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

Optimal. Leaf size=333 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt{a+b x^2}}+\frac{\sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{a^2 \sqrt{b} e^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^2}} \]

[Out]

(-2*A)/(a*e*Sqrt[e*x]*Sqrt[a + b*x^2]) - ((3*A*b - a*B)*(e*x)^(3/2))/(a^2*e^3*Sq
rt[a + b*x^2]) + ((3*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(a^2*Sqrt[b]*e^2*(Sqr
t[a] + Sqrt[b]*x)) - ((3*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^(7/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2]) + ((3*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^(7/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.642021, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt{a+b x^2}}-\frac{(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt{a+b x^2}}+\frac{\sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{a^2 \sqrt{b} e^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(a*e*Sqrt[e*x]*Sqrt[a + b*x^2]) - ((3*A*b - a*B)*(e*x)^(3/2))/(a^2*e^3*Sq
rt[a + b*x^2]) + ((3*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(a^2*Sqrt[b]*e^2*(Sqr
t[a] + Sqrt[b]*x)) - ((3*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^(7/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2]) + ((3*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^(7/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 66.8392, size = 303, normalized size = 0.91 \[ - \frac{2 A}{a e \sqrt{e x} \sqrt{a + b x^{2}}} - \frac{\left (e x\right )^{\frac{3}{2}} \left (3 A b - B a\right )}{a^{2} e^{3} \sqrt{a + b x^{2}}} + \frac{\sqrt{e x} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{a^{2} \sqrt{b} e^{2} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{7}{4}} b^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} b^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{a + b x^{2}}} \]

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)**(3/2),x)

[Out]

-2*A/(a*e*sqrt(e*x)*sqrt(a + b*x**2)) - (e*x)**(3/2)*(3*A*b - B*a)/(a**2*e**3*sq
rt(a + b*x**2)) + sqrt(e*x)*sqrt(a + b*x**2)*(3*A*b - B*a)/(a**2*sqrt(b)*e**2*(s
qrt(a) + sqrt(b)*x)) - sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sq
rt(b)*x)*(3*A*b - B*a)*elliptic_e(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))),
1/2)/(a**(7/4)*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)*(3*A*b - B*a)*elliptic_f(2*atan(b**(1/4)
*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(2*a**(7/4)*b**(3/4)*e**(3/2)*sqrt(a + b*x*
*2))

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Mathematica [C]  time = 0.400567, size = 202, normalized size = 0.61 \[ \frac{x \left (\sqrt{b} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (-2 a A+a B x^2-3 A b x^2\right )+\sqrt{a} x \sqrt{\frac{b x^2}{a}+1} (a B-3 A b) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-\sqrt{a} x \sqrt{\frac{b x^2}{a}+1} (a B-3 A b) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{a^2 \sqrt{b} (e x)^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[b]*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(-2*a*A - 3*A*b*x^2 + a*B*x^2) - Sqrt[a]
*(-3*A*b + a*B)*x*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqr
t[a]]], -1] + Sqrt[a]*(-3*A*b + a*B)*x*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[S
qrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1]))/(a^2*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 = 0.031, size = 386, normalized size = 1.2 \[{\frac{1}{2\,be{a}^{2}} \left ( 6\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-3\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-2\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){a}^{2}-6\,A{x}^{2}{b}^{2}+2\,B{x}^{2}ab-4\,abA \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*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-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-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^2-6*A*x^2*b^2+2*B*x^2*a*b-4*a*b*A)/(b*x^2+a)^(1/2)/b/e/
(e*x)^(1/2)/a^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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