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

Optimal. Leaf size=344 \[ -\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{2 a^2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(e x)^{3/2} (a B+A b)}{2 a^2 b e \sqrt{a+b x^2}}+\frac{(e x)^{3/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.641502, antiderivative size = 344, 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 \[ -\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{2 a^2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(e x)^{3/2} (a B+A b)}{2 a^2 b e \sqrt{a+b x^2}}+\frac{(e x)^{3/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.1822, size = 304, normalized size = 0.88 \[ \frac{\left (e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (A b + B a\right )}{2 a^{2} b e \sqrt{a + b x^{2}}} - \frac{\sqrt{e x} \sqrt{a + b x^{2}} \left (A b + B a\right )}{2 a^{2} b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{\sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{\sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{7}{4}} b^{\frac{7}{4}} \sqrt{a + b x^{2}}} \]

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)**(5/2),x)

[Out]

(e*x)**(3/2)*(A*b - B*a)/(3*a*b*e*(a + b*x**2)**(3/2)) + (e*x)**(3/2)*(A*b + B*a
)/(2*a**2*b*e*sqrt(a + b*x**2)) - sqrt(e*x)*sqrt(a + b*x**2)*(A*b + B*a)/(2*a**2
*b**(3/2)*(sqrt(a) + sqrt(b)*x)) + sqrt(e)*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)/(2*a**(7/4)*b**(7/4)*sqrt(a + b*x**2)) - sqrt(e)*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)/(4*a**(7/4)*b**(7/4)*sqrt(
a + b*x**2))

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Mathematica [C]  time = 0.940555, size = 247, normalized size = 0.72 \[ \frac{e \left (b x^2 \left (a^2 B+a b \left (5 A+3 B x^2\right )+3 A b^2 x^2\right )-\frac{3 \left (a+b x^2\right ) (a B+A b) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{6 a^2 b^2 \sqrt{e x} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(e*(b*x^2*(a^2*B + 3*A*b^2*x^2 + a*b*(5*A + 3*B*x^2)) - (3*(A*b + a*B)*(a + b*x^
2)*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(a + b*x^2) - Sqrt[a]*Sqrt[b]*Sqrt[1 + a/(b*x^2)]*
x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + Sqrt[a]*Sq
rt[b]*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/
Sqrt[x]], -1]))/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(6*a^2*b^2*Sqrt[e*x]*(a + b*x^2)^(3/
2))

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Maple [B]  time = 0.027, size = 764, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(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))*x^2*a*b^2-3*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)*El
lipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2+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))*x^2*a^2*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))*x^2*a^2*b+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^2
*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^2*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^3-3*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
^3-6*A*x^4*b^3-6*B*x^4*a*b^2-10*A*x^2*a*b^2-2*B*x^2*a^2*b)*(e*x)^(1/2)/b^2/a^2/x
/(b*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)*sqrt(e*x)/((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)), 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)**(1/2)/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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