Optimal. Leaf size=68 \[ \frac {\text {RootSum}\left [a^6-a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(618\) vs. \(2(68)=136\).
time = 0.28, antiderivative size = 618, normalized size of antiderivative = 9.09, number of steps
used = 5, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2081, 926, 93}
\begin {gather*} \frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}-\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {a} b \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}+\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {a} b \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-a^{3/2} x\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (a^{3/2} x+\sqrt {b}\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}-\sqrt [3]{x}\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}-\sqrt [3]{x}\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 926
Rule 2081
Rubi steps
\begin {align*} \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^2\right )} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}\right ) \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a^3 x^3}}\\ &=\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{x}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{x}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}-a^{3/2} x\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}+a^{3/2} x\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 106, normalized size = 1.56 \begin {gather*} \frac {x^{2/3} \sqrt [3]{-b+a^3 x} \text {RootSum}\left [a^6-a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{3} x^{2}-b \right ) \left (a^{3} x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.40, size = 2040, normalized size = 30.00 \begin {gather*} \text {Too large to display} \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\right )} \left (a^{3} x^{2} - b\right )}\, dx \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*} \text {could not integrate} \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 (b-a^3\,x^2\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________