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

Optimal. Leaf size=377 \[ \frac{4 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{e x} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (e x)^{3/2} \left (a+b x^2\right )^{3/2} (13 A b-3 a B)}{117 b e}+\frac{4 a (e x)^{3/2} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e} \]

[Out]

(4*a*(13*A*b - 3*a*B)*(e*x)^(3/2)*Sqrt[a + b*x^2])/(195*b*e) + (8*a^2*(13*A*b -
3*a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(195*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) + (2*(13*A
*b - 3*a*B)*(e*x)^(3/2)*(a + b*x^2)^(3/2))/(117*b*e) + (2*B*(e*x)^(3/2)*(a + b*x
^2)^(5/2))/(13*b*e) - (8*a^(9/4)*(13*A*b - 3*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])/(195*b^(7/4)*Sqrt[a + b*x^2]) + (4*a^(9/4)*(13*A*b -
3*a*B)*Sqrt[e]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*E
llipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(195*b^(7/4)*Sqr
t[a + b*x^2])

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Rubi [A]  time = 0.717556, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{4 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{e x} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (e x)^{3/2} \left (a+b x^2\right )^{3/2} (13 A b-3 a B)}{117 b e}+\frac{4 a (e x)^{3/2} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e} \]

Antiderivative was successfully verified.

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

[Out]

(4*a*(13*A*b - 3*a*B)*(e*x)^(3/2)*Sqrt[a + b*x^2])/(195*b*e) + (8*a^2*(13*A*b -
3*a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(195*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) + (2*(13*A
*b - 3*a*B)*(e*x)^(3/2)*(a + b*x^2)^(3/2))/(117*b*e) + (2*B*(e*x)^(3/2)*(a + b*x
^2)^(5/2))/(13*b*e) - (8*a^(9/4)*(13*A*b - 3*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])/(195*b^(7/4)*Sqrt[a + b*x^2]) + (4*a^(9/4)*(13*A*b -
3*a*B)*Sqrt[e]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*E
llipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(195*b^(7/4)*Sqr
t[a + b*x^2])

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Rubi in Sympy [A]  time = 72.7116, size = 352, normalized size = 0.93 \[ \frac{2 B \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{5}{2}}}{13 b e} - \frac{8 a^{\frac{9}{4}} \sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (13 A b - 3 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} + \frac{4 a^{\frac{9}{4}} \sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (13 A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} + \frac{8 a^{2} \sqrt{e x} \sqrt{a + b x^{2}} \left (13 A b - 3 B a\right )}{195 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{4 a \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (13 A b - 3 B a\right )}{195 b e} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (13 A b - 3 B a\right )}{117 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(e*x)**(3/2)*(a + b*x**2)**(5/2)/(13*b*e) - 8*a**(9/4)*sqrt(e)*sqrt((a + b*x
**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(13*A*b - 3*B*a)*elliptic_e
(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(195*b**(7/4)*sqrt(a + b*x*
*2)) + 4*a**(9/4)*sqrt(e)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) +
sqrt(b)*x)*(13*A*b - 3*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt
(e))), 1/2)/(195*b**(7/4)*sqrt(a + b*x**2)) + 8*a**2*sqrt(e*x)*sqrt(a + b*x**2)*
(13*A*b - 3*B*a)/(195*b**(3/2)*(sqrt(a) + sqrt(b)*x)) + 4*a*(e*x)**(3/2)*sqrt(a
+ b*x**2)*(13*A*b - 3*B*a)/(195*b*e) + 2*(e*x)**(3/2)*(a + b*x**2)**(3/2)*(13*A*
b - 3*B*a)/(117*b*e)

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Mathematica [C]  time = 1.46976, size = 214, normalized size = 0.57 \[ \frac{2 \sqrt{x} \sqrt{e x} \left (b \sqrt{x} \left (a+b x^2\right ) \left (12 a^2 B+a b \left (143 A+75 B x^2\right )+5 b^2 x^2 \left (13 A+9 B x^2\right )\right )+12 a^2 (3 a B-13 A b) \left (-\sqrt{x} \left (\frac{a}{x^2}+b\right )+\frac{i a \sqrt{\frac{a}{b x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2}}\right )\right )}{585 b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*Sqrt[e*x]*(b*Sqrt[x]*(a + b*x^2)*(12*a^2*B + 5*b^2*x^2*(13*A + 9*B*x^
2) + a*b*(143*A + 75*B*x^2)) + 12*a^2*(-13*A*b + 3*a*B)*(-((b + a/x^2)*Sqrt[x])
+ (I*a*Sqrt[1 + a/(b*x^2)]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x
]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))/((I*Sqrt
[a])/Sqrt[b])^(3/2))))/(585*b^2*Sqrt[a + b*x^2])

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Maple [A]  time = 0.037, size = 438, normalized size = 1.2 \[{\frac{2}{585\,{b}^{2}x}\sqrt{ex} \left ( 45\,B{x}^{8}{b}^{4}+65\,A{x}^{6}{b}^{4}+120\,B{x}^{6}a{b}^{3}+156\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}b-78\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}b-36\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{4}+18\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{4}+208\,A{x}^{4}a{b}^{3}+87\,B{x}^{4}{a}^{2}{b}^{2}+143\,A{x}^{2}{a}^{2}{b}^{2}+12\,B{x}^{2}{a}^{3}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585/(b*x^2+a)^(1/2)*(e*x)^(1/2)/b^2*(45*B*x^8*b^4+65*A*x^6*b^4+120*B*x^6*a*b^3
+156*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^3*b-78*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^3*b-36*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^4+18*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^4+208*A*x^4*a*b^3+87*B*x^4*a^2*b^2+143*A*x^2*a^2
*b^2+12*B*x^2*a^3*b)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 57.8709, size = 197, normalized size = 0.52 \[ \frac{A a^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{A \sqrt{a} b \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{B a^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{B \sqrt{a} b \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{5} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(3/2)*(e*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*x**2*exp_polar(I
*pi)/a)/(2*e*gamma(7/4)) + A*sqrt(a)*b*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4)
, (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*e**3*gamma(11/4)) + B*a**(3/2)*(e*x)**(7
/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*e**3*gam
ma(11/4)) + B*sqrt(a)*b*(e*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), b
*x**2*exp_polar(I*pi)/a)/(2*e**5*gamma(15/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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