Optimal. Leaf size=44 \[ -3 b^3 \log \left (\frac {b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)+3 b^2 \sqrt [3]{x}-\frac {3}{2} b x^{2/3}+x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 44, 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 = {190, 44} \[ 3 b^2 \sqrt [3]{x}-3 b^3 \log \left (\frac {b}{\sqrt [3]{x}}+1\right )-b^3 \log (x)-\frac {3}{2} b x^{2/3}+x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{1+\frac {b}{\sqrt [3]{x}}} \, dx &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{x^4 (1+b x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname {Subst}\left (\int \left (\frac {1}{x^4}-\frac {b}{x^3}+\frac {b^2}{x^2}-\frac {b^3}{x}+\frac {b^4}{1+b x}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=3 b^2 \sqrt [3]{x}-\frac {3}{2} b x^{2/3}+x-3 b^3 \log \left (1+\frac {b}{\sqrt [3]{x}}\right )-b^3 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 35, normalized size = 0.80 \[ -3 b^3 \log \left (b+\sqrt [3]{x}\right )+3 b^2 \sqrt [3]{x}-\frac {3}{2} b x^{2/3}+x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 27, normalized size = 0.61 \[ -3 \, b^{3} \log \left (b + x^{\frac {1}{3}}\right ) + 3 \, b^{2} x^{\frac {1}{3}} - \frac {3}{2} \, b x^{\frac {2}{3}} + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 28, normalized size = 0.64 \[ -3 \, b^{3} \log \left ({\left | b + x^{\frac {1}{3}} \right |}\right ) + 3 \, b^{2} x^{\frac {1}{3}} - \frac {3}{2} \, b x^{\frac {2}{3}} + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 28, normalized size = 0.64 \[ -3 b^{3} \ln \left (b +x^{\frac {1}{3}}\right )+3 b^{2} x^{\frac {1}{3}}-\frac {3 b \,x^{\frac {2}{3}}}{2}+x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 40, normalized size = 0.91 \[ -b^{3} \log \relax (x) - 3 \, b^{3} \log \left (\frac {b}{x^{\frac {1}{3}}} + 1\right ) + \frac {1}{2} \, {\left (\frac {6 \, b^{2}}{x^{\frac {2}{3}}} - \frac {3 \, b}{x^{\frac {1}{3}}} + 2\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 27, normalized size = 0.61 \[ x-3\,b^3\,\ln \left (b+x^{1/3}\right )-\frac {3\,b\,x^{2/3}}{2}+3\,b^2\,x^{1/3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 34, normalized size = 0.77 \[ - 3 b^{3} \log {\left (b + \sqrt [3]{x} \right )} + 3 b^{2} \sqrt [3]{x} - \frac {3 b x^{\frac {2}{3}}}{2} + x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________