Optimal. Leaf size=332 \[ \frac {(-1)^{5/6} \sqrt {3} \tanh ^{-1}\left (\frac {\frac {i \sqrt [3]{a^3 x^3-b^3}}{\sqrt {3}}+\frac {\left (-\sqrt {3} a-i a\right ) x}{2^{2/3} \sqrt {3}}+\frac {\sqrt {3} b+i b}{2^{2/3} \sqrt {3}}}{\sqrt [3]{a^3 x^3-b^3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\sqrt [3]{-\frac {1}{2}} \log \left ((-1)^{2/3} a^{3/2} \sqrt {b} x-2^{2/3} \sqrt {a} \sqrt {b} \sqrt [3]{a^3 x^3-b^3}-(-1)^{2/3} \sqrt {a} b^{3/2}\right )}{2 a b}-\frac {\sqrt [3]{-\frac {1}{2}} \log \left (-2 \sqrt [3]{2} a b \left (a^3 x^3-b^3\right )^{2/3}+\sqrt [3]{-1} a^3 b x^2-2 \sqrt [3]{-1} a^2 b^2 x+\left ((-2)^{2/3} a b^2-(-2)^{2/3} a^2 b x\right ) \sqrt [3]{a^3 x^3-b^3}+\sqrt [3]{-1} a b^3\right )}{4 a b} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 139, normalized size of antiderivative = 0.42, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2148} \begin {gather*} -\frac {3 \log \left (2^{2/3} a \sqrt [3]{a^3 x^3-b^3}+a (b-a x)\right )}{4 \sqrt [3]{2} a b}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (b-a x)}{\sqrt [3]{a^3 x^3-b^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\log \left ((b-a x) (a x+b)^2\right )}{4 \sqrt [3]{2} a b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2148
Rubi steps
\begin {align*} \int \frac {1}{(b+a x) \sqrt [3]{-b^3+a^3 x^3}} \, dx &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (b-a x)}{\sqrt [3]{-b^3+a^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\log \left ((b-a x) (b+a x)^2\right )}{4 \sqrt [3]{2} a b}-\frac {3 \log \left (a (b-a x)+2^{2/3} a \sqrt [3]{-b^3+a^3 x^3}\right )}{4 \sqrt [3]{2} a b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(b+a x) \sqrt [3]{-b^3+a^3 x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.34, size = 385, normalized size = 1.16 \begin {gather*} \frac {(-1)^{5/6} \sqrt {3} \tanh ^{-1}\left (\frac {\frac {i b+\sqrt {3} b}{2^{2/3} \sqrt {3}}+\frac {\left (-i a-\sqrt {3} a\right ) x}{2^{2/3} \sqrt {3}}+\frac {i \sqrt [3]{-b^3+a^3 x^3}}{\sqrt {3}}}{\sqrt [3]{-b^3+a^3 x^3}}\right )}{2 \sqrt [3]{2} a b}+\frac {\sqrt [3]{-\frac {1}{2}} \log \left (\left (-1+i \sqrt {3}\right ) \sqrt {a} b^{3/2}+a^{3/2} \sqrt {b} x-i \sqrt {3} a^{3/2} \sqrt {b} x+2\ 2^{2/3} \sqrt {a} \sqrt {b} \sqrt [3]{-b^3+a^3 x^3}\right )}{2 a b}-\frac {\sqrt [3]{-\frac {1}{2}} \log \left (-a b^3-i \sqrt {3} a b^3+2 a^2 b^2 x+2 i \sqrt {3} a^2 b^2 x-a^3 b x^2-i \sqrt {3} a^3 b x^2+\left (-2 (-2)^{2/3} a b^2+2 (-2)^{2/3} a^2 b x\right ) \sqrt [3]{-b^3+a^3 x^3}+4 \sqrt [3]{2} a b \left (-b^3+a^3 x^3\right )^{2/3}\right )}{4 a b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} {\left (a x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x +b \right ) \left (a^{3} x^{3}-b^{3}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} {\left (a x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a^3\,x^3-b^3\right )}^{1/3}\,\left (b+a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{\left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )} \left (a x + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________