3.689 \(\int \frac{1}{x^{5/3} (a+b x)^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0501187, antiderivative size = 128, 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} \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/3)*(a + b*x)^2),x]

[Out]

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

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^{5/3} (a+b x)^2} \, dx &=\frac{1}{a x^{2/3} (a+b x)}+\frac{5 \int \frac{1}{x^{5/3} (a+b x)} \, dx}{3 a}\\ &=-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)}-\frac{(5 b) \int \frac{1}{x^{2/3} (a+b x)} \, dx}{3 a^2}\\ &=-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}-\frac{\left (5 \sqrt [3]{b}\right ) \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 a^{7/3}}-\frac{\left (5 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 a^{8/3}}\\ &=-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)}-\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}-\frac{\left (5 b^{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 )}{a^{8/3}}\\ &=-\frac{5}{2 a^2 x^{2/3}}+\frac{1}{a x^{2/3} (a+b x)}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{8/3}}-\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac{5 b^{2/3} \log (a+b x)}{6 a^{8/3}}\\ \end{align*}

Mathematica [C]  time = 0.0050339, size = 27, normalized size = 0.21 \[ -\frac{3 \, _2F_1\left (-\frac{2}{3},2;\frac{1}{3};-\frac{b x}{a}\right )}{2 a^2 x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/3)*(a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 121, normalized size = 1. \begin{align*} -{\frac{3}{2\,{a}^{2}}{x}^{-{\frac{2}{3}}}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }\sqrt [3]{x}}-{\frac{5}{3\,{a}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5}{6\,{a}^{2}}\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}}{3\,{a}^{2}}\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^(5/3)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.45074, size = 456, normalized size = 3.56 \begin{align*} \frac{10 \, \sqrt{3}{\left (b x^{2} + a x\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x^{\frac{1}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - 5 \,{\left (b x^{2} + a x\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + 10 \,{\left (b x^{2} + a x\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{\frac{1}{3}} - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 3 \,{\left (5 \, b x + 3 \, a\right )} x^{\frac{1}{3}}}{6 \,{\left (a^{2} b x^{2} + a^{3} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/3)/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/6*(10*sqrt(3)*(b*x^2 + a*x)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x^(1/3)*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b
) - 5*(b*x^2 + a*x)*(-b^2/a^2)^(1/3)*log(b^2*x^(2/3) + a*b*x^(1/3)*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) +
10*(b*x^2 + a*x)*(-b^2/a^2)^(1/3)*log(b*x^(1/3) - a*(-b^2/a^2)^(1/3)) - 3*(5*b*x + 3*a)*x^(1/3))/(a^2*b*x^2 +
a^3*x)

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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**(5/3)/(b*x+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.08042, size = 185, normalized size = 1.45 \begin{align*} \frac{5 \, b \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 )}{3 \, 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 )}{3 \, a^{3}} - \frac{b x^{\frac{1}{3}}}{{\left (b x + a\right )} a^{2}} - \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 )}{6 \, a^{3}} - \frac{3}{2 \, a^{2} x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/3)/(b*x+a)^2,x, algorithm="giac")

[Out]

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