2 c d (2 n+1) x^n \, _2F_1\left (1+\frac{1}{n},-p;2+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{(n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.55, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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Sympy [C]  time = 23.8943, size = 143, normalized size = 0.71 \begin{align*} \frac{a^{p} c^{2} 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{2 a^{p} c 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 )} + \frac{a^{p} d^{2} x x^{2 n} \Gamma \left (2 + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, 2 + \frac{1}{n} \\ 3 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \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)**2,x)

[Out]

a**p*c**2*x*gamma(1/n)*hyper((1/n, -p), (1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 1/n)) + 2*a**p*c*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)) + a**p*d**
2*x*x**(2*n)*gamma(2 + 1/n)*hyper((-p, 2 + 1/n), (3 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(3 + 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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError