3.52 \(\int \frac{\left (a x+b x^3\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=134 \[ \frac{4 a^{3/4} b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a x+b x^3}}-\frac{2 \left (a x+b x^3\right )^{3/2}}{3 x^3}+\frac{4}{3} b \sqrt{a x+b x^3} \]

[Out]

(4*b*Sqrt[a*x + b*x^3])/3 - (2*(a*x + b*x^3)^(3/2))/(3*x^3) + (4*a^(3/4)*b^(3/4)
*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ellipti
cF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.249916, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{4 a^{3/4} b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt{a x+b x^3}}-\frac{2 \left (a x+b x^3\right )^{3/2}}{3 x^3}+\frac{4}{3} b \sqrt{a x+b x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3)^(3/2)/x^4,x]

[Out]

(4*b*Sqrt[a*x + b*x^3])/3 - (2*(a*x + b*x^3)^(3/2))/(3*x^3) + (4*a^(3/4)*b^(3/4)
*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Ellipti
cF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 23.1007, size = 129, normalized size = 0.96 \[ \frac{4 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{x} \left (a + b x^{2}\right )} + \frac{4 b \sqrt{a x + b x^{3}}}{3} - \frac{2 \left (a x + b x^{3}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(3/4)*b**(3/4)*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)/(3*s
qrt(x)*(a + b*x**2)) + 4*b*sqrt(a*x + b*x**3)/3 - 2*(a*x + b*x**3)**(3/2)/(3*x**
3)

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Mathematica [C]  time = 0.285941, size = 107, normalized size = 0.8 \[ \frac{2 \left (-a^2+\frac{4 i a b x^{5/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}+b^2 x^4\right )}{3 x \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-a^2 + b^2*x^4 + ((4*I)*a*b*Sqrt[1 + a/(b*x^2)]*x^(5/2)*EllipticF[I*ArcSinh[
Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(3*x*Sqrt[x
*(a + b*x^2)])

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Maple [A]  time = 0.027, size = 139, normalized size = 1. \[ -{\frac{2\,a}{3\,{x}^{2}}\sqrt{b{x}^{3}+ax}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+ax}}+{\frac{4\,a}{3}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x)^(3/2)/x^4,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a*x)^(3/2)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(a + b*x**2))**(3/2)/x**4, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a*x)^(3/2)/x^4, x)