3.813 \(\int \frac{A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=176 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}} \]

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*Sqrt[a + b*x^2]) - ((5*A*b - 3*a*B)*Sqrt[e*x])/(3*a^2*
e^3*Sqrt[a + b*x^2]) - ((5*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])/(6*a^(9/4)*b^(1/4)*e^(5/2)*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.303116, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)),x]

[Out]

(-2*A)/(3*a*e*(e*x)^(3/2)*Sqrt[a + b*x^2]) - ((5*A*b - 3*a*B)*Sqrt[e*x])/(3*a^2*
e^3*Sqrt[a + b*x^2]) - ((5*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])/(6*a^(9/4)*b^(1/4)*e^(5/2)*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.44, size = 163, normalized size = 0.93 \[ - \frac{2 A}{3 a e \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}} - \frac{\sqrt{e x} \left (5 A b - 3 B a\right )}{3 a^{2} e^{3} \sqrt{a + b x^{2}}} - \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{9}{4}} \sqrt [4]{b} e^{\frac{5}{2}} \sqrt{a + b x^{2}}} \]

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)**(3/2),x)

[Out]

-2*A/(3*a*e*(e*x)**(3/2)*sqrt(a + b*x**2)) - sqrt(e*x)*(5*A*b - 3*B*a)/(3*a**2*e
**3*sqrt(a + b*x**2)) - sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + s
qrt(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)/(6*a**(9/4)*b**(1/4)*e**(5/2)*sqrt(a + b*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.223779, size = 146, normalized size = 0.83 \[ \frac{x \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (-2 a A+3 a B x^2-5 A b x^2\right )-i x^{5/2} \sqrt{\frac{a}{b x^2}+1} (5 A b-3 a B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 a^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} (e x)^{5/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.032, size = 232, normalized size = 1.3 \[ -{\frac{1}{6\,bx{a}^{2}{e}^{2}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}xb-3\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}xa+10\,A{x}^{2}{b}^{2}-6\,B{x}^{2}ab+4\,abA \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(e*x)^(5/2)/(b*x^2+a)^(3/2),x)

[Out]

-1/6/x*(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)^(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+10*A*x^2*b^2-6*B*x^2*a*b+4*a*b*A)/(b*x^2+a)^(1/2)/b/a^2/e^2/(e*x)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{{\left (b e^{2} x^{4} + a e^{2} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)/((b*e^2*x^4 + a*e^2*x^2)*sqrt(b*x^2 + a)*sqrt(e*x)), x)

_______________________________________________________________________________________

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)**(5/2)/(b*x**2+a)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)), x)