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

Optimal. Leaf size=193 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \sqrt{c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d e^3} \]

[Out]

(-2*b*(5*b*c - 14*a*d)*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*d^2*e) + (2*b^2*(e*x)^(5/2
)*Sqrt[c + d*x^2])/(7*d*e^3) + ((5*b^2*c^2 - 14*a*b*c*d + 21*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])/(21*c^(1/4)*d^(9/4)*Sqrt[e]*Sqrt[c + d*x^
2])

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Rubi [A]  time = 0.413634, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \sqrt{c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*(5*b*c - 14*a*d)*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*d^2*e) + (2*b^2*(e*x)^(5/2
)*Sqrt[c + d*x^2])/(7*d*e^3) + ((5*b^2*c^2 - 14*a*b*c*d + 21*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])/(21*c^(1/4)*d^(9/4)*Sqrt[e]*Sqrt[c + d*x^
2])

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Rubi in Sympy [A]  time = 41.0705, size = 178, normalized size = 0.92 \[ \frac{2 b^{2} \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}{7 d e^{3}} + \frac{2 b \sqrt{e x} \sqrt{c + d x^{2}} \left (14 a d - 5 b c\right )}{21 d^{2} e} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (21 a^{2} d^{2} - b c \left (14 a d - 5 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{21 \sqrt [4]{c} d^{\frac{9}{4}} \sqrt{e} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(5/2)*sqrt(c + d*x**2)/(7*d*e**3) + 2*b*sqrt(e*x)*sqrt(c + d*x**2)
*(14*a*d - 5*b*c)/(21*d**2*e) + sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqr
t(c) + sqrt(d)*x)*(21*a**2*d**2 - b*c*(14*a*d - 5*b*c))*elliptic_f(2*atan(d**(1/
4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(21*c**(1/4)*d**(9/4)*sqrt(e)*sqrt(c + d*
x**2))

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Mathematica [C]  time = 0.373638, size = 148, normalized size = 0.77 \[ \frac{2 x \left (-b \left (c+d x^2\right ) \left (-14 a d+5 b c-3 b d x^2\right )+\frac{i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{21 d^2 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-(b*(c + d*x^2)*(5*b*c - 14*a*d - 3*b*d*x^2)) + (I*(5*b^2*c^2 - 14*a*b*c*d
+ 21*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/
Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(21*d^2*Sqrt[e*x]*Sqrt[c + d
*x^2])

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Maple [A]  time = 0.028, size = 350, normalized size = 1.8 \[{\frac{1}{21\,{d}^{3}} \left ( 21\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{d}^{2}-14\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) abcd+5\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{2}+6\,{x}^{5}{b}^{2}{d}^{3}+28\,{x}^{3}ab{d}^{3}-4\,{x}^{3}{b}^{2}c{d}^{2}+28\,xabc{d}^{2}-10\,x{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21/(d*x^2+c)^(1/2)*(21*(-c*d)^(1/2)*((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)*Ellipti
cF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*d^2-14*(-c*d)^(1/2)*
((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*d+5*(-c*d)^(1/2)*((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)*Elli
pticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2+6*x^5*b^2*d^3
+28*x^3*a*b*d^3-4*x^3*b^2*c*d^2+28*x*a*b*c*d^2-10*x*b^2*c^2*d)/(e*x)^(1/2)/d^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.3917, size = 144, normalized size = 0.75 \[ \frac{a^{2} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{a b x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{c} \sqrt{e} \Gamma \left (\frac{9}{4}\right )} + \frac{b^{2} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} \sqrt{e} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), d*x**2*exp_polar(I*pi)/c)/(2*s
qrt(c)*sqrt(e)*gamma(5/4)) + a*b*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), d
*x**2*exp_polar(I*pi)/c)/(sqrt(c)*sqrt(e)*gamma(9/4)) + b**2*x**(9/2)*gamma(9/4)
*hyper((1/2, 9/4), (13/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*sqrt(e)*gamma(1
3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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