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

Optimal. Leaf size=158 \[ -\frac{4 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{a x+b x^3}}+\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}+\frac{12}{77} a x^2 \sqrt{a x+b x^3} \]

[Out]

(8*a^2*Sqrt[a*x + b*x^3])/(77*b) + (12*a*x^2*Sqrt[a*x + b*x^3])/77 + (2*x*(a*x +
b*x^3)^(3/2))/11 - (4*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])/(77
*b^(5/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.273054, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{4 a^{11/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{a x+b x^3}}+\frac{8 a^2 \sqrt{a x+b x^3}}{77 b}+\frac{2}{11} x \left (a x+b x^3\right )^{3/2}+\frac{12}{77} a x^2 \sqrt{a x+b x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3)^(3/2),x]

[Out]

(8*a^2*Sqrt[a*x + b*x^3])/(77*b) + (12*a*x^2*Sqrt[a*x + b*x^3])/77 + (2*x*(a*x +
b*x^3)^(3/2))/11 - (4*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])/(77
*b^(5/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 25.389, size = 151, normalized size = 0.96 \[ - \frac{4 a^{\frac{11}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{77 b^{\frac{5}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{8 a^{2} \sqrt{a x + b x^{3}}}{77 b} + \frac{12 a x^{2} \sqrt{a x + b x^{3}}}{77} + \frac{2 x \left (a x + b x^{3}\right )^{\frac{3}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*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)/(77*b**(5/4
)*sqrt(x)*(a + b*x**2)) + 8*a**2*sqrt(a*x + b*x**3)/(77*b) + 12*a*x**2*sqrt(a*x
+ b*x**3)/77 + 2*x*(a*x + b*x**3)**(3/2)/11

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Mathematica [C]  time = 0.217735, size = 148, normalized size = 0.94 \[ \frac{2 x \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (4 a^3+17 a^2 b x^2+20 a b^2 x^4+7 b^3 x^6\right )-4 i a^3 \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{77 b \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(4*a^3 + 17*a^2*b*x^2 + 20*a*b^2*x^4 + 7*b^3*x^6
) - (4*I)*a^3*Sqrt[1 + a/(b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/S
qrt[b]]/Sqrt[x]], -1]))/(77*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.023, size = 166, normalized size = 1.1 \[{\frac{2\,b{x}^{4}}{11}\sqrt{b{x}^{3}+ax}}+{\frac{26\,a{x}^{2}}{77}\sqrt{b{x}^{3}+ax}}+{\frac{8\,{a}^{2}}{77\,b}\sqrt{b{x}^{3}+ax}}-{\frac{4\,{a}^{3}}{77\,{b}^{2}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x)^(3/2),x)

[Out]

2/11*b*x^4*(b*x^3+a*x)^(1/2)+26/77*a*x^2*(b*x^3+a*x)^(1/2)+8/77*a^2*(b*x^3+a*x)^
(1/2)/b-4/77/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)*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. \[ \int{\left (b x^{3} + a x\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x + b*x**3)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a*x)^(3/2), x)