Optimal. Leaf size=166 \[ -\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}{\sqrt {a x^3-b}-\sqrt {b}}\right )}{16 \sqrt {2} b^{7/4}}+\frac {a^2 \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^3-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^3-b}}\right )}{16 \sqrt {2} b^{7/4}}+\frac {\left (a x^3-4 b\right ) \sqrt [4]{a x^3-b}}{24 b x^6} \]
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Rubi [A] time = 0.25, antiderivative size = 257, normalized size of antiderivative = 1.55, number of steps used = 13, number of rules used = 10, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {266, 47, 51, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {a^2 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{32 \sqrt {2} b^{7/4}}+\frac {a^2 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{32 \sqrt {2} b^{7/4}}-\frac {a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{16 \sqrt {2} b^{7/4}}+\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{16 \sqrt {2} b^{7/4}}+\frac {a \sqrt [4]{a x^3-b}}{24 b x^3}-\frac {\sqrt [4]{a x^3-b}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 204
Rule 211
Rule 266
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b+a x^3}}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-b+a x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {1}{24} a \operatorname {Subst}\left (\int \frac {1}{x^2 (-b+a x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {a \sqrt [4]{-b+a x^3}}{24 b x^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x (-b+a x)^{3/4}} \, dx,x,x^3\right )}{32 b}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {a \sqrt [4]{-b+a x^3}}{24 b x^3}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{8 b}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {a \sqrt [4]{-b+a x^3}}{24 b x^3}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{16 b^{3/2}}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{16 b^{3/2}}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {a \sqrt [4]{-b+a x^3}}{24 b x^3}-\frac {a^2 \operatorname {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 )}{32 \sqrt {2} b^{7/4}}-\frac {a^2 \operatorname {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 )}{32 \sqrt {2} b^{7/4}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{32 b^{3/2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{32 b^{3/2}}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {a \sqrt [4]{-b+a x^3}}{24 b x^3}-\frac {a^2 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{32 \sqrt {2} b^{7/4}}+\frac {a^2 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{32 \sqrt {2} b^{7/4}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{16 \sqrt {2} b^{7/4}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{16 \sqrt {2} b^{7/4}}\\ &=-\frac {\sqrt [4]{-b+a x^3}}{6 x^6}+\frac {a \sqrt [4]{-b+a x^3}}{24 b x^3}-\frac {a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{16 \sqrt {2} b^{7/4}}+\frac {a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{16 \sqrt {2} b^{7/4}}-\frac {a^2 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{32 \sqrt {2} b^{7/4}}+\frac {a^2 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{32 \sqrt {2} b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.25 \begin {gather*} \frac {4 a^2 \left (a x^3-b\right )^{5/4} \, _2F_1\left (\frac {5}{4},3;\frac {9}{4};1-\frac {a x^3}{b}\right )}{15 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 165, normalized size = 0.99 \begin {gather*} \frac {\left (-4 b+a x^3\right ) \sqrt [4]{-b+a x^3}}{24 b x^6}+\frac {a^2 \tan ^{-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 )}{16 \sqrt {2} b^{7/4}}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{\sqrt {b}+\sqrt {-b+a x^3}}\right )}{16 \sqrt {2} b^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 224, normalized size = 1.35 \begin {gather*} \frac {12 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b x^{6} \arctan \left (-\frac {{\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{2} \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {3}{4}} b^{5} - \sqrt {\sqrt {a x^{3} - b} a^{4} + \sqrt {-\frac {a^{8}}{b^{7}}} b^{4}} \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {3}{4}} b^{5}}{a^{8}}\right ) + 3 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b x^{6} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{2} + \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2}\right ) - 3 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b x^{6} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{2} - \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2}\right ) + 4 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}} {\left (a x^{3} - 4 \, b\right )}}{96 \, b x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 223, normalized size = 1.34 \begin {gather*} \frac {\frac {6 \, \sqrt {2} a^{3} \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 {7}{4}}} + \frac {6 \, \sqrt {2} a^{3} \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 {7}{4}}} + \frac {3 \, \sqrt {2} a^{3} \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 {7}{4}}} - \frac {3 \, \sqrt {2} a^{3} \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 {7}{4}}} + \frac {8 \, {\left ({\left (a x^{3} - b\right )}^{\frac {5}{4}} a^{3} - 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3} b\right )}}{a^{2} b x^{6}}}{192 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}-b \right )^{\frac {1}{4}}}{x^{7}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 247, normalized size = 1.49 \begin {gather*} \frac {{\left (a x^{3} - b\right )}^{\frac {5}{4}} a^{2} - 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{2} b}{24 \, {\left ({\left (a x^{3} - b\right )}^{2} b + 2 \, {\left (a x^{3} - b\right )} b^{2} + b^{3}\right )}} + \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}}}}{64 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 95, normalized size = 0.57 \begin {gather*} \frac {{\left (a\,x^3-b\right )}^{5/4}}{24\,b\,x^6}-\frac {{\left (a\,x^3-b\right )}^{1/4}}{8\,x^6}+\frac {a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{16\,{\left (-b\right )}^{7/4}}-\frac {a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,1{}\mathrm {i}}{16\,{\left (-b\right )}^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.36, size = 44, normalized size = 0.27 \begin {gather*} - \frac {\sqrt [4]{a} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 x^{\frac {21}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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