Optimal. Leaf size=77 \[ \frac {3 b^3}{a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 b^2 \sqrt [3]{x}}{a^4}-\frac {3 b x^{2/3}}{a^3}+\frac {x}{a^2}-\frac {12 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 46}
\begin {gather*} -\frac {12 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5}+\frac {3 b^3}{a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 b^2 \sqrt [3]{x}}{a^4}-\frac {3 b x^{2/3}}{a^3}+\frac {x}{a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 196
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx &=-\left (3 \text {Subst}\left (\int \frac {1}{x^4 (a+b x)^2} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \text {Subst}\left (\int \left (\frac {1}{a^2 x^4}-\frac {2 b}{a^3 x^3}+\frac {3 b^2}{a^4 x^2}-\frac {4 b^3}{a^5 x}+\frac {b^4}{a^4 (a+b x)^2}+\frac {4 b^4}{a^5 (a+b x)}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {3 b^3}{a^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 b^2 \sqrt [3]{x}}{a^4}-\frac {3 b x^{2/3}}{a^3}+\frac {x}{a^2}-\frac {12 b^3 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 80, normalized size = 1.04 \begin {gather*} \frac {-3 b^4+9 a b^3 \sqrt [3]{x}+6 a^2 b^2 x^{2/3}-2 a^3 b x+a^4 x^{4/3}}{a^5 \left (b+a \sqrt [3]{x}\right )}-\frac {12 b^3 \log \left (b+a \sqrt [3]{x}\right )}{a^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 62, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {a^{2} x -3 a b \,x^{\frac {2}{3}}+9 b^{2} x^{\frac {1}{3}}}{a^{4}}-\frac {3 b^{4}}{a^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {12 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{5}}\) | \(62\) |
default | \(\frac {a^{2} x -3 a b \,x^{\frac {2}{3}}+9 b^{2} x^{\frac {1}{3}}}{a^{4}}-\frac {3 b^{4}}{a^{5} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {12 b^{3} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{5}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 76, normalized size = 0.99 \begin {gather*} \frac {a^{3} - \frac {2 \, a^{2} b}{x^{\frac {1}{3}}} + \frac {6 \, a b^{2}}{x^{\frac {2}{3}}} + \frac {12 \, b^{3}}{x}}{\frac {a^{5}}{x} + \frac {a^{4} b}{x^{\frac {4}{3}}}} - \frac {12 \, b^{3} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{5}} - \frac {4 \, b^{3} \log \left (x\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 100, normalized size = 1.30 \begin {gather*} \frac {a^{6} x^{2} + a^{3} b^{3} x - 3 \, b^{6} - 12 \, {\left (a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 3 \, {\left (a^{5} b x + 2 \, a^{2} b^{4}\right )} x^{\frac {2}{3}} + 3 \, {\left (3 \, a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac {1}{3}}}{a^{8} x + a^{5} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs.
\(2 (76) = 152\).
time = 0.23, size = 165, normalized size = 2.14 \begin {gather*} \begin {cases} \frac {a^{4} x^{\frac {4}{3}}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {2 a^{3} b x}{a^{6} \sqrt [3]{x} + a^{5} b} + \frac {6 a^{2} b^{2} x^{\frac {2}{3}}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {12 a b^{3} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {12 b^{4} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac {12 b^{4}}{a^{6} \sqrt [3]{x} + a^{5} b} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {5}{3}}}{5 b^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.53, size = 65, normalized size = 0.84 \begin {gather*} -\frac {12 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{5}} - \frac {3 \, b^{4}}{{\left (a x^{\frac {1}{3}} + b\right )} a^{5}} + \frac {a^{4} x - 3 \, a^{3} b x^{\frac {2}{3}} + 9 \, a^{2} b^{2} x^{\frac {1}{3}}}{a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.13, size = 65, normalized size = 0.84 \begin {gather*} \frac {x}{a^2}-\frac {3\,b^4}{a\,\left (a^4\,b+a^5\,x^{1/3}\right )}-\frac {3\,b\,x^{2/3}}{a^3}-\frac {12\,b^3\,\ln \left (b+a\,x^{1/3}\right )}{a^5}+\frac {9\,b^2\,x^{1/3}}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________