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

Optimal. Leaf size=174 \[ -\frac{a^{3/4} e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{a+b x^2}}+\frac{2 e \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac{2 B (e x)^{5/2} \sqrt{a+b x^2}}{7 b e} \]

[Out]

(2*(7*A*b - 5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b^2) + (2*B*(e*x)^(5/2)*Sqrt
[a + b*x^2])/(7*b*e) - (a^(3/4)*(7*A*b - 5*a*B)*e^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sq
rt[(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])/(21*b^(9/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.293975, antiderivative size = 174, 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 \[ -\frac{a^{3/4} e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{a+b x^2}}+\frac{2 e \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac{2 B (e x)^{5/2} \sqrt{a+b x^2}}{7 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(7*A*b - 5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b^2) + (2*B*(e*x)^(5/2)*Sqrt
[a + b*x^2])/(7*b*e) - (a^(3/4)*(7*A*b - 5*a*B)*e^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sq
rt[(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])/(21*b^(9/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 28.6141, size = 162, normalized size = 0.93 \[ \frac{2 B \left (e x\right )^{\frac{5}{2}} \sqrt{a + b x^{2}}}{7 b e} - \frac{a^{\frac{3}{4}} e^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (7 A b - 5 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{9}{4}} \sqrt{a + b x^{2}}} + \frac{2 e \sqrt{e x} \sqrt{a + b x^{2}} \left (7 A b - 5 B a\right )}{21 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(1/2),x)

[Out]

2*B*(e*x)**(5/2)*sqrt(a + b*x**2)/(7*b*e) - a**(3/4)*e**(3/2)*sqrt((a + b*x**2)/
(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(7*A*b - 5*B*a)*elliptic_f(2*ata
n(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(21*b**(9/4)*sqrt(a + b*x**2)) +
2*e*sqrt(e*x)*sqrt(a + b*x**2)*(7*A*b - 5*B*a)/(21*b**2)

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Mathematica [C]  time = 0.430906, size = 134, normalized size = 0.77 \[ \frac{2 e \sqrt{e x} \left (-\left (a+b x^2\right ) \left (5 a B-7 A b-3 b B x^2\right )+\frac{i a \sqrt{x} \sqrt{\frac{a}{b x^2}+1} (5 a B-7 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{21 b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[e*x]*(-((a + b*x^2)*(-7*A*b + 5*a*B - 3*b*B*x^2)) + (I*a*(-7*A*b + 5*a
*B)*Sqrt[1 + a/(b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sq
rt[x]], -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(21*b^2*Sqrt[a + b*x^2])

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Maple [A]  time = 0.039, size = 250, normalized size = 1.4 \[ -{\frac{e}{21\,x{b}^{3}}\sqrt{ex} \left ( 7\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}ab-5\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{2}-6\,B{x}^{5}{b}^{3}-14\,A{x}^{3}{b}^{3}+4\,B{x}^{3}a{b}^{2}-14\,Axa{b}^{2}+10\,Bx{a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)*(B*x^2+A)/(b*x^2+a)^(1/2),x)

[Out]

-1/21*e/x*(e*x)^(1/2)/(b*x^2+a)^(1/2)*(7*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))*(-a*b)^(1/2)*a*b-5*
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)*a^2-6*B*x^5*b^3-14*A*x^3*b^3+4*B*x^3*a*b^2-14*A
*x*a*b^2+10*B*x*a^2*b)/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 75.1032, size = 94, normalized size = 0.54 \[ \frac{A e^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} + \frac{B e^{\frac{3}{2}} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(1/2),x)

[Out]

A*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/
a)/(2*sqrt(a)*gamma(9/4)) + B*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((1/2, 9/4), (13
/4,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(13/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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