3.854 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=393 \[ -\frac{(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt{c+d x^2}}+\frac{\sqrt{e x} \sqrt{c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.861488, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt{c+d x^2}}+\frac{\sqrt{e x} \sqrt{c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{2 a^2}{c e \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 84.9013, size = 359, normalized size = 0.91 \[ - \frac{2 a^{2}}{c e \sqrt{e x} \sqrt{c + d x^{2}}} - \frac{\left (e x\right )^{\frac{3}{2}} \left (a d \left (3 a d - 2 b c\right ) + b^{2} c^{2}\right )}{c^{2} d e^{3} \sqrt{c + d x^{2}}} + \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (a d \left (3 a d - 2 b c\right ) + 3 b^{2} c^{2}\right )}{c^{2} d^{\frac{3}{2}} e^{2} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (3 a d - 2 b c\right ) + 3 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{7}{4}} d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (3 a d - 2 b c\right ) + 3 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 c^{\frac{7}{4}} d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(c*e*sqrt(e*x)*sqrt(c + d*x**2)) - (e*x)**(3/2)*(a*d*(3*a*d - 2*b*c) + b
**2*c**2)/(c**2*d*e**3*sqrt(c + d*x**2)) + sqrt(e*x)*sqrt(c + d*x**2)*(a*d*(3*a*
d - 2*b*c) + 3*b**2*c**2)/(c**2*d**(3/2)*e**2*(sqrt(c) + sqrt(d)*x)) - sqrt((c +
d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(a*d*(3*a*d - 2*b*c) +
3*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(c**
(7/4)*d**(7/4)*e**(3/2)*sqrt(c + d*x**2)) + sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)
*x)**2)*(sqrt(c) + sqrt(d)*x)*(a*d*(3*a*d - 2*b*c) + 3*b**2*c**2)*elliptic_f(2*a
tan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(2*c**(7/4)*d**(7/4)*e**(3/2)*s
qrt(c + d*x**2))

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Mathematica [C]  time = 0.568719, size = 250, normalized size = 0.64 \[ \frac{x \left (-\sqrt{c} x \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-\sqrt{d} \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \left (a^2 d \left (2 c+3 d x^2\right )-2 a b c d x^2+b^2 c^2 x^2\right )\right )}{c^2 d^{3/2} (e x)^{3/2} \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-(Sqrt[d]*Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]*(b^2*c^2*x^2 - 2*a*b*c*d*x^2 + a^2*d*(
2*c + 3*d*x^2))) + Sqrt[c]*(3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (d*x^2
)/c]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1] - Sqrt[c]*(3*b^2*c^2
- 2*a*b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[
d]*x)/Sqrt[c]]], -1]))/(c^2*d^(3/2)*Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]*(e*x)^(3/2)*Sqrt
[c + d*x^2])

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Maple [A]  time = 0.034, size = 594, normalized size = 1.5 \[{\frac{1}{2\,{d}^{2}e{c}^{2}} \left ( 6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-4\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+2\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-6\,{x}^{2}{a}^{2}{d}^{3}+4\,{x}^{2}abc{d}^{2}-2\,{x}^{2}{b}^{2}{c}^{2}d-4\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*
d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(
1/2))^(1/2),1/2*2^(1/2))*a^2*c*d^2-4*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(
1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*Elliptic
E(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*d+6*((d*x+(-c*d)^
(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/
(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1
/2))*b^2*c^3-3*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/
2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/
(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c*d^2+2*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(
1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*
EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*d-3*((d*x
+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1
/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),
1/2*2^(1/2))*b^2*c^3-6*x^2*a^2*d^3+4*x^2*a*b*c*d^2-2*x^2*b^2*c^2*d-4*a^2*c*d^2)/
(d*x^2+c)^(1/2)/d^2/e/(e*x)^(1/2)/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}{{\left (d e x^{3} + c e x\right )} \sqrt{d x^{2} + c} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/((d*e*x^3 + c*e*x)*sqrt(d*x^2 + c)*sqrt(e*x
)), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(3/2)), x)