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

Optimal. Leaf size=299 \[ \frac{2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-3 a B)}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{\sqrt [4]{a} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^2}}-\frac{2 \sqrt [4]{a} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^2}}+\frac{2 B (e x)^{3/2} \sqrt{a+b x^2}}{5 b e} \]

[Out]

(2*B*(e*x)^(3/2)*Sqrt[a + b*x^2])/(5*b*e) + (2*(5*A*b - 3*a*B)*Sqrt[e*x]*Sqrt[a
+ b*x^2])/(5*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) - (2*a^(1/4)*(5*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*Arc
Tan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*b^(7/4)*Sqrt[a + b*x^2]) +
(a^(1/4)*(5*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]
)/(5*b^(7/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.546146, antiderivative size = 299, 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 \[ \frac{2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-3 a B)}{5 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{\sqrt [4]{a} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^2}}-\frac{2 \sqrt [4]{a} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{a+b x^2}}+\frac{2 B (e x)^{3/2} \sqrt{a+b x^2}}{5 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*(e*x)^(3/2)*Sqrt[a + b*x^2])/(5*b*e) + (2*(5*A*b - 3*a*B)*Sqrt[e*x]*Sqrt[a
+ b*x^2])/(5*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) - (2*a^(1/4)*(5*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*Arc
Tan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*b^(7/4)*Sqrt[a + b*x^2]) +
(a^(1/4)*(5*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]
)/(5*b^(7/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 53.9259, size = 277, normalized size = 0.93 \[ \frac{2 B \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{5 b e} - \frac{2 \sqrt [4]{a} \sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 3 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} + \frac{\sqrt [4]{a} \sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{e x} \sqrt{a + b x^{2}} \left (5 A b - 3 B a\right )}{5 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} \]

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*(e*x)**(3/2)*sqrt(a + b*x**2)/(5*b*e) - 2*a**(1/4)*sqrt(e)*sqrt((a + b*x**2)
/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(5*A*b - 3*B*a)*elliptic_e(2*at
an(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(5*b**(7/4)*sqrt(a + b*x**2)) +
a**(1/4)*sqrt(e)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*
x)*(5*A*b - 3*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2
)/(5*b**(7/4)*sqrt(a + b*x**2)) + 2*sqrt(e*x)*sqrt(a + b*x**2)*(5*A*b - 3*B*a)/(
5*b**(3/2)*(sqrt(a) + sqrt(b)*x))

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Mathematica [C]  time = 1.09915, size = 181, normalized size = 0.61 \[ \frac{2 \sqrt{e x} \left (B x^{3/2} \sqrt{a+b x^2}-\frac{x (5 A b-3 a B) \left (-\sqrt{x} \left (\frac{a}{x^2}+b\right )+\frac{i a \sqrt{\frac{a}{b x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2}}\right )}{b \sqrt{a+b x^2}}\right )}{5 b \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e*x]*(B*x^(3/2)*Sqrt[a + b*x^2] - ((5*A*b - 3*a*B)*x*(-((b + a/x^2)*Sqrt
[x]) + (I*a*Sqrt[1 + a/(b*x^2)]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/S
qrt[x]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))/((I
*Sqrt[a])/Sqrt[b])^(3/2)))/(b*Sqrt[a + b*x^2])))/(5*b*Sqrt[x])

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Maple [A]  time = 0.022, size = 379, normalized size = 1.3 \[{\frac{1}{5\,{b}^{2}x}\sqrt{ex} \left ( 10\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-6\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+3\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+2\,{b}^{2}B{x}^{4}+2\,B{x}^{2}ab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(e*x)^(1/2)/(b*x^2+a)^(1/2)/b^2*(10*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)*Ell
ipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b-5*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)*Ellip
ticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^2+2*b^2*B*x^4+2*B*x^
2*a*b)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.37777, size = 92, normalized size = 0.31 \[ \frac{A \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e \Gamma \left (\frac{7}{4}\right )} + \frac{B \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{3} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(e*x)**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2
*sqrt(a)*e*gamma(7/4)) + B*(e*x)**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), b*
x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*e**3*gamma(11/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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