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

Optimal. Leaf size=551 \[ \frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e (e x)^{3/2} (a d+4 b c)}{2 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e (e x)^{3/2}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{e (e x)^{3/2}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 3.56544, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ \frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e (e x)^{3/2} (a d+4 b c)}{2 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e (e x)^{3/2}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{e (e x)^{3/2}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(5*d*e*(e*x)^(3/2))/(6*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (e*(e*x)^(3/2))/(2*(b*
c - a*d)*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(4*b*c + a*d)*e*(e*x)^(3/2))/(2*c*(
b*c - a*d)^3*Sqrt[c - d*x^2]) - (d^(1/4)*(4*b*c + a*d)*e^(5/2)*Sqrt[1 - (d*x^2)/
c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*c^(1/4)*(b*c
- a*d)^3*Sqrt[c - d*x^2]) + (d^(1/4)*(4*b*c + a*d)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*
EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*c^(1/4)*(b*c -
a*d)^3*Sqrt[c - d*x^2]) + (Sqrt[b]*c^(1/4)*(3*b*c + 7*a*d)*e^(5/2)*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*Sqrt[a]*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2])
- (Sqrt[b]*c^(1/4)*(3*b*c + 7*a*d)*e^(5/2)*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*Sqrt[a]*d^(1/4)*(b*c - a*d)^3*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((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 = 1.87978, size = 568, normalized size = 1.03 \[ \frac{e (e x)^{3/2} \left (\frac{49 a \left (a^2 d^2+11 a b c d+3 b^2 c^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) (a d-b c)^3 \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{-14 x^2 \left (a^2 d^2 \left (c-3 d x^2\right )+a b d \left (11 c^2-10 c d x^2+3 d^2 x^4\right )+b^2 c \left (3 c^2-17 c d x^2+12 d^2 x^4\right )\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-11 a c \left (7 a^2 d^2 \left (c-3 d x^2\right )+a b d \left (77 c^2-67 c d x^2+18 d^2 x^4\right )+b^2 c \left (21 c^2-107 c d x^2+72 d^2 x^4\right )\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{c \left (b x^2-a\right ) \left (c-d x^2\right ) (b c-a d)^3 \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}\right )}{42 \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(e*(e*x)^(3/2)*((49*a*(3*b^2*c^2 + 11*a*b*c*d + a^2*d^2)*AppellF1[3/4, 1/2, 1, 7
/4, (d*x^2)/c, (b*x^2)/a])/((-(b*c) + a*d)^3*(a - b*x^2)*(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]))) + (
-11*a*c*(7*a^2*d^2*(c - 3*d*x^2) + a*b*d*(77*c^2 - 67*c*d*x^2 + 18*d^2*x^4) + b^
2*c*(21*c^2 - 107*c*d*x^2 + 72*d^2*x^4))*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c,
(b*x^2)/a] - 14*x^2*(a^2*d^2*(c - 3*d*x^2) + a*b*d*(11*c^2 - 10*c*d*x^2 + 3*d^2*
x^4) + b^2*c*(3*c^2 - 17*c*d*x^2 + 12*d^2*x^4))*(2*b*c*AppellF1[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]))/(c*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)*(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*AppellF1[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*Sqr
t[c - d*x^2])

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Maple [B]  time = 0.063, size = 5078, normalized size = 9.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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