Optimal. Leaf size=150 \[ -\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{-\sqrt {b}+\sqrt {-b+a x^3}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^3}}\right )}{6 \sqrt {2} b^{3/4}} \]
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Rubi [A]
time = 0.15, antiderivative size = 225, normalized size of antiderivative = 1.50, number of steps
used = 12, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {272, 43, 65,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {a \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{6 \sqrt {2} b^{3/4}}-\frac {a \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{12 \sqrt {2} b^{3/4}}-\frac {\sqrt [4]{a x^3-b}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 210
Rule 217
Rule 272
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b+a x^3}}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [4]{-b+a x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}+\frac {1}{12} a \text {Subst}\left (\int \frac {1}{x (-b+a x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )\\ &=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}+\frac {\text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{6 \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{6 \sqrt {b}}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {b}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {b}}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}-\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{3 x^3}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}+\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{3/4}}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 145, normalized size = 0.97 \begin {gather*} \frac {-4 b^{3/4} \sqrt [4]{-b+a x^3}+\sqrt {2} a x^3 \text {ArcTan}\left (\frac {-\sqrt {b}+\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}\right )+\sqrt {2} a x^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{\sqrt {b}+\sqrt {-b+a x^3}}\right )}{12 b^{3/4} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}-b \right )^{\frac {1}{4}}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 182, normalized size = 1.21 \begin {gather*} \frac {\sqrt {2} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{12 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{12 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{24 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} a \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{24 \, b^{\frac {3}{4}}} - \frac {{\left (a x^{3} - b\right )}^{\frac {1}{4}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 198, normalized size = 1.32 \begin {gather*} \frac {4 \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \arctan \left (-\frac {{\left (a x^{3} - b\right )}^{\frac {1}{4}} a \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {3}{4}} b^{2} - \sqrt {\sqrt {a x^{3} - b} a^{2} + \sqrt {-\frac {a^{4}}{b^{3}}} b^{2}} \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {3}{4}} b^{2}}{a^{4}}\right ) + \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a + \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} x^{3} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a - \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b\right ) - 4 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.72, size = 44, normalized size = 0.29 \begin {gather*} - \frac {\sqrt [4]{a} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 x^{\frac {9}{4}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 195, normalized size = 1.30 \begin {gather*} \frac {\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {\sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {8 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}} a}{x^{3}}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 69, normalized size = 0.46 \begin {gather*} -\frac {{\left (a\,x^3-b\right )}^{1/4}}{3\,x^3}-\frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{3/4}}-\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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