3.354 \(\int x^{3/2} \sqrt{b x^2+c x^4} \, dx\)

Optimal. Leaf size=293 \[ -\frac{2 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4}}{45 c}+\frac{2}{9} x^{5/2} \sqrt{b x^2+c x^4} \]

[Out]

(-4*b^2*x^(3/2)*(b + c*x^2))/(15*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^
4]) + (4*b*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(45*c) + (2*x^(5/2)*Sqrt[b*x^2 + c*x^4])
/9 + (4*b^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2
]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x^2 +
c*x^4]) - (2*b^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]
*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x
^2 + c*x^4])

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Rubi [A]  time = 0.621729, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{2 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{9/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b^2 x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{4 b \sqrt{x} \sqrt{b x^2+c x^4}}{45 c}+\frac{2}{9} x^{5/2} \sqrt{b x^2+c x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*Sqrt[b*x^2 + c*x^4],x]

[Out]

(-4*b^2*x^(3/2)*(b + c*x^2))/(15*c^(3/2)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[b*x^2 + c*x^
4]) + (4*b*Sqrt[x]*Sqrt[b*x^2 + c*x^4])/(45*c) + (2*x^(5/2)*Sqrt[b*x^2 + c*x^4])
/9 + (4*b^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2
]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x^2 +
c*x^4]) - (2*b^(9/4)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]
*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(15*c^(7/4)*Sqrt[b*x
^2 + c*x^4])

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Rubi in Sympy [A]  time = 56.5116, size = 275, normalized size = 0.94 \[ \frac{4 b^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{2 b^{\frac{9}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{7}{4}} x \left (b + c x^{2}\right )} - \frac{4 b^{2} \sqrt{b x^{2} + c x^{4}}}{15 c^{\frac{3}{2}} \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} + \frac{4 b \sqrt{x} \sqrt{b x^{2} + c x^{4}}}{45 c} + \frac{2 x^{\frac{5}{2}} \sqrt{b x^{2} + c x^{4}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(c*x**4+b*x**2)**(1/2),x)

[Out]

4*b**(9/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqrt(b) + sqrt(c)*x)*sqr
t(b*x**2 + c*x**4)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/b**(1/4)), 1/2)/(15*c**(7/
4)*x*(b + c*x**2)) - 2*b**(9/4)*sqrt((b + c*x**2)/(sqrt(b) + sqrt(c)*x)**2)*(sqr
t(b) + sqrt(c)*x)*sqrt(b*x**2 + c*x**4)*elliptic_f(2*atan(c**(1/4)*sqrt(x)/b**(1
/4)), 1/2)/(15*c**(7/4)*x*(b + c*x**2)) - 4*b**2*sqrt(b*x**2 + c*x**4)/(15*c**(3
/2)*sqrt(x)*(sqrt(b) + sqrt(c)*x)) + 4*b*sqrt(x)*sqrt(b*x**2 + c*x**4)/(45*c) +
2*x**(5/2)*sqrt(b*x**2 + c*x**4)/9

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Mathematica [C]  time = 0.314554, size = 190, normalized size = 0.65 \[ \frac{2 x^{3/2} \left (6 b^{5/2} \sqrt{\frac{c x^2}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )-6 b^{5/2} \sqrt{\frac{c x^2}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}}\right )\right |-1\right )+\sqrt{c} x \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \left (2 b^2+7 b c x^2+5 c^2 x^4\right )\right )}{45 c^{3/2} \sqrt{\frac{i \sqrt{c} x}{\sqrt{b}}} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*Sqrt[b*x^2 + c*x^4],x]

[Out]

(2*x^(3/2)*(Sqrt[c]*x*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*(2*b^2 + 7*b*c*x^2 + 5*c^2*x^4
) - 6*b^(5/2)*Sqrt[1 + (c*x^2)/b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/Sqrt[b]
]], -1] + 6*b^(5/2)*Sqrt[1 + (c*x^2)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]*x)/S
qrt[b]]], -1]))/(45*c^(3/2)*Sqrt[(I*Sqrt[c]*x)/Sqrt[b]]*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.033, size = 226, normalized size = 0.8 \[ -{\frac{2}{ \left ( 45\,c{x}^{2}+45\,b \right ){c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -5\,{c}^{3}{x}^{6}+6\,{b}^{3}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -3\,{b}^{3}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) -7\,b{c}^{2}{x}^{4}-2\,{b}^{2}c{x}^{2} \right ){x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(c*x^4+b*x^2)^(1/2),x)

[Out]

-2/45*(c*x^4+b*x^2)^(1/2)/x^(3/2)/(c*x^2+b)/c^2*(-5*c^3*x^6+6*b^3*((c*x+(-b*c)^(
1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c
/(-b*c)^(1/2))^(1/2)*EllipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/
2))-3*b^3*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(
-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c
)^(1/2))^(1/2),1/2*2^(1/2))-7*b*c^2*x^4-2*b^2*c*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*x^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^2)*x^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*x^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2)*x^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{\frac{3}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x**(3/2)*sqrt(x**2*(b + c*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2}} x^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*x^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2)*x^(3/2), x)