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

Optimal. Leaf size=625 \[ -\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{2 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b (e x)^{3/2}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (e x)^{3/2} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d)*(e*x)^(3/2))/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*(
e*x)^(3/2))/(2*a*(b*c - a*d)*e*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 5*
a*b*c*d - a^2*d^2)*(e*x)^(3/2))/(2*a*c^2*(b*c - a*d)^3*e*Sqrt[c - d*x^2]) - (d^(
1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSi
n[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*c^(5/4)*(b*c - a*d)^3*Sqrt[c
- d*x^2]) + (d^(1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c
]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*c^(5/4)*(b*
c - a*d)^3*Sqrt[c - d*x^2]) - (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*Sqrt[e]*Sqrt[1 - (
d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqr
t[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2
]) + (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqr
t[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
-1])/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 3.87844, antiderivative size = 625, 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 \[ -\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{2 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b (e x)^{3/2}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (e x)^{3/2} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(3*b*c + 2*a*d)*(e*x)^(3/2))/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*(
e*x)^(3/2))/(2*a*(b*c - a*d)*e*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 5*
a*b*c*d - a^2*d^2)*(e*x)^(3/2))/(2*a*c^2*(b*c - a*d)^3*e*Sqrt[c - d*x^2]) - (d^(
1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSi
n[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*c^(5/4)*(b*c - a*d)^3*Sqrt[c
- d*x^2]) + (d^(1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c
]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*c^(5/4)*(b*
c - a*d)^3*Sqrt[c - d*x^2]) - (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*Sqrt[e]*Sqrt[1 - (
d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqr
t[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(3/2)*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2
]) + (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqr
t[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
-1])/(4*a^(3/2)*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)**(1/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 2.42885, size = 626, normalized size = 1. \[ \frac{x \sqrt{e x} \left (\frac{14 x^2 \left (a^3 d^3 \left (3 d x^2-5 c\right )+a^2 b d^2 \left (17 c^2-10 c d x^2-3 d^2 x^4\right )+a b^2 c d^2 x^2 \left (15 d x^2-17 c\right )+3 b^3 c^2 \left (c-d x^2\right )^2\right ) \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-11 a c \left (7 a^3 d^3 \left (5 c-3 d x^2\right )+a^2 b d^2 \left (-119 c^2+73 c d x^2+18 d^2 x^4\right )+2 a b^2 c d^2 x^2 \left (52 c-45 d x^2\right )-3 b^3 c^2 \left (7 c^2-13 c d x^2+6 d^2 x^4\right )\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{a \left (d x^2-c\right ) (a d-b c)^3 \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{49 c \left (a^3 d^3-5 a^2 b c d^2-12 a b^2 c^2 d+b^3 c^3\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{(b c-a d)^3 \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}\right )}{42 c^2 \left (a-b x^2\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*Sqrt[e*x]*((49*c*(b^3*c^3 - 12*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*AppellF
1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a])/((b*c - a*d)^3*(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*(2*a*b^2*c*d^2*x^2*(52*c - 45*d*x^2) + 7*a^3*d^3*(5*c - 3*d*x^2) - 3*b^
3*c^2*(7*c^2 - 13*c*d*x^2 + 6*d^2*x^4) + a^2*b*d^2*(-119*c^2 + 73*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*(3*b^3*c^2*(
c - d*x^2)^2 + a^3*d^3*(-5*c + 3*d*x^2) + a*b^2*c*d^2*x^2*(-17*c + 15*d*x^2) + a
^2*b*d^2*(17*c^2 - 10*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]))
/(a*(-(b*c) + a*d)^3*(-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*c^2*(a - b*x^2)*
Sqrt[c - d*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.066, size = 5689, normalized size = 9.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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