3.932 \(\int \frac{1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=735 \[ -\frac{5 b^{5/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 b^{5/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{6 a c^2 e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right )}{2 a^2 c^3 e \sqrt{e x} (b c-a d)^3}+\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 c^{9/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 c^{9/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e \sqrt{e x} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e \sqrt{e x} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*Sqrt[e*x]*(c - d*x^2)^(3/2)) + b/(2*a
*(b*c - a*d)*e*Sqrt[e*x]*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 19*a*b
*c*d - 7*a^2*d^2))/(6*a*c^2*(b*c - a*d)^3*e*Sqrt[e*x]*Sqrt[c - d*x^2]) - ((5*b^3
*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*Sqrt[c - d*x^2])/(2*a^2*c^3*
(b*c - a*d)^3*e*Sqrt[e*x]) - (d^(1/4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d
^2 - 7*a^3*d^3)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4
)*Sqrt[e])], -1])/(2*a^2*c^(9/4)*(b*c - a*d)^3*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/
4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*Sqrt[1 - (d*x^2)/c]
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*c^(9/4)*(b
*c - a*d)^3*e^(3/2)*Sqrt[c - d*x^2]) - (5*b^(5/2)*c^(1/4)*(b*c - 3*a*d)*Sqrt[1 -
(d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*S
qrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^3*e^(3/2)*Sqrt
[c - d*x^2]) + (5*b^(5/2)*c^(1/4)*(b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(
Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])
], -1])/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^3*e^(3/2)*Sqrt[c - d*x^2])

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Rubi [A]  time = 4.75141, antiderivative size = 735, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{5 b^{5/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 b^{5/2} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-3 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{6 a c^2 e \sqrt{e x} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right )}{2 a^2 c^3 e \sqrt{e x} (b c-a d)^3}+\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 c^{9/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 c^{9/4} e^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e \sqrt{e x} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e \sqrt{e x} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*Sqrt[e*x]*(c - d*x^2)^(3/2)) + b/(2*a
*(b*c - a*d)*e*Sqrt[e*x]*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(3*b^2*c^2 + 19*a*b
*c*d - 7*a^2*d^2))/(6*a*c^2*(b*c - a*d)^3*e*Sqrt[e*x]*Sqrt[c - d*x^2]) - ((5*b^3
*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*Sqrt[c - d*x^2])/(2*a^2*c^3*
(b*c - a*d)^3*e*Sqrt[e*x]) - (d^(1/4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d
^2 - 7*a^3*d^3)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4
)*Sqrt[e])], -1])/(2*a^2*c^(9/4)*(b*c - a*d)^3*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/
4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*Sqrt[1 - (d*x^2)/c]
*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*c^(9/4)*(b
*c - a*d)^3*e^(3/2)*Sqrt[c - d*x^2]) - (5*b^(5/2)*c^(1/4)*(b*c - 3*a*d)*Sqrt[1 -
(d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*S
qrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^3*e^(3/2)*Sqrt
[c - d*x^2]) + (5*b^(5/2)*c^(1/4)*(b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(
Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])
], -1])/(4*a^(5/2)*d^(1/4)*(b*c - a*d)^3*e^(3/2)*Sqrt[c - d*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 3.59308, size = 582, normalized size = 0.79 \[ \frac{x \left (\frac{33 a b c d x^4 \left (7 a^3 d^3-19 a^2 b c d^2+12 a b^2 c^2 d-5 b^3 c^3\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{49 a c x^2 \left (7 a^4 d^4-19 a^3 b c d^3+12 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+5 b^4 c^4\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{7 \left (a^4 d^3 \left (12 c^2-35 c d x^2+21 d^2 x^4\right )-a^3 b d^2 \left (36 c^3-83 c^2 d x^2+22 c d^2 x^4+21 d^3 x^6\right )+a^2 b^2 c d \left (36 c^3-36 c^2 d x^2-59 c d^2 x^4+57 d^3 x^6\right )-12 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+15 b^4 c^3 x^2 \left (c-d x^2\right )^2\right )}{c-d x^2}\right )}{42 a^2 c^3 (e x)^{3/2} \left (a-b x^2\right ) \sqrt{c-d x^2} (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

(x*((-7*(15*b^4*c^3*x^2*(c - d*x^2)^2 - 12*a*b^3*c^2*(c - d*x^2)^2*(c + 3*d*x^2)
+ a^4*d^3*(12*c^2 - 35*c*d*x^2 + 21*d^2*x^4) - a^3*b*d^2*(36*c^3 - 83*c^2*d*x^2
+ 22*c*d^2*x^4 + 21*d^3*x^6) + a^2*b^2*c*d*(36*c^3 - 36*c^2*d*x^2 - 59*c*d^2*x^
4 + 57*d^3*x^6)))/(c - d*x^2) - (49*a*c*(5*b^4*c^4 - 20*a*b^3*c^3*d + 12*a^2*b^2
*c^2*d^2 - 19*a^3*b*c*d^3 + 7*a^4*d^4)*x^2*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c,
(b*x^2)/a])/(7*a*c*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*
b*c*AppellF1[7/4, 1/2, 2, 11/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[7/4, 3/2, 1
, 11/4, (d*x^2)/c, (b*x^2)/a])) + (33*a*b*c*d*(-5*b^3*c^3 + 12*a*b^2*c^2*d - 19*
a^2*b*c*d^2 + 7*a^3*d^3)*x^4*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])/
(11*a*c*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF
1[11/4, 1/2, 2, 15/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[11/4, 3/2, 1, 15/4, (
d*x^2)/c, (b*x^2)/a]))))/(42*a^2*c^3*(-(b*c) + a*d)^3*(e*x)^(3/2)*(a - b*x^2)*Sq
rt[c - d*x^2])

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Maple [B]  time = 0.079, size = 6334, normalized size = 8.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(3/2)), 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)^2*(-d*x^2 + c)^(5/2)*(e*x)^(3/2)),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.2428, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(3/2)),x, algorithm="giac")

[Out]

Done