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

Optimal. Leaf size=143 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac{\log (a+b x)}{18 a^{4/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^{2/3}}{3 a b (a+b x)}-\frac{x^{2/3}}{2 b (a+b x)^2} \]

[Out]

-x^(2/3)/(2*b*(a + b*x)^2) + x^(2/3)/(3*a*b*(a + b*x)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3)
)]/(3*Sqrt[3]*a^(4/3)*b^(5/3)) - Log[a^(1/3) + b^(1/3)*x^(1/3)]/(6*a^(4/3)*b^(5/3)) + Log[a + b*x]/(18*a^(4/3)
*b^(5/3))

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Rubi [A]  time = 0.0499361, antiderivative size = 143, 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, Rules used = {47, 51, 56, 617, 204, 31} \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac{\log (a+b x)}{18 a^{4/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^{2/3}}{3 a b (a+b x)}-\frac{x^{2/3}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(2/3)/(a + b*x)^3,x]

[Out]

-x^(2/3)/(2*b*(a + b*x)^2) + x^(2/3)/(3*a*b*(a + b*x)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3)
)]/(3*Sqrt[3]*a^(4/3)*b^(5/3)) - Log[a^(1/3) + b^(1/3)*x^(1/3)]/(6*a^(4/3)*b^(5/3)) + Log[a + b*x]/(18*a^(4/3)
*b^(5/3))

Rule 47

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

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 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)^3} \, dx &=-\frac{x^{2/3}}{2 b (a+b x)^2}+\frac{\int \frac{1}{\sqrt [3]{x} (a+b x)^2} \, dx}{3 b}\\ &=-\frac{x^{2/3}}{2 b (a+b x)^2}+\frac{x^{2/3}}{3 a b (a+b x)}+\frac{\int \frac{1}{\sqrt [3]{x} (a+b x)} \, dx}{9 a b}\\ &=-\frac{x^{2/3}}{2 b (a+b x)^2}+\frac{x^{2/3}}{3 a b (a+b x)}+\frac{\log (a+b x)}{18 a^{4/3} b^{5/3}}+\frac{\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 b^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}\\ &=-\frac{x^{2/3}}{2 b (a+b x)^2}+\frac{x^{2/3}}{3 a b (a+b x)}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac{\log (a+b x)}{18 a^{4/3} b^{5/3}}+\frac{\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^{4/3} b^{5/3}}\\ &=-\frac{x^{2/3}}{2 b (a+b x)^2}+\frac{x^{2/3}}{3 a b (a+b x)}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{4/3} b^{5/3}}+\frac{\log (a+b x)}{18 a^{4/3} b^{5/3}}\\ \end{align*}

Mathematica [C]  time = 0.0048033, size = 27, normalized size = 0.19 \[ \frac{3 x^{5/3} \, _2F_1\left (\frac{5}{3},3;\frac{8}{3};-\frac{b x}{a}\right )}{5 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)/(a + b*x)^3,x]

[Out]

(3*x^(5/3)*Hypergeometric2F1[5/3, 3, 8/3, -((b*x)/a)])/(5*a^3)

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Maple [A]  time = 0.01, size = 132, normalized size = 0.9 \begin{align*} 3\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( 1/9\,{\frac{{x}^{5/3}}{a}}-1/18\,{\frac{{x}^{2/3}}{b}} \right ) }-{\frac{1}{9\,{b}^{2}a}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{18\,{b}^{2}a}\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{\sqrt{3}}{9\,{b}^{2}a}\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)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66037, size = 1208, normalized size = 8.45 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} a + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{2}{3}}\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{\frac{1}{3}}}{b x + a}\right ) +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{\frac{2}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{\frac{1}{3}} - \left (-a b^{2}\right )^{\frac{1}{3}}\right ) + 3 \,{\left (2 \, a b^{3} x - a^{2} b^{2}\right )} x^{\frac{2}{3}}}{18 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, \frac{6 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{\frac{2}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{\frac{1}{3}} - \left (-a b^{2}\right )^{\frac{1}{3}}\right ) + 3 \,{\left (2 \, a b^{3} x - a^{2} b^{2}\right )} x^{\frac{2}{3}}}{18 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.09467, size = 201, normalized size = 1.41 \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 )}{9 \, a^{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 )}{9 \, a^{2} b^{3}} + \frac{2 \, b x^{\frac{5}{3}} - a x^{\frac{2}{3}}}{6 \,{\left (b x + a\right )}^{2} a b} + \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 )}{18 \, a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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