3.1146 \(\int x^3 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

Optimal. Leaf size=146 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+2)}-\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q (a d (q+1)+b c (p+1)) \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^2 d (p+1) (p+q+2)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.318021, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+2)}-\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q (a d (q+1)+b c (p+1)) \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^2 d (p+1) (p+q+2)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.5865, size = 117, normalized size = 0.8 \[ \frac{\left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q + 1}}{2 b d \left (p + q + 2\right )} - \frac{\left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q} \left (a d \left (q + 1\right ) + b c \left (p + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - q, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{d \left (a + b x^{2}\right )}{a d - b c}} \right )}}{2 b^{2} d \left (p + 1\right ) \left (p + q + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.410162, size = 159, normalized size = 1.09 \[ \frac{3 a c x^4 \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (2;-p,-q;3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{4 \left (x^2 \left (b c p F_1\left (3;1-p,-q;4;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q F_1\left (3;-p,1-q;4;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+3 a c F_1\left (2;-p,-q;3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^3*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

(3*a*c*x^4*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[2, -p, -q, 3, -((b*x^2)/a), -((d
*x^2)/c)])/(4*(3*a*c*AppellF1[2, -p, -q, 3, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(b
*c*p*AppellF1[3, 1 - p, -q, 4, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q*AppellF1[3, -
p, 1 - q, 4, -((b*x^2)/a), -((d*x^2)/c)])))

_______________________________________________________________________________________

Maple [F]  time = 0.088, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^p*(d*x^2+c)^q,x)

[Out]

int(x^3*(b*x^2+a)^p*(d*x^2+c)^q,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^3,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^3, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^3,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q*x^3, 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**3*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^3,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^3, x)