3.926 \(\int \frac{(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=568 \[ \frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 d^{3/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 d^{3/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+7 b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+7 b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 (e x)^{3/2} (4 a d+b c)}{2 \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 (e x)^{3/2} (3 a d+2 b c)}{6 b \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{a e^3 (e x)^{3/2}}{2 b \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 3.70999, antiderivative size = 568, 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 \[ \frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 d^{3/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{c^{3/4} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (4 a d+b c) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 d^{3/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+7 b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{a} \sqrt [4]{c} e^{9/2} \sqrt{1-\frac{d x^2}{c}} (3 a d+7 b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 (e x)^{3/2} (4 a d+b c)}{2 \sqrt{c-d x^2} (b c-a d)^3}+\frac{e^3 (e x)^{3/2} (3 a d+2 b c)}{6 b \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{a e^3 (e x)^{3/2}}{2 b \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

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

_______________________________________________________________________________________

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)**(9/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 1.48643, size = 522, normalized size = 0.92 \[ \frac{(e x)^{9/2} \left (\frac{14 x^2 \left (a^2 d \left (7 c-9 d x^2\right )+4 a b \left (2 c^2-4 c d x^2+3 d^2 x^4\right )+b^2 c x^2 \left (3 d x^2-5 c\right )\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c \left (7 a^2 d \left (7 c-9 d x^2\right )+4 a b \left (14 c^2-25 c d x^2+18 d^2 x^4\right )+2 b^2 c x^2 \left (9 d x^2-16 c\right )\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (d x^2-c\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{49 a^2 c (7 a d+8 b c) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{42 x^3 \left (a-b x^2\right ) \sqrt{c-d x^2} (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

((e*x)^(9/2)*((49*a^2*c*(8*b*c + 7*a*d)*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, 1
1/4, (d*x^2)/c, (b*x^2)/a])) + (11*a*c*(7*a^2*d*(7*c - 9*d*x^2) + 2*b^2*c*x^2*(-
16*c + 9*d*x^2) + 4*a*b*(14*c^2 - 25*c*d*x^2 + 18*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*(7*c - 9*d*x^2) + b^2*c*x^2*(-5*c
+ 3*d*x^2) + 4*a*b*(2*c^2 - 4*c*d*x^2 + 3*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 + 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*AppellF
1[11/4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a])))))/(42*(-(b*c) + a*d)^3*x^3*(a - b
*x^2)*Sqrt[c - d*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.09, size = 5126, normalized size = 9. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{9}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="maxima")

[Out]

integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)

_______________________________________________________________________________________

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((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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)**(9/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{9}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)),x, algorithm="giac")

[Out]

integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)