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

Optimal. Leaf size=292 \[ -\frac{d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]

[Out]

-(d*x*(a + b*x^2)^(1/4))/(2*c*(b*c - a*d)*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*(1 + (
b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*(b*c - a*d)*(a
+ b*x^2)^(3/4)) + (a^(1/4)*(5*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqr
t[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*
(b*c - a*d)^2*x) + (a^(1/4)*(5*b*c - 2*a*d)*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])/(4*c*(b*
c - a*d)^2*x)

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Rubi [A]  time = 0.646343, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)}-\frac{\sqrt{a} \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (5 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x*(a + b*x^2)^(1/4))/(2*c*(b*c - a*d)*(c + d*x^2)) - (Sqrt[a]*Sqrt[b]*(1 + (
b*x^2)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*c*(b*c - a*d)*(a
+ b*x^2)^(3/4)) + (a^(1/4)*(5*b*c - 2*a*d)*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqr
t[a]*Sqrt[d])/Sqrt[-(b*c) + a*d]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(4*c*
(b*c - a*d)^2*x) + (a^(1/4)*(5*b*c - 2*a*d)*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])/(4*c*(b*
c - a*d)^2*x)

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Rubi in Sympy [A]  time = 113.161, size = 250, normalized size = 0.86 \[ - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - \frac{5 b c}{2}\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c x \left (a d - b c\right )^{2}} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - \frac{5 b c}{2}\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 c x \left (a d - b c\right )^{2}} + \frac{\sqrt{a} \sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 c \left (a + b x^{2}\right )^{\frac{3}{4}} \left (a d - b c\right )} + \frac{d x \sqrt [4]{a + b x^{2}}}{2 c \left (c + d x^{2}\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*sqrt(-b*x**2/a)*(a*d - 5*b*c/2)*elliptic_pi(-sqrt(a)*sqrt(d)/sqrt(a*d
- b*c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(2*c*x*(a*d - b*c)**2) - a**(1/4
)*sqrt(-b*x**2/a)*(a*d - 5*b*c/2)*elliptic_pi(sqrt(a)*sqrt(d)/sqrt(a*d - b*c), a
sin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(2*c*x*(a*d - b*c)**2) + sqrt(a)*sqrt(b)*
(1 + b*x**2/a)**(3/4)*elliptic_f(atan(sqrt(b)*x/sqrt(a))/2, 2)/(2*c*(a + b*x**2)
**(3/4)*(a*d - b*c)) + d*x*(a + b*x**2)**(1/4)/(2*c*(c + d*x**2)*(a*d - b*c))

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Mathematica [C]  time = 0.811733, size = 340, normalized size = 1.16 \[ \frac{x \left (\frac{5 a b d x^2 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{18 a (a d-2 b c) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}-\frac{3 d \left (a+b x^2\right )}{c}\right )}{6 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) (b c-a d)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-3*d*(a + b*x^2))/c + (18*a*(-2*b*c + a*d)*AppellF1[1/2, 3/4, 1, 3/2, -((b*
x^2)/a), -((d*x^2)/c)])/(-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*AppellF1[3/2, 7/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) + (5*a*b*d*x^2*Appel
lF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/(-10*a*c*AppellF1[3/2, 3/4, 1
, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*AppellF1[5/2, 3/4, 2, 7/2, -((b*
x^2)/a), -((d*x^2)/c)] + 3*b*c*AppellF1[5/2, 7/4, 1, 7/2, -((b*x^2)/a), -((d*x^2
)/c)]))))/(6*(b*c - a*d)*(a + b*x^2)^(3/4)*(c + d*x^2))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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