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

Optimal. Leaf size=338 \[ -\frac{a^{5/4} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (9 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^2}}+\frac{2 a^{5/4} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (9 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^2}}-\frac{2 a e^2 \sqrt{e x} \sqrt{a+b x^2} (9 A b-7 a B)}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 e (e x)^{3/2} \sqrt{a+b x^2} (9 A b-7 a B)}{45 b^2}+\frac{2 B (e x)^{7/2} \sqrt{a+b x^2}}{9 b e} \]

[Out]

(2*(9*A*b - 7*a*B)*e*(e*x)^(3/2)*Sqrt[a + b*x^2])/(45*b^2) + (2*B*(e*x)^(7/2)*Sq
rt[a + b*x^2])/(9*b*e) - (2*a*(9*A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(15
*b^(5/2)*(Sqrt[a] + Sqrt[b]*x)) + (2*a^(5/4)*(9*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])/(15*b^(11/4)*Sqrt[a + b*x^2]) - (a^(5/4)*(
9*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])/(15*b^(1
1/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.649639, antiderivative size = 338, 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 \[ -\frac{a^{5/4} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (9 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^2}}+\frac{2 a^{5/4} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (9 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^2}}-\frac{2 a e^2 \sqrt{e x} \sqrt{a+b x^2} (9 A b-7 a B)}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 e (e x)^{3/2} \sqrt{a+b x^2} (9 A b-7 a B)}{45 b^2}+\frac{2 B (e x)^{7/2} \sqrt{a+b x^2}}{9 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(9*A*b - 7*a*B)*e*(e*x)^(3/2)*Sqrt[a + b*x^2])/(45*b^2) + (2*B*(e*x)^(7/2)*Sq
rt[a + b*x^2])/(9*b*e) - (2*a*(9*A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(15
*b^(5/2)*(Sqrt[a] + Sqrt[b]*x)) + (2*a^(5/4)*(9*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])/(15*b^(11/4)*Sqrt[a + b*x^2]) - (a^(5/4)*(
9*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])/(15*b^(1
1/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 64.9764, size = 318, normalized size = 0.94 \[ \frac{2 B \left (e x\right )^{\frac{7}{2}} \sqrt{a + b x^{2}}}{9 b e} + \frac{2 a^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (9 A b - 7 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{11}{4}} \sqrt{a + b x^{2}}} - \frac{a^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (9 A b - 7 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{11}{4}} \sqrt{a + b x^{2}}} - \frac{2 a e^{2} \sqrt{e x} \sqrt{a + b x^{2}} \left (9 A b - 7 B a\right )}{15 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{2 e \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (9 A b - 7 B a\right )}{45 b^{2}} \]

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)**(1/2),x)

[Out]

2*B*(e*x)**(7/2)*sqrt(a + b*x**2)/(9*b*e) + 2*a**(5/4)*e**(5/2)*sqrt((a + b*x**2
)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(9*A*b - 7*B*a)*elliptic_e(2*a
tan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(15*b**(11/4)*sqrt(a + b*x**2))
- a**(5/4)*e**(5/2)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt
(b)*x)*(9*A*b - 7*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))),
1/2)/(15*b**(11/4)*sqrt(a + b*x**2)) - 2*a*e**2*sqrt(e*x)*sqrt(a + b*x**2)*(9*A
*b - 7*B*a)/(15*b**(5/2)*(sqrt(a) + sqrt(b)*x)) + 2*e*(e*x)**(3/2)*sqrt(a + b*x*
*2)*(9*A*b - 7*B*a)/(45*b**2)

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Mathematica [C]  time = 1.51681, size = 237, normalized size = 0.7 \[ \frac{2 (e x)^{5/2} \left (b x^2 \left (a+b x^2\right ) \left (-7 a B+9 A b+5 b B x^2\right )+\frac{3 a (7 a B-9 A b) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{45 b^3 x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*x)^(5/2)*(b*x^2*(a + b*x^2)*(9*A*b - 7*a*B + 5*b*B*x^2) + (3*a*(-9*A*b + 7
*a*B)*(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]]))/(45*b^3*x^3*Sqrt[a + b*x^2])

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Maple [A]  time = 0.041, size = 417, normalized size = 1.2 \[ -{\frac{{e}^{2}}{45\,x{b}^{3}}\sqrt{ex} \left ( -10\,B{x}^{6}{b}^{3}+54\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}b-27\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}b-42\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}+21\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}-18\,A{x}^{4}{b}^{3}+4\,B{x}^{4}a{b}^{2}-18\,A{x}^{2}a{b}^{2}+14\,B{x}^{2}{a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45/x*e^2*(e*x)^(1/2)/(b*x^2+a)^(1/2)/b^3*(-10*B*x^6*b^3+54*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-27*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)*Ell
ipticE(((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^3-18*A*x^4*b^3+4*B*x^4*a*b^2-18*A*x^2*a*b^2+14*B*x^2*a^2*b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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