3.803 \(\int \frac{A+B x^2}{\sqrt{e x} \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=139 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e} \]

[Out]

(2*B*Sqrt[e*x]*Sqrt[a + b*x^2])/(3*b*e) + ((3*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*S
qrt[(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])/(3*a^(1/4)*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.234162, antiderivative size = 139, 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 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[e*x]*Sqrt[a + b*x^2])/(3*b*e) + ((3*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*S
qrt[(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])/(3*a^(1/4)*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 22.1353, size = 124, normalized size = 0.89 \[ \frac{2 B \sqrt{e x} \sqrt{a + b x^{2}}}{3 b e} + \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{a} b^{\frac{5}{4}} \sqrt{e} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(e*x)*sqrt(a + b*x**2)/(3*b*e) + sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)
**2)*(sqrt(a) + sqrt(b)*x)*(3*A*b - B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a
**(1/4)*sqrt(e))), 1/2)/(3*a**(1/4)*b**(5/4)*sqrt(e)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.470076, size = 116, normalized size = 0.83 \[ \frac{2 x \left (B \left (a+b x^2\right )-\frac{i \sqrt{x} \sqrt{\frac{a}{b x^2}+1} (a B-3 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{3 b \sqrt{e x} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 214, normalized size = 1.5 \[{\frac{1}{3\,{b}^{2}} \left ( 3\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}b-B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}a+2\,{b}^{2}B{x}^{3}+2\,Bxab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.68379, size = 94, normalized size = 0.68 \[ \frac{A \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \sqrt{e} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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