3.805 \(\int \frac{A+B x^2}{(e x)^{5/2} \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 a^{5/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{2 A \sqrt{a+b x^2}}{3 a e (e x)^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.238926, antiderivative size = 138, 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 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 a^{5/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{2 A \sqrt{a+b x^2}}{3 a e (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/((e*x)^(5/2)*Sqrt[a + b*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.1752, size = 126, normalized size = 0.91 \[ - \frac{2 A \sqrt{a + b x^{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} - \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3 a^{\frac{5}{4}} \sqrt [4]{b} e^{\frac{5}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(e*x)**(5/2)/(b*x**2+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.437697, size = 118, normalized size = 0.86 \[ \frac{2 x \left (-A \left (a+b x^2\right )+\frac{i x^{5/2} \sqrt{\frac{a}{b x^2}+1} (3 a B-A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{3 a (e x)^{5/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/((e*x)^(5/2)*Sqrt[a + b*x^2]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 223, normalized size = 1.6 \[ -{\frac{1}{3\,abx{e}^{2}} \left ( A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}xb-3\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}xa+2\,A{x}^{2}{b}^{2}+2\,abA \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(e*x)^(5/2)/(b*x^2+a)^(1/2),x)

[Out]

-1/3/(b*x^2+a)^(1/2)/x*(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)*x*b-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*b)^(1/2)*x*a+2*A*x^2*b^2+2*a*b*A)/b/a/e^2/(e*x)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{b x^{2} + a} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(5/2)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 151.268, size = 97, normalized size = 0.7 \[ \frac{A \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} + \frac{B \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} e^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/(e*x)**(5/2)/(b*x**2+a)**(1/2),x)

[Out]

A*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*e*
*(5/2)*x**(3/2)*gamma(1/4)) + B*sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x
**2*exp_polar(I*pi)/a)/(2*sqrt(a)*e**(5/2)*gamma(5/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{b x^{2} + a} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(5/2)), x)