3.38 \(\int x^3 \sqrt{a x+b x^3} \, dx\)

Optimal. Leaf size=163 \[ \frac{10 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a x+b x^3}}-\frac{20 a^2 \sqrt{a x+b x^3}}{231 b^2}+\frac{2}{11} x^4 \sqrt{a x+b x^3}+\frac{4 a x^2 \sqrt{a x+b x^3}}{77 b} \]

[Out]

(-20*a^2*Sqrt[a*x + b*x^3])/(231*b^2) + (4*a*x^2*Sqrt[a*x + b*x^3])/(77*b) + (2*
x^4*Sqrt[a*x + b*x^3])/11 + (10*a^(11/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a +
b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)],
1/2])/(231*b^(9/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.343981, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{10 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a x+b x^3}}-\frac{20 a^2 \sqrt{a x+b x^3}}{231 b^2}+\frac{2}{11} x^4 \sqrt{a x+b x^3}+\frac{4 a x^2 \sqrt{a x+b x^3}}{77 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[a*x + b*x^3],x]

[Out]

(-20*a^2*Sqrt[a*x + b*x^3])/(231*b^2) + (4*a*x^2*Sqrt[a*x + b*x^3])/(77*b) + (2*
x^4*Sqrt[a*x + b*x^3])/11 + (10*a^(11/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a +
b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)],
1/2])/(231*b^(9/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 31.9539, size = 156, normalized size = 0.96 \[ \frac{10 a^{\frac{11}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{20 a^{2} \sqrt{a x + b x^{3}}}{231 b^{2}} + \frac{4 a x^{2} \sqrt{a x + b x^{3}}}{77 b} + \frac{2 x^{4} \sqrt{a x + b x^{3}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*a**(11/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*s
qrt(a*x + b*x**3)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(231*b**(9/
4)*sqrt(x)*(a + b*x**2)) - 20*a**2*sqrt(a*x + b*x**3)/(231*b**2) + 4*a*x**2*sqrt
(a*x + b*x**3)/(77*b) + 2*x**4*sqrt(a*x + b*x**3)/11

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Mathematica [C]  time = 0.24161, size = 148, normalized size = 0.91 \[ \frac{2 x \left (10 i a^3 \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (-10 a^3-4 a^2 b x^2+27 a b^2 x^4+21 b^3 x^6\right )\right )}{231 b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[a*x + b*x^3],x]

[Out]

(2*x*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(-10*a^3 - 4*a^2*b*x^2 + 27*a*b^2*x^4 + 21*b^3*x
^6) + (10*I)*a^3*Sqrt[1 + a/(b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a]
)/Sqrt[b]]/Sqrt[x]], -1]))/(231*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b^2*Sqrt[x*(a + b*x^2)
])

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Maple [A]  time = 0.03, size = 168, normalized size = 1. \[{\frac{2\,{x}^{4}}{11}\sqrt{b{x}^{3}+ax}}+{\frac{4\,a{x}^{2}}{77\,b}\sqrt{b{x}^{3}+ax}}-{\frac{20\,{a}^{2}}{231\,{b}^{2}}\sqrt{b{x}^{3}+ax}}+{\frac{10\,{a}^{3}}{231\,{b}^{3}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^3+a*x)^(1/2),x)

[Out]

2/11*x^4*(b*x^3+a*x)^(1/2)+4/77*a*x^2*(b*x^3+a*x)^(1/2)/b-20/231*a^2*(b*x^3+a*x)
^(1/2)/b^2+10/231*a^3/b^3*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/
2)*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)/(b*x
^3+a*x)^(1/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a*x)*x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^3 + a*x)*x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a*x)*x^3,x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a*x)*x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3*sqrt(x*(a + b*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a*x)*x^3,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^3 + a*x)*x^3, x)