3.334 \(\int \frac{1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=314 \[ -\frac{d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (a d+4 b c) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} c \sqrt [4]{a+b x^2} (b c-a d)^2}-\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}}+\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}} \]

[Out]

-(d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)) + (Sqrt[b]*(4*b*c + a*d)*
(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*Sqrt[a]*c*
(b*c - a*d)^2*(a + b*x^2)^(1/4)) - (a^(1/4)*Sqrt[d]*(7*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)^(5/2)*x) + (a^(1/4)*Sqrt[d]*(7*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)^(5/2)*x)

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Rubi [A]  time = 1.04017, antiderivative size = 314, 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 \[ -\frac{d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (a d+4 b c) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} c \sqrt [4]{a+b x^2} (b c-a d)^2}-\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}}+\frac{\sqrt [4]{a} \sqrt{d} \sqrt{-\frac{b x^2}{a}} (7 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (a d-b c)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]

[Out]

-(d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2)) + (Sqrt[b]*(4*b*c + a*d)*
(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(2*Sqrt[a]*c*
(b*c - a*d)^2*(a + b*x^2)^(1/4)) - (a^(1/4)*Sqrt[d]*(7*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)^(5/2)*x) + (a^(1/4)*Sqrt[d]*(7*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)^(5/2)*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [4]{a} \sqrt{d} \sqrt{- \frac{b x^{2}}{a}} \left (2 a d - 7 b c\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{4 c x \left (a d - b c\right )^{\frac{5}{2}}} - \frac{\sqrt [4]{a} \sqrt{d} \sqrt{- \frac{b x^{2}}{a}} \left (2 a d - 7 b c\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{4 c x \left (a d - b c\right )^{\frac{5}{2}}} + \frac{d x}{2 c \sqrt [4]{a + b x^{2}} \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{b x \left (a d + 4 b c\right )}{2 a c \sqrt [4]{a + b x^{2}} \left (a d - b c\right )^{2}} - \frac{b \left (a d + 4 b c\right ) \int \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{4 a c \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(5/4)/(d*x**2+c)**2,x)

[Out]

a**(1/4)*sqrt(d)*sqrt(-b*x**2/a)*(2*a*d - 7*b*c)*elliptic_pi(-sqrt(a)*sqrt(d)/sq
rt(a*d - b*c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(4*c*x*(a*d - b*c)**(5/2)
) - a**(1/4)*sqrt(d)*sqrt(-b*x**2/a)*(2*a*d - 7*b*c)*elliptic_pi(sqrt(a)*sqrt(d)
/sqrt(a*d - b*c), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(4*c*x*(a*d - b*c)**(5
/2)) + d*x/(2*c*(a + b*x**2)**(1/4)*(c + d*x**2)*(a*d - b*c)) + b*x*(a*d + 4*b*c
)/(2*a*c*(a + b*x**2)**(1/4)*(a*d - b*c)**2) - b*(a*d + 4*b*c)*Integral((a + b*x
**2)**(-1/4), x)/(4*a*c*(a*d - b*c)**2)

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Mathematica [C]  time = 0.972185, size = 480, normalized size = 1.53 \[ \frac{x \left (\frac{18 \left (-a^2 d^2+4 a b c d+2 b^2 c^2\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{5 a c \left (6 a^2 d^2+5 a b d^2 x^2+4 b^2 c \left (6 c+5 d x^2\right )\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-3 x^2 \left (a^2 d^2+a b d^2 x^2+4 b^2 c \left (c+d x^2\right )\right ) \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{a c \left (10 a c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}\right )}{6 \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x^2)^(5/4)*(c + d*x^2)^2),x]

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x)

[Out]

int(1/(b*x^2+a)^(5/4)/(d*x^2+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(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)^(5/4)*(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)**(5/4)/(d*x**2+c)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + c)^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(d*x^2 + c)^2), x)