3.47 \(\int x \left (a x+b x^3\right )^{3/2} \, dx\)

Optimal. Leaf size=304 \[ -\frac{4 a^{13/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a x+b x^3}}+\frac{8 a^{13/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a x+b x^3}}-\frac{8 a^3 x \left (a+b x^2\right )}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{8 a^2 x \sqrt{a x+b x^3}}{195 b}+\frac{4}{39} a x^3 \sqrt{a x+b x^3}+\frac{2}{13} x^2 \left (a x+b x^3\right )^{3/2} \]

[Out]

(-8*a^3*x*(a + b*x^2))/(65*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) + (8
*a^2*x*Sqrt[a*x + b*x^3])/(195*b) + (4*a*x^3*Sqrt[a*x + b*x^3])/39 + (2*x^2*(a*x
+ b*x^3)^(3/2))/13 + (8*a^(13/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)
/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(
65*b^(7/4)*Sqrt[a*x + b*x^3]) - (4*a^(13/4)*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])/(65*b^(7/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.595158, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{4 a^{13/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a x+b x^3}}+\frac{8 a^{13/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{65 b^{7/4} \sqrt{a x+b x^3}}-\frac{8 a^3 x \left (a+b x^2\right )}{65 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{8 a^2 x \sqrt{a x+b x^3}}{195 b}+\frac{4}{39} a x^3 \sqrt{a x+b x^3}+\frac{2}{13} x^2 \left (a x+b x^3\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a*x + b*x^3)^(3/2),x]

[Out]

(-8*a^3*x*(a + b*x^2))/(65*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) + (8
*a^2*x*Sqrt[a*x + b*x^3])/(195*b) + (4*a*x^3*Sqrt[a*x + b*x^3])/39 + (2*x^2*(a*x
+ b*x^3)^(3/2))/13 + (8*a^(13/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)
/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(
65*b^(7/4)*Sqrt[a*x + b*x^3]) - (4*a^(13/4)*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])/(65*b^(7/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 60.2336, size = 286, normalized size = 0.94 \[ \frac{8 a^{\frac{13}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{4 a^{\frac{13}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{65 b^{\frac{7}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{8 a^{3} \sqrt{a x + b x^{3}}}{65 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{8 a^{2} x \sqrt{a x + b x^{3}}}{195 b} + \frac{4 a x^{3} \sqrt{a x + b x^{3}}}{39} + \frac{2 x^{2} \left (a x + b x^{3}\right )^{\frac{3}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a**(13/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sq
rt(a*x + b*x**3)*elliptic_e(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(65*b**(7/4)
*sqrt(x)*(a + b*x**2)) - 4*a**(13/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)
*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x**3)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**
(1/4)), 1/2)/(65*b**(7/4)*sqrt(x)*(a + b*x**2)) - 8*a**3*sqrt(a*x + b*x**3)/(65*
b**(3/2)*(sqrt(a) + sqrt(b)*x)) + 8*a**2*x*sqrt(a*x + b*x**3)/(195*b) + 4*a*x**3
*sqrt(a*x + b*x**3)/39 + 2*x**2*(a*x + b*x**3)**(3/2)/13

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Mathematica [C]  time = 0.49274, size = 195, normalized size = 0.64 \[ \frac{2 x \left (12 a^{7/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-12 a^{7/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (4 a^3+29 a^2 b x^2+40 a b^2 x^4+15 b^3 x^6\right )\right )}{195 b^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(Sqrt[b]*x*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(4*a^3 + 29*a^2*b*x^2 + 40*a*b^2*x^4
+ 15*b^3*x^6) - 12*a^(7/2)*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt
[b]*x)/Sqrt[a]]], -1] + 12*a^(7/2)*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[
(I*Sqrt[b]*x)/Sqrt[a]]], -1]))/(195*b^(3/2)*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[x*(
a + b*x^2)])

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Maple [A]  time = 0.025, size = 217, normalized size = 0.7 \[{\frac{2\,b{x}^{5}}{13}\sqrt{b{x}^{3}+ax}}+{\frac{10\,a{x}^{3}}{39}\sqrt{b{x}^{3}+ax}}+{\frac{8\,{a}^{2}x}{195\,b}\sqrt{b{x}^{3}+ax}}-{\frac{4\,{a}^{3}}{65\,{b}^{2}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a*x)^(3/2),x)

[Out]

2/13*b*x^5*(b*x^3+a*x)^(1/2)+10/39*a*x^3*(b*x^3+a*x)^(1/2)+8/195*a^2*x*(b*x^3+a*
x)^(1/2)/b-4/65/b^2*a^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)*(-2/b*(-a*b)^(1/2)*EllipticE(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(
1/2),1/2*2^(1/2))+1/b*(-a*b)^(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. \[ \int{\left (b x^{3} + a x\right )}^{\frac{3}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a*x)^(3/2)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a*x)^(3/2)*x, x)