3.323 \(\int \frac{1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=152 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)} \]

[Out]

(a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]),
ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)*x) + (a^(1/4)*Sqrt[-((b*x^2
)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/
a^(1/4)], -1])/((b*c - a*d)*x)

_______________________________________________________________________________________

Rubi [A]  time = 0.365383, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)),x]

[Out]

(a^(1/4)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]),
ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((b*c - a*d)*x) + (a^(1/4)*Sqrt[-((b*x^2
)/a)]*EllipticPi[(Sqrt[a]*Sqrt[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/
a^(1/4)], -1])/((b*c - a*d)*x)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 66.3259, size = 131, normalized size = 0.86 \[ - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{x \left (a d - b c\right )} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{x \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c),x)

[Out]

-a**(1/4)*sqrt(-b*x**2/a)*elliptic_pi(-sqrt(a)*sqrt(d)/sqrt(a*d - b*c), asin((a
+ b*x**2)**(1/4)/a**(1/4)), -1)/(x*(a*d - b*c)) - a**(1/4)*sqrt(-b*x**2/a)*ellip
tic_pi(sqrt(a)*sqrt(d)/sqrt(a*d - b*c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/
(x*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0898739, size = 161, normalized size = 1.06 \[ -\frac{6 a c x F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)),x]

[Out]

(-6*a*c*x*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)])/((a + b*x^2)^(
3/4)*(c + d*x^2)*(-6*a*c*AppellF1[1/2, 3/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)]
+ x^2*(4*a*d*AppellF1[3/2, 3/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 3*b*c*Appe
llF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(3/4)/(d*x^2+c),x)

[Out]

int(1/(b*x^2+a)^(3/4)/(d*x^2+c),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}{\left (d x^{2} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{3}{4}} \left (c + d x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c),x)

[Out]

Integral(1/((a + b*x**2)**(3/4)*(c + d*x**2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}{\left (d x^{2} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)), x)