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

Optimal. Leaf size=290 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{2 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{a \sqrt{b} e^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 A \sqrt{a+b x^2}}{a e \sqrt{e x}} \]

[Out]

(-2*A*Sqrt[a + b*x^2])/(a*e*Sqrt[e*x]) + (2*(A*b + a*B)*Sqrt[e*x]*Sqrt[a + b*x^2
])/(a*Sqrt[b]*e^2*(Sqrt[a] + Sqrt[b]*x)) - (2*(A*b + a*B)*(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^(3/4)*e^(3/2)*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])/(a^(3/4)*b^(3/4)*e^(3/2)*Sq
rt[a + b*x^2])

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Rubi [A]  time = 0.552984, antiderivative size = 290, 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 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{2 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{a \sqrt{b} e^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 A \sqrt{a+b x^2}}{a e \sqrt{e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*Sqrt[a + b*x^2])/(a*e*Sqrt[e*x]) + (2*(A*b + a*B)*Sqrt[e*x]*Sqrt[a + b*x^2
])/(a*Sqrt[b]*e^2*(Sqrt[a] + Sqrt[b]*x)) - (2*(A*b + a*B)*(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^(3/4)*e^(3/2)*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])/(a^(3/4)*b^(3/4)*e^(3/2)*Sq
rt[a + b*x^2])

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Rubi in Sympy [A]  time = 55.4627, size = 265, normalized size = 0.91 \[ - \frac{2 A \sqrt{a + b x^{2}}}{a e \sqrt{e x}} + \frac{2 \sqrt{e x} \sqrt{a + b x^{2}} \left (A b + B a\right )}{a \sqrt{b} e^{2} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{2 \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} b^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} b^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.412106, size = 193, normalized size = 0.67 \[ \frac{x \left (2 \sqrt{a} x \sqrt{\frac{b x^2}{a}+1} (a B+A b) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-2 \left (\sqrt{a} x \sqrt{\frac{b x^2}{a}+1} (a B+A b) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+A \sqrt{b} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (a+b x^2\right )\right )\right )}{a \sqrt{b} (e x)^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 378, normalized size = 1.3 \[{\frac{1}{bea} \left ( 2\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) ab+2\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}-B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){a}^{2}-2\,A{x}^{2}{b}^{2}-2\,abA \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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+2*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-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*a*b*A)/(b*x^2+a)^(1/2)/b/e/(e*x)^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.3657, size = 97, normalized size = 0.33 \[ \frac{A \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{B x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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