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

Optimal. Leaf size=349 \[ -\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt{a+b x^2}}+\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt{a+b x^2}}-\frac{e^2 \sqrt{e x} \sqrt{a+b x^2} (A b-7 a B)}{2 a b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{e (e x)^{3/2} (A b-7 a B)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]

[Out]

((A*b - a*B)*(e*x)^(7/2))/(3*a*b*e*(a + b*x^2)^(3/2)) + ((A*b - 7*a*B)*e*(e*x)^(
3/2))/(6*a*b^2*Sqrt[a + b*x^2]) - ((A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/
(2*a*b^(5/2)*(Sqrt[a] + Sqrt[b]*x)) + ((A*b - 7*a*B)*e^(5/2)*(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^(3/4)*b^(11/4)*Sqrt[a + b*x^2]) - ((A*b - 7*a
*B)*e^(5/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Elli
pticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(4*a^(3/4)*b^(11/4)
*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.66431, antiderivative size = 349, 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 \[ -\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt{a+b x^2}}+\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt{a+b x^2}}-\frac{e^2 \sqrt{e x} \sqrt{a+b x^2} (A b-7 a B)}{2 a b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{e (e x)^{3/2} (A b-7 a B)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(e*x)^(7/2))/(3*a*b*e*(a + b*x^2)^(3/2)) + ((A*b - 7*a*B)*e*(e*x)^(
3/2))/(6*a*b^2*Sqrt[a + b*x^2]) - ((A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/
(2*a*b^(5/2)*(Sqrt[a] + Sqrt[b]*x)) + ((A*b - 7*a*B)*e^(5/2)*(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^(3/4)*b^(11/4)*Sqrt[a + b*x^2]) - ((A*b - 7*a
*B)*e^(5/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Elli
pticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(4*a^(3/4)*b^(11/4)
*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 69.7875, size = 313, normalized size = 0.9 \[ \frac{\left (e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{e \left (e x\right )^{\frac{3}{2}} \left (A b - 7 B a\right )}{6 a b^{2} \sqrt{a + b x^{2}}} - \frac{e^{2} \sqrt{e x} \sqrt{a + b x^{2}} \left (A b - 7 B a\right )}{2 a b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 7 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} b^{\frac{11}{4}} \sqrt{a + b x^{2}}} - \frac{e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 7 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{3}{4}} b^{\frac{11}{4}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x)**(7/2)*(A*b - B*a)/(3*a*b*e*(a + b*x**2)**(3/2)) + e*(e*x)**(3/2)*(A*b - 7
*B*a)/(6*a*b**2*sqrt(a + b*x**2)) - e**2*sqrt(e*x)*sqrt(a + b*x**2)*(A*b - 7*B*a
)/(2*a*b**(5/2)*(sqrt(a) + sqrt(b)*x)) + e**(5/2)*sqrt((a + b*x**2)/(sqrt(a) + s
qrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b - 7*B*a)*elliptic_e(2*atan(b**(1/4)*sqr
t(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(2*a**(3/4)*b**(11/4)*sqrt(a + b*x**2)) - e**(5
/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b - 7*B
*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(4*a**(3/4)*b
**(11/4)*sqrt(a + b*x**2))

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Mathematica [C]  time = 1.02553, size = 249, normalized size = 0.71 \[ \frac{(e x)^{5/2} \left (b x^2 \left (-7 a^2 B+a b \left (A-9 B x^2\right )+3 A b^2 x^2\right )-\frac{3 \left (a+b x^2\right ) (A b-7 a B) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{6 a b^3 x^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(5/2)*(b*x^2*(-7*a^2*B + 3*A*b^2*x^2 + a*b*(A - 9*B*x^2)) - (3*(A*b - 7*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]*Sqrt[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*b^3*x^3*(a + b*x^2
)^(3/2))

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Maple [B]  time = 0.064, size = 767, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(B*x^2+A)/(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^2-42*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+21*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-42*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)*Ellipti
cE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^3+21*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+18*B*x^4*a*b^2-2*A*x^2*a*b^2+14*B*x^2*a^2*b)/x*e^2*(e*x)^(1/2
)/b^3/a/(b*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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