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

Optimal. Leaf size=337 \[ \frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^2}}-\frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a+b x^2}}+\frac{3 e^2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-7 a B)}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{e (e x)^{3/2} (5 A b-7 a B)}{5 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{7/2}}{5 b e \sqrt{a+b x^2}} \]

[Out]

-((5*A*b - 7*a*B)*e*(e*x)^(3/2))/(5*b^2*Sqrt[a + b*x^2]) + (2*B*(e*x)^(7/2))/(5*
b*e*Sqrt[a + b*x^2]) + (3*(5*A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*b^(5
/2)*(Sqrt[a] + Sqrt[b]*x)) - (3*a^(1/4)*(5*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])/(5*b^(11/4)*Sqrt[a + b*x^2]) + (3*a^(1/4)*(5*A*
b - 7*a*B)*e^(5/2)*(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])/(10*b^(11/4)
*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.645411, antiderivative size = 337, 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 \[ \frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^2}}-\frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a+b x^2}}+\frac{3 e^2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-7 a B)}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{e (e x)^{3/2} (5 A b-7 a B)}{5 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{7/2}}{5 b e \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-((5*A*b - 7*a*B)*e*(e*x)^(3/2))/(5*b^2*Sqrt[a + b*x^2]) + (2*B*(e*x)^(7/2))/(5*
b*e*Sqrt[a + b*x^2]) + (3*(5*A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*b^(5
/2)*(Sqrt[a] + Sqrt[b]*x)) - (3*a^(1/4)*(5*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])/(5*b^(11/4)*Sqrt[a + b*x^2]) + (3*a^(1/4)*(5*A*
b - 7*a*B)*e^(5/2)*(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])/(10*b^(11/4)
*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 65.6036, size = 316, normalized size = 0.94 \[ \frac{2 B \left (e x\right )^{\frac{7}{2}}}{5 b e \sqrt{a + b x^{2}}} - \frac{3 \sqrt [4]{a} e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 7 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{11}{4}} \sqrt{a + b x^{2}}} + \frac{3 \sqrt [4]{a} e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 7 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{11}{4}} \sqrt{a + b x^{2}}} - \frac{e \left (e x\right )^{\frac{3}{2}} \left (5 A b - 7 B a\right )}{5 b^{2} \sqrt{a + b x^{2}}} + \frac{3 e^{2} \sqrt{e x} \sqrt{a + b x^{2}} \left (5 A b - 7 B a\right )}{5 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} \]

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)**(3/2),x)

[Out]

2*B*(e*x)**(7/2)/(5*b*e*sqrt(a + b*x**2)) - 3*a**(1/4)*e**(5/2)*sqrt((a + b*x**2
)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(5*A*b - 7*B*a)*elliptic_e(2*a
tan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(5*b**(11/4)*sqrt(a + b*x**2))
+ 3*a**(1/4)*e**(5/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqr
t(b)*x)*(5*A*b - 7*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e)))
, 1/2)/(10*b**(11/4)*sqrt(a + b*x**2)) - e*(e*x)**(3/2)*(5*A*b - 7*B*a)/(5*b**2*
sqrt(a + b*x**2)) + 3*e**2*sqrt(e*x)*sqrt(a + b*x**2)*(5*A*b - 7*B*a)/(5*b**(5/2
)*(sqrt(a) + sqrt(b)*x))

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Mathematica [C]  time = 1.44467, size = 229, normalized size = 0.68 \[ \frac{(e x)^{5/2} \left (b x^2 \left (7 a B-5 A b+2 b B x^2\right )+\frac{3 (5 A b-7 a B) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{5 b^3 x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(5/2)*(b*x^2*(-5*A*b + 7*a*B + 2*b*B*x^2) + (3*(5*A*b - 7*a*B)*(Sqrt[(I*S
qrt[a])/Sqrt[b]]*(a + b*x^2) - Sqrt[a]*Sqrt[b]*Sqrt[1 + a/(b*x^2)]*x^(3/2)*Ellip
ticE[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]]))/(5*b^3*x^3*Sqrt[a + b*x^2])

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Maple [A]  time = 0.05, size = 391, normalized size = 1.2 \[{\frac{{e}^{2}}{10\,x{b}^{3}}\sqrt{ex} \left ( 30\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-15\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-42\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+21\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+4\,{b}^{2}B{x}^{4}-10\,A{x}^{2}{b}^{2}+14\,B{x}^{2}ab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10/x*e^2*(e*x)^(1/2)*(30*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-15*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-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))*a^2+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^2+4*b^2*B*x^4-10*A*x^2*b^2+14*
B*x^2*a*b)/(b*x^2+a)^(1/2)/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{2} x^{4} + A e^{2} x^{2}\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*e^2*x^4 + A*e^2*x^2)*sqrt(e*x)/(b*x^2 + a)^(3/2), 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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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