3.811 \(\int \frac{A+B x^2}{\sqrt{e x} \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{5/4} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{\sqrt{e x} (A b-a B)}{a b e \sqrt{a+b x^2}} \]

[Out]

((A*b - a*B)*Sqrt[e*x])/(a*b*e*Sqrt[a + b*x^2]) + ((A*b + a*B)*(Sqrt[a] + Sqrt[b
]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[
e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*a^(5/4)*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.247807, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{5/4} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{\sqrt{e x} (A b-a B)}{a b e \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*Sqrt[e*x])/(a*b*e*Sqrt[a + b*x^2]) + ((A*b + a*B)*(Sqrt[a] + Sqrt[b
]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[
e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*a^(5/4)*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 23.7532, size = 126, normalized size = 0.88 \[ \frac{\sqrt{e x} \left (A b - B a\right )}{a b e \sqrt{a + b x^{2}}} + \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{5}{4}} b^{\frac{5}{4}} \sqrt{e} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(e*x)*(A*b - B*a)/(a*b*e*sqrt(a + b*x**2)) + sqrt((a + b*x**2)/(sqrt(a) + sq
rt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b + B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e
*x)/(a**(1/4)*sqrt(e))), 1/2)/(2*a**(5/4)*b**(5/4)*sqrt(e)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.154984, size = 133, normalized size = 0.92 \[ \frac{i x^{3/2} \sqrt{\frac{a}{b x^2}+1} (a B+A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )+x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} (A b-a B)}{a b \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{e x} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(A*b - a*B)*x + I*(A*b + a*B)*Sqrt[1 + a/(b*x^2)]*x^(
3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/(a*Sqrt[(I*Sqr
t[a])/Sqrt[b]]*b*Sqrt[e*x]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.029, size = 213, normalized size = 1.5 \[{\frac{1}{2\,a{b}^{2}} \left ( A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}b+B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}a+2\,Ax{b}^{2}-2\,Bxab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*
b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(
1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*b+B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2
)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*Ell
ipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a+2*A*x
*b^2-2*B*x*a*b)/(b*x^2+a)^(1/2)/a/(e*x)^(1/2)/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*sqrt(e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)/((b*x^2 + a)^(3/2)*sqrt(e*x)), x)

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Sympy [A]  time = 71.0387, size = 94, normalized size = 0.65 \[ \frac{A \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{e} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*sqrt(x)*gamma(1/4)*hyper((1/4, 3/2), (5/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(
3/2)*sqrt(e)*gamma(5/4)) + B*x**(5/2)*gamma(5/4)*hyper((5/4, 3/2), (9/4,), b*x**
2*exp_polar(I*pi)/a)/(2*a**(3/2)*sqrt(e)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*sqrt(e*x)), x)