3.397 \(\int \frac{1}{x^2 \sqrt [3]{a+b x}} \, dx\)

Optimal. Leaf size=100 \[ \frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{(a+b x)^{2/3}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0334924, antiderivative size = 100, 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 = {51, 55, 617, 204, 31} \[ \frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{(a+b x)^{2/3}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^(1/3)),x]

[Out]

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

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 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[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] && PosQ
[(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 \sqrt [3]{a+b x}} \, dx &=-\frac{(a+b x)^{2/3}}{a x}-\frac{b \int \frac{1}{x \sqrt [3]{a+b x}} \, dx}{3 a}\\ &=-\frac{(a+b x)^{2/3}}{a x}+\frac{b \log (x)}{6 a^{4/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a}\\ &=-\frac{(a+b x)^{2/3}}{a x}+\frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{4/3}}\\ &=-\frac{(a+b x)^{2/3}}{a x}-\frac{b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.0078275, size = 33, normalized size = 0.33 \[ \frac{3 b (a+b x)^{2/3} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{b x}{a}+1\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^(1/3)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 95, normalized size = 1. \begin{align*} -{\frac{1}{ax} \left ( bx+a \right ) ^{{\frac{2}{3}}}}-{\frac{b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{4}{3}}}}+{\frac{b}{6}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{4}{3}}}}-{\frac{b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(2/3)/a/x-1/3*b/a^(4/3)*ln((b*x+a)^(1/3)-a^(1/3))+1/6*b/a^(4/3)*ln((b*x+a)^(2/3)+a^(1/3)*(b*x+a)^(1/3
)+a^(2/3))-1/3*b/a^(4/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(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/(b*x+a)^(1/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.98536, size = 867, normalized size = 8.67 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b x \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b x - 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} a + \left (-a\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} - 3 \,{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + 3 \, a}{x}\right ) + \left (-a\right )^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a\right )^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{6 \, a^{2} x}, -\frac{6 \, \sqrt{\frac{1}{3}} a b x \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \arctan \left (\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}}\right ) - \left (-a\right )^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) + 2 \, \left (-a\right )^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) + 6 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{6 \, a^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 3.08162, size = 831, normalized size = 8.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**(5/3)*b**(7/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(2/3)/(9*
a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma
(5/3)) - 2*a**(5/3)*b**(7/3)*(a/b + x)**(4/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*p
i/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b +
x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) - 2*a**(5/3)*b**(7/3)*(a/b + x)**(4/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*e
xp_polar(4*I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b*
*(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) + 2*a**(2/3)*b**(10/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*log(1
- b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) -
9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) + 2*a**(2/3)*b**(10/3)*(a/b + x)**(7/3)*exp(-2*I*pi
/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/
3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) + 2*a**(2/3)*b**(10/3
)*(a/b + x)**(7/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(2/3)/(9*a**3*b**(4/3)
*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3)) + 6*a*
b**3*(a/b + x)**2*exp(2*I*pi/3)*gamma(2/3)/(9*a**3*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) - 9*a**2
*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3))

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Giac [A]  time = 2.13989, size = 147, normalized size = 1.47 \begin{align*} -\frac{\frac{2 \, \sqrt{3} b^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{4}{3}}} - \frac{b^{2} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{4}{3}}} + \frac{2 \, b^{2} \log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{4}{3}}} + \frac{6 \,{\left (b x + a\right )}^{\frac{2}{3}} b}{a x}}{6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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