Optimal. Leaf size=82 \[ \frac {\text {RootSum}\left [\text {$\#$1}^{12}-4 \text {$\#$1}^9 a+6 \text {$\#$1}^6 a^2-4 \text {$\#$1}^3 a^3+a^4-a b^3\& ,\frac {\log \left (\sqrt [3]{a x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{4 b} \]
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Rubi [B] time = 1.60, antiderivative size = 1208, normalized size of antiderivative = 14.73, number of steps used = 11, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2056, 6725, 912, 91} \begin {gather*} \frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt {-\sqrt {a}} x+\sqrt [4]{b}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{a} x+\sqrt [4]{b}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 91
Rule 912
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-b+a x^4\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}-\sqrt {a} x^2\right )}-\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}-\sqrt {a} x^2\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}+\sqrt {a} x^2\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}}+\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}}+\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}-\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}+\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}}}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}}}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}}}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}}}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}\\ \end {align*}
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Mathematica [C] time = 0.21, size = 167, normalized size = 2.04 \begin {gather*} -\frac {3 x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a x-\sqrt [4]{a} b^{3/4} x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a x-i \sqrt [4]{a} b^{3/4} x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a x+i \sqrt [4]{a} b^{3/4} x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a x+\sqrt [4]{a} b^{3/4} x}{a x-b}\right )\right )}{4 b \sqrt [3]{x^2 (a x-b)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.92, size = 82, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^6-4 a \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b} \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{4} - b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{3}} \left (a \,x^{4}-b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{4} - b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{\left (b-a\,x^4\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x^{4} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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