3.695 \(\int \frac{1}{x^{2/3} (a+b x)^3} \, dx\)

Optimal. Leaf size=140 \[ \frac{5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac{5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{\sqrt [3]{x}}{2 a (a+b x)^2} \]

[Out]

x^(1/3)/(2*a*(a + b*x)^2) + (5*x^(1/3))/(6*a^2*(a + b*x)) - (5*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a
^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^(1/3)) + (5*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(6*a^(8/3)*b^(1/3)) - (5*Log[a + b*
x])/(18*a^(8/3)*b^(1/3))

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Rubi [A]  time = 0.0496618, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {51, 58, 617, 204, 31} \[ \frac{5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac{5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{\sqrt [3]{x}}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(2/3)*(a + b*x)^3),x]

[Out]

x^(1/3)/(2*a*(a + b*x)^2) + (5*x^(1/3))/(6*a^2*(a + b*x)) - (5*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a
^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^(1/3)) + (5*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(6*a^(8/3)*b^(1/3)) - (5*Log[a + b*
x])/(18*a^(8/3)*b^(1/3))

Rule 51

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

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(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{1}{x^{2/3} (a+b x)^3} \, dx &=\frac{\sqrt [3]{x}}{2 a (a+b x)^2}+\frac{5 \int \frac{1}{x^{2/3} (a+b x)^2} \, dx}{6 a}\\ &=\frac{\sqrt [3]{x}}{2 a (a+b x)^2}+\frac{5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac{5 \int \frac{1}{x^{2/3} (a+b x)} \, dx}{9 a^2}\\ &=\frac{\sqrt [3]{x}}{2 a (a+b x)^2}+\frac{5 \sqrt [3]{x}}{6 a^2 (a+b x)}-\frac{5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}+\frac{5 \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 )}{6 a^{7/3} b^{2/3}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}\\ &=\frac{\sqrt [3]{x}}{2 a (a+b x)^2}+\frac{5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac{5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{8/3} \sqrt [3]{b}}\\ &=\frac{\sqrt [3]{x}}{2 a (a+b x)^2}+\frac{5 \sqrt [3]{x}}{6 a^2 (a+b x)}-\frac{5 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac{5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [C]  time = 0.0039178, size = 25, normalized size = 0.18 \[ \frac{3 \sqrt [3]{x} \, _2F_1\left (\frac{1}{3},3;\frac{4}{3};-\frac{b x}{a}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(2/3)*(a + b*x)^3),x]

[Out]

(3*x^(1/3)*Hypergeometric2F1[1/3, 3, 4/3, -((b*x)/a)])/a^3

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Maple [A]  time = 0.006, size = 136, normalized size = 1. \begin{align*}{\frac{1}{2\,a \left ( bx+a \right ) ^{2}}\sqrt [3]{x}}+{\frac{5}{6\,{a}^{2} \left ( bx+a \right ) }\sqrt [3]{x}}+{\frac{5}{9\,{a}^{2}b}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5}{18\,{a}^{2}b}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}}{9\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(2/3)/(b*x+a)^3,x)

[Out]

1/2*x^(1/3)/a/(b*x+a)^2+5/6*x^(1/3)/a^2/(b*x+a)+5/9/a^2/b/(1/b*a)^(2/3)*ln(x^(1/3)+(1/b*a)^(1/3))-5/18/a^2/b/(
1/b*a)^(2/3)*ln(x^(2/3)-(1/b*a)^(1/3)*x^(1/3)+(1/b*a)^(2/3))+5/9/a^2/b/(1/b*a)^(2/3)*3^(1/2)*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(1/x^(2/3)/(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.6851, size = 1226, normalized size = 8.76 \begin{align*} \left [\frac{15 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}} a + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}}}{b x + a}\right ) - 5 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 10 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 3 \,{\left (5 \, a^{2} b^{2} x + 8 \, a^{3} b\right )} x^{\frac{1}{3}}}{18 \,{\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}}, \frac{30 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (a^{2} b\right )^{\frac{1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - 5 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 10 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 3 \,{\left (5 \, a^{2} b^{2} x + 8 \, a^{3} b\right )} x^{\frac{1}{3}}}{18 \,{\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(2/3)/(b*x+a)^3,x, algorithm="fricas")

[Out]

[1/18*(15*sqrt(1/3)*(a*b^3*x^2 + 2*a^2*b^2*x + a^3*b)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x - a^2 + 3*sqrt(1/3)*
(2*a*b*x^(2/3) - (a^2*b)^(1/3)*a + (a^2*b)^(2/3)*x^(1/3))*sqrt(-(a^2*b)^(1/3)/b) - 3*(a^2*b)^(1/3)*a*x^(1/3))/
(b*x + a)) - 5*(b^2*x^2 + 2*a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a^2*b)^(2/3)*x^(1/
3)) + 10*(b^2*x^2 + 2*a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(1/3) + (a^2*b)^(2/3)) + 3*(5*a^2*b^2*x + 8*a^3*b)*
x^(1/3))/(a^4*b^3*x^2 + 2*a^5*b^2*x + a^6*b), 1/18*(30*sqrt(1/3)*(a*b^3*x^2 + 2*a^2*b^2*x + a^3*b)*sqrt((a^2*b
)^(1/3)/b)*arctan(-sqrt(1/3)*((a^2*b)^(1/3)*a - 2*(a^2*b)^(2/3)*x^(1/3))*sqrt((a^2*b)^(1/3)/b)/a^2) - 5*(b^2*x
^2 + 2*a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a^2*b)^(2/3)*x^(1/3)) + 10*(b^2*x^2 + 2
*a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(1/3) + (a^2*b)^(2/3)) + 3*(5*a^2*b^2*x + 8*a^3*b)*x^(1/3))/(a^4*b^3*x^2
+ 2*a^5*b^2*x + a^6*b)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(2/3)/(b*x+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.07824, size = 193, normalized size = 1.38 \begin{align*} -\frac{5 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} + \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{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 )}{9 \, a^{3} b} + \frac{5 \, \left (-a b^{2}\right )^{\frac{1}{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 )}{18 \, a^{3} b} + \frac{5 \, b x^{\frac{4}{3}} + 8 \, a x^{\frac{1}{3}}}{6 \,{\left (b x + a\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(2/3)/(b*x+a)^3,x, algorithm="giac")

[Out]

-5/9*(-a/b)^(1/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 + 5/9*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x^(1
/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) + 5/18*(-a*b^2)^(1/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2
/3))/(a^3*b) + 1/6*(5*b*x^(4/3) + 8*a*x^(1/3))/((b*x + a)^2*a^2)