Optimal. Leaf size=68 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+a b^3\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{2 b} \]
________________________________________________________________________________________
Rubi [B] time = 0.34, antiderivative size = 633, normalized size of antiderivative = 9.31, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2056, 912, 91} \begin {gather*} \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {-a} x+\sqrt {b}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 91
Rule 912
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (b+a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+a x^2\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+\sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{2 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}+\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 103, normalized size = 1.51 \begin {gather*} \frac {3 x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3-\sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3+\sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )\right )}{2 b \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 68, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 2062, normalized size = 30.32
result too large to display
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^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{2} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{2}+b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{2} + 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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (a\,x^2+b\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]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{2} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________