3.346 \(\int \left (a+b x^2\right )^{-1-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac{a d}{2 b c-2 a d}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x \left (a+b x^2\right )^{-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}}}{a c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0575358, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02 \[ \frac{x \left (a+b x^2\right )^{-\frac{b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}}}{a c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(-1 - (b*c)/(2*b*c - 2*a*d))*(c + d*x^2)^(-1 + (a*d)/(2*b*c - 2*a*d)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.1508, size = 44, normalized size = 0.83 \[ \frac{x \left (a + b x^{2}\right )^{\frac{b c}{2 \left (a d - b c\right )}} \left (c + d x^{2}\right )^{- \frac{a d}{2 \left (a d - b c\right )}}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(-1-b*c/(-2*a*d+2*b*c))*(d*x**2+c)**(-1+a*d/(-2*a*d+2*b*c)),x)

[Out]

x*(a + b*x**2)**(b*c/(2*(a*d - b*c)))*(c + d*x**2)**(-a*d/(2*(a*d - b*c)))/(a*c)

_______________________________________________________________________________________

Mathematica [C]  time = 2.40049, size = 594, normalized size = 11.21 \[ 3 a c x \left (a+b x^2\right )^{\frac{b c}{2 a d-2 b c}} \left (c+d x^2\right )^{\frac{a d}{2 b c-2 a d}} \left (\frac{b F_1\left (\frac{1}{2};\frac{b c}{2 b c-2 a d}+1,\frac{a d}{2 a d-2 b c};\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (a+b x^2\right ) \left (x^2 \left (a^2 d^2 F_1\left (\frac{3}{2};\frac{b c}{2 b c-2 a d}+1,\frac{a d}{2 a d-2 b c}+1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c (2 a d-3 b c) F_1\left (\frac{3}{2};\frac{b c}{2 b c-2 a d}+2,\frac{a d}{2 a d-2 b c};\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+3 a c (b c-a d) F_1\left (\frac{1}{2};\frac{b c}{2 b c-2 a d}+1,\frac{a d}{2 a d-2 b c};\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}+\frac{d F_1\left (\frac{1}{2};\frac{b c}{2 b c-2 a d},\frac{a d}{2 a d-2 b c}+1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (x^2 \left (b^2 c^2 F_1\left (\frac{3}{2};\frac{b c}{2 b c-2 a d}+1,\frac{a d}{2 a d-2 b c}+1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+a d (2 b c-3 a d) F_1\left (\frac{3}{2};\frac{b c}{2 b c-2 a d},\frac{a d}{2 a d-2 b c}+2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )+3 a c (a d-b c) F_1\left (\frac{1}{2};\frac{b c}{2 b c-2 a d},\frac{a d}{2 a d-2 b c}+1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2)^(-1 - (b*c)/(2*b*c - 2*a*d))*(c + d*x^2)^(-1 + (a*d)/(2*b*c - 2*a*d)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 71, normalized size = 1.3 \[{\frac{x}{ac} \left ( b{x}^{2}+a \right ) ^{1-{\frac{2\,ad-3\,bc}{2\,ad-2\,bc}}} \left ( d{x}^{2}+c \right ) ^{1-{\frac{3\,ad-2\,bc}{2\,ad-2\,bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(-1-b*c/(-2*a*d+2*b*c))*(d*x^2+c)^(-1+a*d/(-2*a*d+2*b*c)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{-\frac{b c}{2 \,{\left (b c - a d\right )}} - 1}{\left (d x^{2} + c\right )}^{\frac{a d}{2 \,{\left (b c - a d\right )}} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d) - 1),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d
) - 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.277354, size = 123, normalized size = 2.32 \[ \frac{b d x^{5} +{\left (b c + a d\right )} x^{3} + a c x}{{\left (b x^{2} + a\right )}^{\frac{3 \, b c - 2 \, a d}{2 \,{\left (b c - a d\right )}}}{\left (d x^{2} + c\right )}^{\frac{2 \, b c - 3 \, a d}{2 \,{\left (b c - a d\right )}}} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d) - 1),x, algorithm="fricas")

[Out]

(b*d*x^5 + (b*c + a*d)*x^3 + a*c*x)/((b*x^2 + a)^(1/2*(3*b*c - 2*a*d)/(b*c - a*d
))*(d*x^2 + c)^(1/2*(2*b*c - 3*a*d)/(b*c - a*d))*a*c)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{-\frac{b c}{2 \,{\left (b c - a d\right )}} - 1}{\left (d x^{2} + c\right )}^{\frac{a d}{2 \,{\left (b c - a d\right )}} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d) - 1),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d
) - 1), x)