3.326 \(\int \frac{1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=274 \[ \frac{2 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (3 b c-8 a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^2} (b c-a d)^2}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}+\frac{2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)} \]

[Out]

(2*b*x)/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)) + (2*Sqrt[b]*(3*b*c - 8*a*d)*(1 + (b
*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*(b*c - a*
d)^2*(a + b*x^2)^(1/4)) + (a^(1/4)*d^(3/2)*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)^(5/2)*x) - (a^(1/4)*d^(3/2)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqr
t[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((-(b*c) + a*d
)^(5/2)*x)

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Rubi [A]  time = 0.964766, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{2 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (3 b c-8 a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^2} (b c-a d)^2}+\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}-\frac{\sqrt [4]{a} d^{3/2} \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{x (a d-b c)^{5/2}}+\frac{2 b x}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^(9/4)*(c + d*x^2)),x]

[Out]

(2*b*x)/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)) + (2*Sqrt[b]*(3*b*c - 8*a*d)*(1 + (b
*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*(b*c - a*
d)^2*(a + b*x^2)^(1/4)) + (a^(1/4)*d^(3/2)*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)^(5/2)*x) - (a^(1/4)*d^(3/2)*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[a]*Sqr
t[d])/Sqrt[-(b*c) + a*d], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/((-(b*c) + a*d
)^(5/2)*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [4]{a} d^{\frac{3}{2}} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{x \left (a d - b c\right )^{\frac{5}{2}}} - \frac{\sqrt [4]{a} d^{\frac{3}{2}} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{x \left (a d - b c\right )^{\frac{5}{2}}} - \frac{2 b x}{5 a \left (a + b x^{2}\right )^{\frac{5}{4}} \left (a d - b c\right )} - \frac{2 b x \left (8 a d - 3 b c\right )}{5 a^{2} \sqrt [4]{a + b x^{2}} \left (a d - b c\right )^{2}} + \frac{b \left (8 a d - 3 b c\right ) \int \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{5 a^{2} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(9/4)/(d*x**2+c),x)

[Out]

a**(1/4)*d**(3/2)*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)**(5/2)) - a**(1/4)*d**(3/
2)*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)**(5/2)) - 2*b*x/(5*a*(a + b*x**2)**(5/4)*
(a*d - b*c)) - 2*b*x*(8*a*d - 3*b*c)/(5*a**2*(a + b*x**2)**(1/4)*(a*d - b*c)**2)
+ b*(8*a*d - 3*b*c)*Integral((a + b*x**2)**(-1/4), x)/(5*a**2*(a*d - b*c)**2)

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Mathematica [C]  time = 1.62462, size = 404, normalized size = 1.47 \[ \frac{2 x \left (\frac{9 a c \left (5 a^2 d^2+8 a b c d-3 b^2 c^2\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}+\frac{3 b \left (-9 a^2 d+4 a b \left (c-2 d x^2\right )+3 b^2 c x^2\right )}{a+b x^2}-\frac{5 a b c d x^2 (8 a d-3 b c) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{\left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}\right )}{15 a^2 \sqrt [4]{a+b x^2} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^2)^(9/4)*(c + d*x^2)),x]

[Out]

(2*x*((3*b*(-9*a^2*d + 3*b^2*c*x^2 + 4*a*b*(c - 2*d*x^2)))/(a + b*x^2) + (9*a*c*
(-3*b^2*c^2 + 8*a*b*c*d + 5*a^2*d^2)*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -(
(d*x^2)/c)])/((c + d*x^2)*(6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x
^2)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*
c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)]))) - (5*a*b*c*d*(-3*b*c
+ 8*a*d)*x^2*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/((c + d*x^
2)*(-10*a*c*AppellF1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*
AppellF1[5/2, 1/4, 2, 7/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[5/2, 5/4,
1, 7/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(1/4)
)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(9/4)/(d*x^2+c),x)

[Out]

int(1/(b*x^2+a)^(9/4)/(d*x^2+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}{\left (d x^{2} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)), 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)^(9/4)*(d*x^2 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{9}{4}} \left (c + d x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(9/4)/(d*x**2+c),x)

[Out]

Integral(1/((a + b*x**2)**(9/4)*(c + d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}{\left (d x^{2} + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(9/4)*(d*x^2 + c)), x)