3.1145 \(\int x^5 \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx\)

Optimal. Leaf size=242 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (a^2 d^2 \left (q^2+3 q+2\right )+2 a b c d (p+1) (q+1)+b^2 c^2 \left (p^2+3 p+2\right )\right ) \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^3 d^2 (p+1) (p+q+2) (p+q+3)}-\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d (q+2)+b c (p+2))}{2 b^2 d^2 (p+q+2) (p+q+3)}+\frac{x^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+3)} \]

[Out]

-((b*c*(2 + p) + a*d*(2 + q))*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b^2*d^
2*(2 + p + q)*(3 + p + q)) + (x^2*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b*
d*(3 + p + q)) + ((b^2*c^2*(2 + 3*p + p^2) + 2*a*b*c*d*(1 + p)*(1 + q) + a^2*d^2
*(2 + 3*q + q^2))*(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^3*d^2*(1 + p)*(2 + p + q)*(3 + p +
q)*((b*(c + d*x^2))/(b*c - a*d))^q)

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Rubi [A]  time = 0.717857, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (a^2 d^2 \left (q^2+3 q+2\right )+2 a b c d (p+1) (q+1)+b^2 c^2 \left (p^2+3 p+2\right )\right ) \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b^3 d^2 (p+1) (p+q+2) (p+q+3)}-\frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1} (a d (q+2)+b c (p+2))}{2 b^2 d^2 (p+q+2) (p+q+3)}+\frac{x^2 \left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^{q+1}}{2 b d (p+q+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

-((b*c*(2 + p) + a*d*(2 + q))*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b^2*d^
2*(2 + p + q)*(3 + p + q)) + (x^2*(a + b*x^2)^(1 + p)*(c + d*x^2)^(1 + q))/(2*b*
d*(3 + p + q)) + ((b^2*c^2*(2 + 3*p + p^2) + 2*a*b*c*d*(1 + p)*(1 + q) + a^2*d^2
*(2 + 3*q + q^2))*(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^3*d^2*(1 + p)*(2 + p + q)*(3 + p +
q)*((b*(c + d*x^2))/(b*c - a*d))^q)

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Rubi in Sympy [A]  time = 101.978, size = 206, normalized size = 0.85 \[ \frac{x^{2} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q + 1}}{2 b d \left (p + q + 3\right )} - \frac{\left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q + 1} \left (a d \left (q + 2\right ) + b c \left (p + 2\right )\right )}{2 b^{2} d^{2} \left (p + q + 2\right ) \left (p + q + 3\right )} + \frac{\left (\frac{b \left (- c - d x^{2}\right )}{a d - b c}\right )^{- q} \left (a + b x^{2}\right )^{p + 1} \left (c + d x^{2}\right )^{q} \left (- a b c d \left (p + q + 2\right ) + \left (a d \left (q + 1\right ) + b c \left (p + 1\right )\right ) \left (a d \left (q + 2\right ) + b c \left (p + 2\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - q, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{d \left (a + b x^{2}\right )}{a d - b c}} \right )}}{2 b^{3} d^{2} \left (p + 1\right ) \left (p + q + 2\right ) \left (p + q + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**p*(d*x**2+c)**q,x)

[Out]

x**2*(a + b*x**2)**(p + 1)*(c + d*x**2)**(q + 1)/(2*b*d*(p + q + 3)) - (a + b*x*
*2)**(p + 1)*(c + d*x**2)**(q + 1)*(a*d*(q + 2) + b*c*(p + 2))/(2*b**2*d**2*(p +
q + 2)*(p + q + 3)) + (b*(-c - d*x**2)/(a*d - b*c))**(-q)*(a + b*x**2)**(p + 1)
*(c + d*x**2)**q*(-a*b*c*d*(p + q + 2) + (a*d*(q + 1) + b*c*(p + 1))*(a*d*(q + 2
) + b*c*(p + 2)))*hyper((-q, p + 1), (p + 2,), d*(a + b*x**2)/(a*d - b*c))/(2*b*
*3*d**2*(p + 1)*(p + q + 2)*(p + q + 3))

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Mathematica [C]  time = 0.442012, size = 160, normalized size = 0.66 \[ \frac{2 a c x^6 \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (3;-p,-q;4;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{3 \left (b c p x^2 F_1\left (4;1-p,-q;5;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d q x^2 F_1\left (4;-p,1-q;5;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+4 a c F_1\left (3;-p,-q;4;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^5*(a + b*x^2)^p*(c + d*x^2)^q,x]

[Out]

(2*a*c*x^6*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[3, -p, -q, 4, -((b*x^2)/a), -((d
*x^2)/c)])/(3*(4*a*c*AppellF1[3, -p, -q, 4, -((b*x^2)/a), -((d*x^2)/c)] + b*c*p*
x^2*AppellF1[4, 1 - p, -q, 5, -((b*x^2)/a), -((d*x^2)/c)] + a*d*q*x^2*AppellF1[4
, -p, 1 - q, 5, -((b*x^2)/a), -((d*x^2)/c)]))

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^p*(d*x^2+c)^q,x)

[Out]

int(x^5*(b*x^2+a)^p*(d*x^2+c)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^5,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^5,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^p*(d*x^2 + c)^q*x^5, x)

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^5,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^p*(d*x^2 + c)^q*x^5, x)