3.1147 \(\int x \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

Optimal. Leaf size=85 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b (p+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.160307, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.0313, size = 65, normalized size = 0.76 \[ \frac{\left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{d \left (a + b x^{2}\right )}{a d - b c}} \right )}}{2 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

(b*(-c - d*x**2)/(a*d - b*c))**(-q)*(a + b*x**2)**(p + 1)*(c + d*x**2)**q*hyper(
(-q, p + 1), (p + 2,), d*(a + b*x**2)/(a*d - b*c))/(2*b*(p + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.115828, size = 84, normalized size = 0.99 \[ \frac{\left (a+b x^2\right )^p \left (c+d x^2\right )^{q+1} \left (\frac{d \left (a+b x^2\right )}{a d-b c}\right )^{-p} \, _2F_1\left (-p,q+1;q+2;\frac{b \left (d x^2+c\right )}{b c-a d}\right )}{2 d (q+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int x \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^p*(d*x^2+c)^q,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x, x)