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

Optimal. Leaf size=372 \[ -\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-5 a d (a d+2 b c)\right )}{5 c d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3} \]

[Out]

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

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Rubi [A]  time = 0.826278, antiderivative size = 372, 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 \[ -\frac{2 a^2 \sqrt{c+d x^2}}{c e \sqrt{e x}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (3 b^2 c^2-5 a d (a d+2 b c)\right )}{5 c d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (3 b^2 c^2-5 a d (a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 81.2749, size = 347, normalized size = 0.93 \[ - \frac{2 a^{2} \sqrt{c + d x^{2}}}{c e \sqrt{e x}} + \frac{2 b^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{5 d e^{3}} - \frac{2 \sqrt{e x} \sqrt{c + d x^{2}} \left (- 5 a d \left (a d + 2 b c\right ) + 3 b^{2} c^{2}\right )}{5 c d^{\frac{3}{2}} e^{2} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 5 a d \left (a d + 2 b c\right ) + 3 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 5 a d \left (a d + 2 b c\right ) + 3 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} \]

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)**(1/2),x)

[Out]

-2*a**2*sqrt(c + d*x**2)/(c*e*sqrt(e*x)) + 2*b**2*(e*x)**(3/2)*sqrt(c + d*x**2)/
(5*d*e**3) - 2*sqrt(e*x)*sqrt(c + d*x**2)*(-5*a*d*(a*d + 2*b*c) + 3*b**2*c**2)/(
5*c*d**(3/2)*e**2*(sqrt(c) + sqrt(d)*x)) + 2*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d
)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-5*a*d*(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)/(5*c**(3/4)*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)*(-5*a*d*(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)/(5*c**(3/4)*d**(7/4)*e**(3/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.70417, size = 200, normalized size = 0.54 \[ \frac{2 x \left (d \left (c+d x^2\right ) \left (b^2 c x^2-5 a^2 d\right )+x^{3/2} \left (-5 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )\right )}{5 c d^2 (e x)^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*(-5*a^2*d + b^2*c*x^2)*(c + d*x^2) + (3*b^2*c^2 - 10*a*b*c*d - 5*a^2*d^2
)*x^(3/2)*(-((d + c/x^2)*Sqrt[x]) + (I*c*Sqrt[1 + c/(d*x^2)]*(EllipticE[I*ArcSin
h[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])
/Sqrt[d]]/Sqrt[x]], -1]))/((I*Sqrt[c])/Sqrt[d])^(3/2))))/(5*c*d^2*(e*x)^(3/2)*Sq
rt[c + d*x^2])

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Maple [A]  time = 0.032, size = 595, normalized size = 1.6 \[{\frac{1}{5\,e{d}^{2}c} \left ( 10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+20\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+2\,{x}^{4}{b}^{2}c{d}^{2}-10\,{x}^{2}{a}^{2}{d}^{3}+2\,{x}^{2}{b}^{2}{c}^{2}d-10\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

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)^(1/2),x)

[Out]

1/5*(10*((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+20*((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)*Ellipt
icE(((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^3-5*((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^2-10*((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+2*x^4*b^2*c*d^2-10*x^2*a^2*d^3+2*x^2*b^2*c^2*d-10*a^2*c*
d^2)/(d*x^2+c)^(1/2)/d^2/e/(e*x)^(1/2)/c

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

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 = 25.1243, size = 148, normalized size = 0.4 \[ \frac{a^{2} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{a b x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{c} e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{b^{2} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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