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

Optimal. Leaf size=301 \[ -\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (A b-3 a B)}{a b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(e x)^{3/2} (A b-a B)}{a b e \sqrt{a+b x^2}} \]

[Out]

((A*b - a*B)*(e*x)^(3/2))/(a*b*e*Sqrt[a + b*x^2]) - ((A*b - 3*a*B)*Sqrt[e*x]*Sqr
t[a + b*x^2])/(a*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) + ((A*b - 3*a*B)*Sqrt[e]*(Sqrt[a
] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(
1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(a^(3/4)*b^(7/4)*Sqrt[a + b*x^2]) - ((
A*b - 3*a*B)*Sqrt[e]*(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^(3/4)
*b^(7/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.583796, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (A b-3 a B)}{a b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(e x)^{3/2} (A b-a B)}{a b e \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(e*x)^(3/2))/(a*b*e*Sqrt[a + b*x^2]) - ((A*b - 3*a*B)*Sqrt[e*x]*Sqr
t[a + b*x^2])/(a*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) + ((A*b - 3*a*B)*Sqrt[e]*(Sqrt[a
] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(
1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(a^(3/4)*b^(7/4)*Sqrt[a + b*x^2]) - ((
A*b - 3*a*B)*Sqrt[e]*(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^(3/4)
*b^(7/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 56.7525, size = 270, normalized size = 0.9 \[ \frac{\left (e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{a b e \sqrt{a + b x^{2}}} - \frac{\sqrt{e x} \sqrt{a + b x^{2}} \left (A b - 3 B a\right )}{a b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{\sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 3 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{\sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} b^{\frac{7}{4}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x)**(3/2)*(A*b - B*a)/(a*b*e*sqrt(a + b*x**2)) - sqrt(e*x)*sqrt(a + b*x**2)*(
A*b - 3*B*a)/(a*b**(3/2)*(sqrt(a) + sqrt(b)*x)) + sqrt(e)*sqrt((a + b*x**2)/(sqr
t(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b - 3*B*a)*elliptic_e(2*atan(b**(
1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(a**(3/4)*b**(7/4)*sqrt(a + b*x**2)) -
sqrt(e)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b -
3*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(2*a**(3/
4)*b**(7/4)*sqrt(a + b*x**2))

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Mathematica [C]  time = 1.01997, size = 216, normalized size = 0.72 \[ \frac{i e \left (\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (3 a B-A b+2 b B x^2\right )-\sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} (A b-3 a B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} (A b-3 a B) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{b^{5/2} \left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2} \sqrt{e x} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(I*e*(Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[b]]*(-(A*b) + 3*a*B + 2*b*B*x^2) + Sqrt[b]*(
A*b - 3*a*B)*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sq
rt[b]]/Sqrt[x]], -1] - Sqrt[b]*(A*b - 3*a*B)*Sqrt[1 + a/(b*x^2)]*x^(3/2)*Ellipti
cF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))/(((I*Sqrt[a])/Sqrt[b])^(3
/2)*b^(5/2)*Sqrt[e*x]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.026, size = 382, normalized size = 1.3 \[ -{\frac{1}{2\,{b}^{2}xa}\sqrt{ex} \left ( 2\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) ab-6\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+3\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}-2\,A{x}^{2}{b}^{2}+2\,B{x}^{2}ab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(e*x)^(1/2)*(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)*EllipticE(((b*x+(-a*b)^
(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b-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-6*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)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2
*2^(1/2))*a^2+3*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)*EllipticF(((b*x+(-a*b)^(1/2
))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^2-2*A*x^2*b^2+2*B*x^2*a*b)/(b*x^2+a)^(1/2)
/b^2/x/a

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 46.2766, size = 94, normalized size = 0.31 \[ \frac{A \sqrt{e} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} + \frac{B \sqrt{e} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((3/4, 3/2), (7/4,), b*x**2*exp_polar(I*pi)/a
)/(2*a**(3/2)*gamma(7/4)) + B*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((3/2, 7/4), (11/
4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*gamma(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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