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

Optimal. Leaf size=476 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt{e x}}-\frac{8 c \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{117 c d e^3}-\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d e^3}-\frac{4 c^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{8 c^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3} \]

[Out]

(-4*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(195*d*e^3
) - (8*c*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^
(3/2)*e^2*(Sqrt[c] + Sqrt[d]*x)) - (2*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)
^(3/2)*(c + d*x^2)^(3/2))/(117*c*d*e^3) - (2*a^2*(c + d*x^2)^(5/2))/(c*e*Sqrt[e*
x]) + (2*b^2*(e*x)^(3/2)*(c + d*x^2)^(5/2))/(13*d*e^3) + (8*c^(5/4)*(3*b^2*c^2 -
13*a*d*(2*b*c + 9*a*d))*(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])/(195*d
^(7/4)*e^(3/2)*Sqrt[c + d*x^2]) - (4*c^(5/4)*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d)
)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(195*d^(7/4)*e^(3/2)*Sqrt[c +
d*x^2])

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Rubi [A]  time = 1.08154, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt{e x}}-\frac{8 c \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{117 c d e^3}-\frac{4 (e x)^{3/2} \sqrt{c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d e^3}-\frac{4 c^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{8 c^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(195*d*e^3
) - (8*c*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^
(3/2)*e^2*(Sqrt[c] + Sqrt[d]*x)) - (2*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)
^(3/2)*(c + d*x^2)^(3/2))/(117*c*d*e^3) - (2*a^2*(c + d*x^2)^(5/2))/(c*e*Sqrt[e*
x]) + (2*b^2*(e*x)^(3/2)*(c + d*x^2)^(5/2))/(13*d*e^3) + (8*c^(5/4)*(3*b^2*c^2 -
13*a*d*(2*b*c + 9*a*d))*(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])/(195*d
^(7/4)*e^(3/2)*Sqrt[c + d*x^2]) - (4*c^(5/4)*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d)
)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(195*d^(7/4)*e^(3/2)*Sqrt[c +
d*x^2])

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Rubi in Sympy [A]  time = 105.974, size = 454, normalized size = 0.95 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{c e \sqrt{e x}} + \frac{2 b^{2} \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{5}{2}}}{13 d e^{3}} + \frac{8 c^{\frac{5}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 13 a d \left (9 a d + 2 b c\right ) + 3 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{4 c^{\frac{5}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 13 a d \left (9 a d + 2 b c\right ) + 3 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{8 c \sqrt{e x} \sqrt{c + d x^{2}} \left (- 13 a d \left (9 a d + 2 b c\right ) + 3 b^{2} c^{2}\right )}{195 d^{\frac{3}{2}} e^{2} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{4 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (- 13 a d \left (9 a d + 2 b c\right ) + 3 b^{2} c^{2}\right )}{195 d e^{3}} - \frac{2 \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (- 13 a d \left (9 a d + 2 b c\right ) + 3 b^{2} c^{2}\right )}{117 c d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*(c + d*x**2)**(5/2)/(c*e*sqrt(e*x)) + 2*b**2*(e*x)**(3/2)*(c + d*x**2)**
(5/2)/(13*d*e**3) + 8*c**(5/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt
(c) + sqrt(d)*x)*(-13*a*d*(9*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)/(195*d**(7/4)*e**(3/2)*sqrt(c + d*x**2))
- 4*c**(5/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*
(-13*a*d*(9*a*d + 2*b*c) + 3*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c*
*(1/4)*sqrt(e))), 1/2)/(195*d**(7/4)*e**(3/2)*sqrt(c + d*x**2)) - 8*c*sqrt(e*x)*
sqrt(c + d*x**2)*(-13*a*d*(9*a*d + 2*b*c) + 3*b**2*c**2)/(195*d**(3/2)*e**2*(sqr
t(c) + sqrt(d)*x)) - 4*(e*x)**(3/2)*sqrt(c + d*x**2)*(-13*a*d*(9*a*d + 2*b*c) +
3*b**2*c**2)/(195*d*e**3) - 2*(e*x)**(3/2)*(c + d*x**2)**(3/2)*(-13*a*d*(9*a*d +
2*b*c) + 3*b**2*c**2)/(117*c*d*e**3)

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Mathematica [C]  time = 1.65051, size = 261, normalized size = 0.55 \[ \frac{x^{3/2} \left (\frac{2 \sqrt{c+d x^2} \left (117 a^2 d \left (d x^2-5 c\right )+26 a b d x^2 \left (11 c+5 d x^2\right )+3 b^2 x^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )\right )}{3 d \sqrt{x}}-\frac{8 c x \left (117 a^2 d^2+26 a b c d-3 b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )}{d^2 \sqrt{c+d x^2}}\right )}{195 (e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(3/2)*((2*Sqrt[c + d*x^2]*(117*a^2*d*(-5*c + d*x^2) + 26*a*b*d*x^2*(11*c + 5*
d*x^2) + 3*b^2*x^2*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4)))/(3*d*Sqrt[x]) - (8*c*(-3*
b^2*c^2 + 26*a*b*c*d + 117*a^2*d^2)*x*(-((d + c/x^2)*Sqrt[x]) + (I*c*Sqrt[1 + c/
(d*x^2)]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] - Elliptic
F[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/((I*Sqrt[c])/Sqrt[d])^(3/2
)))/(d^2*Sqrt[c + d*x^2])))/(195*(e*x)^(3/2))

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Maple [A]  time = 0.032, size = 669, normalized size = 1.4 \[{\frac{2}{585\,e{d}^{2}} \left ( 45\,{x}^{8}{b}^{2}{d}^{4}+130\,{x}^{6}ab{d}^{4}+120\,{x}^{6}{b}^{2}c{d}^{3}+1404\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}+312\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d-36\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}-702\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}-156\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d+18\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}+117\,{x}^{4}{a}^{2}{d}^{4}+416\,{x}^{4}abc{d}^{3}+87\,{x}^{4}{b}^{2}{c}^{2}{d}^{2}-468\,{x}^{2}{a}^{2}c{d}^{3}+286\,{x}^{2}ab{c}^{2}{d}^{2}+12\,{x}^{2}{b}^{2}{c}^{3}d-585\,{a}^{2}{c}^{2}{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*(45*x^8*b^2*d^4+130*x^6*a*b*d^4+120*x^6*b^2*c*d^3+1404*((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^2*d^2+312*((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*b*c^3*d-36*((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^4-702*((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^2*d^2-156*((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^3*d+18*((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^4+117*x^4*a^2*d^4+416*x^4*a*b*c*d^3+87*x^4*b^2*c^2*d^2-468*x^2*a^2*c*d^3+28
6*x^2*a*b*c^2*d^2+12*x^2*b^2*c^3*d-585*a^2*c^2*d^2)/(d*x^2+c)^(1/2)/d^2/e/(e*x)^
(1/2)

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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. \[{\rm integral}\left (\frac{{\left (b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x} e x}, x\right ) \]

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*d*x^6 + (b^2*c + 2*a*b*d)*x^4 + a^2*c + (2*a*b*c + a^2*d)*x^2)*sqr
t(d*x^2 + c)/(sqrt(e*x)*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*(d*x**2+c)**(3/2)/(e*x)**(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)