3.676 \(\int \frac{x^{2/3}}{a+b x} \, dx\)

Optimal. Leaf size=111 \[ \frac{3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac{a^{2/3} \log (a+b x)}{2 b^{5/3}}+\frac{\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac{3 x^{2/3}}{2 b} \]

[Out]

(3*x^(2/3))/(2*b) + (Sqrt[3]*a^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/b^(5/3) + (3*a^(
2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*b^(5/3)) - (a^(2/3)*Log[a + b*x])/(2*b^(5/3))

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Rubi [A]  time = 0.0396263, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {50, 56, 617, 204, 31} \[ \frac{3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac{a^{2/3} \log (a+b x)}{2 b^{5/3}}+\frac{\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac{3 x^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^(2/3)/(a + b*x),x]

[Out]

(3*x^(2/3))/(2*b) + (Sqrt[3]*a^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/b^(5/3) + (3*a^(
2/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*b^(5/3)) - (a^(2/3)*Log[a + b*x])/(2*b^(5/3))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^{2/3}}{a+b x} \, dx &=\frac{3 x^{2/3}}{2 b}-\frac{a \int \frac{1}{\sqrt [3]{x} (a+b x)} \, dx}{b}\\ &=\frac{3 x^{2/3}}{2 b}-\frac{a^{2/3} \log (a+b x)}{2 b^{5/3}}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{\frac{a^{2/3}}{b^{2/3}}-\frac{\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{2 b^2}+\frac{\left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3}}\\ &=\frac{3 x^{2/3}}{2 b}+\frac{3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac{a^{2/3} \log (a+b x)}{2 b^{5/3}}-\frac{\left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{5/3}}\\ &=\frac{3 x^{2/3}}{2 b}+\frac{\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{5/3}}+\frac{3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac{a^{2/3} \log (a+b x)}{2 b^{5/3}}\\ \end{align*}

Mathematica [C]  time = 0.0066705, size = 29, normalized size = 0.26 \[ -\frac{3 x^{2/3} \left (\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x}{a}\right )-1\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)/(a + b*x),x]

[Out]

(-3*x^(2/3)*(-1 + Hypergeometric2F1[2/3, 1, 5/3, -((b*x)/a)]))/(2*b)

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Maple [A]  time = 0.005, size = 107, normalized size = 1. \begin{align*}{\frac{3}{2\,b}{x}^{{\frac{2}{3}}}}+{\frac{a}{{b}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a}{2\,{b}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}}{{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x^(2/3)/b+1/b^2*a/(1/b*a)^(1/3)*ln(x^(1/3)+(1/b*a)^(1/3))-1/2/b^2*a/(1/b*a)^(1/3)*ln(x^(2/3)-(1/b*a)^(1/3)
*x^(1/3)+(1/b*a)^(2/3))-1/b^2*a*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x^(1/3)-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78864, size = 336, normalized size = 3.03 \begin{align*} -\frac{2 \, \sqrt{3} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x^{\frac{1}{3}} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (-b x^{\frac{1}{3}} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a x^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a x^{\frac{1}{3}}\right ) - 3 \, x^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(3)*(a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x^(1/3)*(a^2/b^2)^(1/3) - sqrt(3)*a)/a) + (a^2/b^2)^(1
/3)*log(-b*x^(1/3)*(a^2/b^2)^(2/3) + a*x^(2/3) + a*(a^2/b^2)^(1/3)) - 2*(a^2/b^2)^(1/3)*log(b*(a^2/b^2)^(2/3)
+ a*x^(1/3)) - 3*x^(2/3))/b

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Sympy [A]  time = 12.8211, size = 228, normalized size = 2.05 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{2}{3}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{3 x^{\frac{5}{3}}}{5 a} & \text{for}\: b = 0 \\\frac{3 x^{\frac{2}{3}}}{2 b} & \text{for}\: a = 0 \\\frac{\left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + \sqrt [3]{x} \right )}}{b^{7} \left (\frac{1}{b}\right )^{\frac{16}{3}}} - \frac{\left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac{1}{b}} + 4 x^{\frac{2}{3}} \right )}}{2 b^{7} \left (\frac{1}{b}\right )^{\frac{16}{3}}} + \frac{\left (-1\right )^{\frac{2}{3}} \sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} \right )}}{b^{7} \left (\frac{1}{b}\right )^{\frac{16}{3}}} + \frac{3 x^{\frac{2}{3}}}{2 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x**(2/3), Eq(a, 0) & Eq(b, 0)), (3*x**(5/3)/(5*a), Eq(b, 0)), (3*x**(2/3)/(2*b), Eq(a, 0)), ((-
1)**(2/3)*a**(2/3)*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x**(1/3))/(b**7*(1/b)**(16/3)) - (-1)**(2/3)*a**(2
/3)*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x**(1/3)*(1/b)**(1/3) + 4*x**(2/3))/(2*b*
*7*(1/b)**(16/3)) + (-1)**(2/3)*sqrt(3)*a**(2/3)*atan(sqrt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*x**(1/3)/(3*a**(1/3)*(
1/b)**(1/3)))/(b**7*(1/b)**(16/3)) + 3*x**(2/3)/(2*b), True))

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Giac [A]  time = 1.0966, size = 159, normalized size = 1.43 \begin{align*} \frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{b} + \frac{3 \, x^{\frac{2}{3}}}{2 \, b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{b^{3}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/b + 3/2*x^(2/3)/b + sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2
*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/b^3 - 1/2*(-a*b^2)^(2/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2
/3))/b^3