3.677 \(\int \frac {\sqrt [3]{x}}{a+b x} \, dx\)

Optimal. Leaf size=109 \[ -\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{4/3}}+\frac {\sqrt [3]{a} \log (a+b x)}{2 b^{4/3}}+\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{4/3}}+\frac {3 \sqrt [3]{x}}{b} \]

[Out]

3*x^(1/3)/b-3/2*a^(1/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/b^(4/3)+1/2*a^(1/3)*ln(b*x+a)/b^(4/3)+a^(1/3)*arctan(1/3*(
a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)/b^(4/3)

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Rubi [A]  time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, 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, 58, 617, 204, 31} \[ -\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{4/3}}+\frac {\sqrt [3]{a} \log (a+b x)}{2 b^{4/3}}+\frac {\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{4/3}}+\frac {3 \sqrt [3]{x}}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^(1/3)/(a + b*x),x]

[Out]

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

Rule 31

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

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 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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 126, normalized size = 1.16 \[ \frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+6 \sqrt [3]{b} \sqrt [3]{x}}{2 b^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1/3)/(a + b*x),x]

[Out]

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

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fricas [A]  time = 0.49, size = 114, normalized size = 1.05 \[ \frac {2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 6 \, x^{\frac {1}{3}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.15, size = 119, normalized size = 1.09 \[ \frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2}} + \frac {3 \, x^{\frac {1}{3}}}{b} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 108, normalized size = 0.99 \[ -\frac {\sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {a \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {a \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {3 x^{\frac {1}{3}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.93, size = 115, normalized size = 1.06 \[ -\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3 \, x^{\frac {1}{3}}}{b} + \frac {a \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 126, normalized size = 1.16 \[ \frac {3\,x^{1/3}}{b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (9\,{\left (-a\right )}^{4/3}\,b^{2/3}+9\,a\,b\,x^{1/3}\right )}{b^{4/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (9\,{\left (-a\right )}^{4/3}\,b^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+9\,a\,b\,x^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{4/3}}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (9\,{\left (-a\right )}^{4/3}\,b^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-9\,a\,b\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/3)/(a + b*x),x)

[Out]

(3*x^(1/3))/b + ((-a)^(1/3)*log(9*(-a)^(4/3)*b^(2/3) + 9*a*b*x^(1/3)))/b^(4/3) + ((-a)^(1/3)*log(9*(-a)^(4/3)*
b^(2/3)*((3^(1/2)*1i)/2 - 1/2) + 9*a*b*x^(1/3))*((3^(1/2)*1i)/2 - 1/2))/b^(4/3) - ((-a)^(1/3)*log(9*(-a)^(4/3)
*b^(2/3)*((3^(1/2)*1i)/2 + 1/2) - 9*a*b*x^(1/3))*((3^(1/2)*1i)/2 + 1/2))/b^(4/3)

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sympy [A]  time = 6.10, size = 219, normalized size = 2.01 \[ \begin {cases} \tilde {\infty } \sqrt [3]{x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {4}{3}}}{4 a} & \text {for}\: b = 0 \\\frac {3 \sqrt [3]{x}}{b} & \text {for}\: a = 0 \\\frac {\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sqrt [3]{x} \right )}}{b} - \frac {\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac {1}{b}} + 4 x^{\frac {2}{3}} \right )}}{2 b} - \frac {\sqrt [3]{-1} \sqrt {3} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )}}{b} + \frac {3 \sqrt [3]{x}}{b} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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