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

Optimal. Leaf size=421 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt{e x}}-\frac{2 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 c d e^3}+\frac{2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3} \]

[Out]

(-2*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(15*c*d*e^3)
- (4*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*d^(3/2)*e^
2*(Sqrt[c] + Sqrt[d]*x)) - (2*a^2*(c + d*x^2)^(3/2))/(c*e*Sqrt[e*x]) + (2*b^2*(e
*x)^(3/2)*(c + d*x^2)^(3/2))/(9*d*e^3) + (4*c^(1/4)*(b^2*c^2 - 3*a*d*(2*b*c + 5*
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])/(15*d^(7/4)*e^(3/2)*Sqrt[
c + d*x^2]) - (2*c^(1/4)*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(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])/(15*d^(7/4)*e^(3/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.946531, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt{e x}}-\frac{2 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{4 \sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{4 \sqrt{e x} \sqrt{c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 (e x)^{3/2} \sqrt{c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 c d e^3}+\frac{2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(15*c*d*e^3)
- (4*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*d^(3/2)*e^
2*(Sqrt[c] + Sqrt[d]*x)) - (2*a^2*(c + d*x^2)^(3/2))/(c*e*Sqrt[e*x]) + (2*b^2*(e
*x)^(3/2)*(c + d*x^2)^(3/2))/(9*d*e^3) + (4*c^(1/4)*(b^2*c^2 - 3*a*d*(2*b*c + 5*
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])/(15*d^(7/4)*e^(3/2)*Sqrt[
c + d*x^2]) - (2*c^(1/4)*(b^2*c^2 - 3*a*d*(2*b*c + 5*a*d))*(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])/(15*d^(7/4)*e^(3/2)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 94.0651, size = 396, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{c e \sqrt{e x}} + \frac{2 b^{2} \left (e x\right )^{\frac{3}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}{9 d e^{3}} + \frac{4 \sqrt [4]{c} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 3 a d \left (5 a d + 2 b c\right ) + b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{2 \sqrt [4]{c} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 3 a d \left (5 a d + 2 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{4 \sqrt{e x} \sqrt{c + d x^{2}} \left (- 3 a d \left (5 a d + 2 b c\right ) + b^{2} c^{2}\right )}{15 d^{\frac{3}{2}} e^{2} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (- 3 a d \left (5 a d + 2 b c\right ) + b^{2} c^{2}\right )}{15 c d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/(e*x)**(3/2),x)

[Out]

-2*a**2*(c + d*x**2)**(3/2)/(c*e*sqrt(e*x)) + 2*b**2*(e*x)**(3/2)*(c + d*x**2)**
(3/2)/(9*d*e**3) + 4*c**(1/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(
c) + sqrt(d)*x)*(-3*a*d*(5*a*d + 2*b*c) + b**2*c**2)*elliptic_e(2*atan(d**(1/4)*
sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(15*d**(7/4)*e**(3/2)*sqrt(c + d*x**2)) - 2*
c**(1/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-3*a
*d*(5*a*d + 2*b*c) + b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*s
qrt(e))), 1/2)/(15*d**(7/4)*e**(3/2)*sqrt(c + d*x**2)) - 4*sqrt(e*x)*sqrt(c + d*
x**2)*(-3*a*d*(5*a*d + 2*b*c) + b**2*c**2)/(15*d**(3/2)*e**2*(sqrt(c) + sqrt(d)*
x)) - 2*(e*x)**(3/2)*sqrt(c + d*x**2)*(-3*a*d*(5*a*d + 2*b*c) + b**2*c**2)/(15*c
*d*e**3)

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Mathematica [C]  time = 0.851718, size = 260, normalized size = 0.62 \[ \frac{2 x \left (d \left (c+d x^2\right ) \left (-45 a^2 d+18 a b d x^2+b^2 x^2 \left (2 c+5 d x^2\right )\right )-\frac{6 \left (-15 a^2 d^2-6 a b c d+b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{45 d^2 (e x)^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*(c + d*x^2)*(-45*a^2*d + 18*a*b*d*x^2 + b^2*x^2*(2*c + 5*d*x^2)) - (6*(b
^2*c^2 - 6*a*b*c*d - 15*a^2*d^2)*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(c + d*x^2) - Sqrt[c
]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[
d]]/Sqrt[x]], -1] + Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticF[I*ArcS
inh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(45*d^
2*(e*x)^(3/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.055, size = 624, normalized size = 1.5 \[{\frac{2}{45\,e{d}^{2}} \left ( 5\,{x}^{6}{b}^{2}{d}^{3}+90\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+36\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-45\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-18\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+18\,{x}^{4}ab{d}^{3}+7\,{x}^{4}{b}^{2}c{d}^{2}-45\,{x}^{2}{a}^{2}{d}^{3}+18\,{x}^{2}abc{d}^{2}+2\,{x}^{2}{b}^{2}{c}^{2}d-45\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(3/2),x)

[Out]

2/45*(5*x^6*b^2*d^3+90*((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+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))*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^3-45*((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^2-18*((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)
)*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+18*x^4*a*b*d^3+7*x^4*b^2*c*d^2-45*x^2*a^
2*d^3+18*x^2*a*b*c*d^2+2*x^2*b^2*c^2*d-45*a^2*c*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. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{\left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 30.7029, size = 153, normalized size = 0.36 \[ \frac{a^{2} \sqrt{c} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{a b \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{b^{2} \sqrt{c} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/(e*x)**(3/2),x)

[Out]

a**2*sqrt(c)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), d*x**2*exp_polar(I*pi)/c)/(
2*e**(3/2)*sqrt(x)*gamma(3/4)) + a*b*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/
4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(e**(3/2)*gamma(7/4)) + b**2*sqrt(c)*x**(7
/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(3/2)
*gamma(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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