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

Optimal. Leaf size=606 \[ \frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{2 a c^2 e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 c^{11/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*(e*x)^(3/2)*(c - d*x^2)^(3/2)) + b/(2
*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 7*a*
b*c*d - 3*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*e*(e*x)^(3/2)*Sqrt[c - d*x^2]) - ((7*
b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c - d*x^2])/(6*a^2*
c^3*(b*c - a*d)^3*e*(e*x)^(3/2)) + (d^(3/4)*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2
*b*c*d^2 - 15*a^3*d^3)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/
(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(11/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2])
+ (b^3*c^(1/4)*(7*b*c - 17*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^
3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2]) + (b^3*c^(1/4)*(7*b*c - 17*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^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sq
rt[c - d*x^2])

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Rubi [A]  time = 3.82499, antiderivative size = 606, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{2 a c^2 e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 c^{11/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((e*x)^(5/2)*(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*(e*x)^(3/2)*(c - d*x^2)^(3/2)) + b/(2
*a*(b*c - a*d)*e*(e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 7*a*
b*c*d - 3*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*e*(e*x)^(3/2)*Sqrt[c - d*x^2]) - ((7*
b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c - d*x^2])/(6*a^2*
c^3*(b*c - a*d)^3*e*(e*x)^(3/2)) + (d^(3/4)*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2
*b*c*d^2 - 15*a^3*d^3)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/
(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(11/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2])
+ (b^3*c^(1/4)*(7*b*c - 17*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^
3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2]) + (b^3*c^(1/4)*(7*b*c - 17*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^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sq
rt[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)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 3.75623, size = 582, normalized size = 0.96 \[ \frac{x \left (\frac{9 a b c d x^4 \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{25 a c x^2 \left (15 a^4 d^4-35 a^3 b c d^3+12 a^2 b^2 c^2 d^2+44 a b^3 c^3 d-21 b^4 c^4\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{5 \left (a^4 d^3 \left (4 c^2-21 c d x^2+15 d^2 x^4\right )-a^3 b d^2 \left (12 c^3-45 c^2 d x^2+14 c d^2 x^4+15 d^3 x^6\right )+a^2 b^2 c d \left (12 c^3-12 c^2 d x^2-37 c d^2 x^4+35 d^3 x^6\right )-4 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+7 b^4 c^3 x^2 \left (c-d x^2\right )^2\right )}{c-d x^2}\right )}{30 a^2 c^3 (e x)^{5/2} \left (a-b x^2\right ) \sqrt{c-d x^2} (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-5*(7*b^4*c^3*x^2*(c - d*x^2)^2 - 4*a*b^3*c^2*(c - d*x^2)^2*(c + 3*d*x^2) +
a^4*d^3*(4*c^2 - 21*c*d*x^2 + 15*d^2*x^4) - a^3*b*d^2*(12*c^3 - 45*c^2*d*x^2 +
14*c*d^2*x^4 + 15*d^3*x^6) + a^2*b^2*c*d*(12*c^3 - 12*c^2*d*x^2 - 37*c*d^2*x^4 +
35*d^3*x^6)))/(c - d*x^2) + (25*a*c*(-21*b^4*c^4 + 44*a*b^3*c^3*d + 12*a^2*b^2*
c^2*d^2 - 35*a^3*b*c*d^3 + 15*a^4*d^4)*x^2*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c,
(b*x^2)/a])/(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*
b*c*AppellF1[5/4, 1/2, 2, 9/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[5/4, 3/2, 1,
9/4, (d*x^2)/c, (b*x^2)/a])) + (9*a*b*c*d*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*
b*c*d^2 - 15*a^3*d^3)*x^4*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(9*a
*c*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[9/4,
1/2, 2, 13/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1[9/4, 3/2, 1, 13/4, (d*x^2)/c
, (b*x^2)/a]))))/(30*a^2*c^3*(-(b*c) + a*d)^3*(e*x)^(5/2)*(a - b*x^2)*Sqrt[c - d
*x^2])

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Maple [B]  time = 0.079, size = 5248, normalized size = 8.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x)^(5/2)/(-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. \[ \int \frac{1}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

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)^(5/2)),x, algorithm="maxima")

[Out]

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

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

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)^(5/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.25053, 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)^(5/2)),x, algorithm="giac")

[Out]

Done