3.6.35 \(\int \frac {(a+b x^3)^{2/3}}{x^2} \, dx\) [535]

Optimal. Leaf size=36 \[ -\frac {\left (a+b x^3\right )^{5/3} \, _2F_1\left (1,\frac {4}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{a x} \]

[Out]

-(b*x^3+a)^(5/3)*hypergeom([1, 4/3],[2/3],-b*x^3/a)/a/x

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \begin {gather*} -\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \left (\frac {b x^3}{a}+1\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3)/x^2,x]

[Out]

-(((a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, -1/3, 2/3, -((b*x^3)/a)])/(x*(1 + (b*x^3)/a)^(2/3)))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^2} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^2} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 10.02, size = 49, normalized size = 1.36 \begin {gather*} -\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \left (1+\frac {b x^3}{a}\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(2/3)/x^2,x]

[Out]

-(((a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, -1/3, 2/3, -((b*x^3)/a)])/(x*(1 + (b*x^3)/a)^(2/3)))

________________________________________________________________________________________

Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3)/x^2, x)

________________________________________________________________________________________

Fricas [F]
time = 0.37, size = 15, normalized size = 0.42 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3 + a)^(2/3)/x^2, x)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.42, size = 41, normalized size = 1.14 \begin {gather*} \frac {a^{\frac {2}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(2/3)*gamma(-1/3)*hyper((-2/3, -1/3), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3)/x^2, x)

________________________________________________________________________________________

Mupad [B]
time = 1.26, size = 39, normalized size = 1.08 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},-\frac {1}{3};\ \frac {2}{3};\ -\frac {a}{b\,x^3}\right )}{x\,{\left (\frac {a}{b\,x^3}+1\right )}^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(2/3)/x^2,x)

[Out]

((a + b*x^3)^(2/3)*hypergeom([-2/3, -1/3], 2/3, -a/(b*x^3)))/(x*(a/(b*x^3) + 1)^(2/3))

________________________________________________________________________________________