Optimal. Leaf size=167 \[ -\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}{\sqrt {a x^5-b}-\sqrt {b}}\right )}{80 \sqrt {2} b^{11/4}}+\frac {21 a^2 \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^5-b}}\right )}{80 \sqrt {2} b^{11/4}}+\frac {\sqrt [4]{a x^5-b} \left (7 a x^5+4 b\right )}{40 b^2 x^{10}} \]
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Rubi [A] time = 0.25, antiderivative size = 260, normalized size of antiderivative = 1.56, number of steps used = 13, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {266, 51, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {21 a^2 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {a x^5-b}+\sqrt {b}\right )}{160 \sqrt {2} b^{11/4}}+\frac {21 a^2 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {a x^5-b}+\sqrt {b}\right )}{160 \sqrt {2} b^{11/4}}-\frac {21 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{80 \sqrt {2} b^{11/4}}+\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{80 \sqrt {2} b^{11/4}}+\frac {7 a \sqrt [4]{a x^5-b}}{40 b^2 x^5}+\frac {\sqrt [4]{a x^5-b}}{10 b x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 204
Rule 211
Rule 266
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^{11} \left (-b+a x^5\right )^{3/4}} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x^3 (-b+a x)^{3/4}} \, dx,x,x^5\right )\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {(7 a) \operatorname {Subst}\left (\int \frac {1}{x^2 (-b+a x)^{3/4}} \, dx,x,x^5\right )}{40 b}\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {7 a \sqrt [4]{-b+a x^5}}{40 b^2 x^5}+\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (-b+a x)^{3/4}} \, dx,x,x^5\right )}{160 b^2}\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {7 a \sqrt [4]{-b+a x^5}}{40 b^2 x^5}+\frac {(21 a) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{40 b^2}\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {7 a \sqrt [4]{-b+a x^5}}{40 b^2 x^5}+\frac {(21 a) \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{80 b^{5/2}}+\frac {(21 a) \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{80 b^{5/2}}\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {7 a \sqrt [4]{-b+a x^5}}{40 b^2 x^5}-\frac {\left (21 a^2\right ) \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^5}\right )}{160 \sqrt {2} b^{11/4}}-\frac {\left (21 a^2\right ) \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^5}\right )}{160 \sqrt {2} b^{11/4}}+\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{160 b^{5/2}}+\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{160 b^{5/2}}\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {7 a \sqrt [4]{-b+a x^5}}{40 b^2 x^5}-\frac {21 a^2 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{160 \sqrt {2} b^{11/4}}+\frac {21 a^2 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{160 \sqrt {2} b^{11/4}}+\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{80 \sqrt {2} b^{11/4}}-\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{80 \sqrt {2} b^{11/4}}\\ &=\frac {\sqrt [4]{-b+a x^5}}{10 b x^{10}}+\frac {7 a \sqrt [4]{-b+a x^5}}{40 b^2 x^5}-\frac {21 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{80 \sqrt {2} b^{11/4}}+\frac {21 a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{80 \sqrt {2} b^{11/4}}-\frac {21 a^2 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{160 \sqrt {2} b^{11/4}}+\frac {21 a^2 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{160 \sqrt {2} b^{11/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.25 \begin {gather*} \frac {4 a^2 \sqrt [4]{a x^5-b} \, _2F_1\left (\frac {1}{4},3;\frac {5}{4};1-\frac {a x^5}{b}\right )}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 166, normalized size = 0.99 \begin {gather*} \frac {\sqrt [4]{-b+a x^5} \left (4 b+7 a x^5\right )}{40 b^2 x^{10}}+\frac {21 a^2 \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^5}}\right )}{80 \sqrt {2} b^{11/4}}+\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}{\sqrt {b}+\sqrt {-b+a x^5}}\right )}{80 \sqrt {2} b^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 234, normalized size = 1.40 \begin {gather*} \frac {84 \, b^{2} x^{10} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{2} b^{8} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {3}{4}} - \sqrt {b^{6} \sqrt {-\frac {a^{8}}{b^{11}}} + \sqrt {a x^{5} - b} a^{4}} b^{8} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {3}{4}}}{a^{8}}\right ) + 21 \, b^{2} x^{10} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (21 \, b^{3} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{2}\right ) - 21 \, b^{2} x^{10} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-21 \, b^{3} \left (-\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{2}\right ) + 4 \, {\left (7 \, a x^{5} + 4 \, b\right )} {\left (a x^{5} - b\right )}^{\frac {1}{4}}}{160 \, b^{2} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 224, normalized size = 1.34 \begin {gather*} \frac {\frac {42 \, \sqrt {2} a^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {11}{4}}} + \frac {42 \, \sqrt {2} a^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {11}{4}}} + \frac {21 \, \sqrt {2} a^{3} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {11}{4}}} - \frac {21 \, \sqrt {2} a^{3} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {11}{4}}} + \frac {8 \, {\left (7 \, {\left (a x^{5} - b\right )}^{\frac {5}{4}} a^{3} + 11 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{3} b\right )}}{a^{2} b^{2} x^{10}}}{320 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{11} \left (a \,x^{5}-b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 250, normalized size = 1.50 \begin {gather*} \frac {7 \, {\left (a x^{5} - b\right )}^{\frac {5}{4}} a^{2} + 11 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{2} b}{40 \, {\left ({\left (a x^{5} - b\right )}^{2} b^{2} + 2 \, {\left (a x^{5} - b\right )} b^{3} + b^{4}\right )}} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - 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^{5} - 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^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {3}{4}}}\right )}}{320 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 98, normalized size = 0.59 \begin {gather*} \frac {11\,{\left (a\,x^5-b\right )}^{1/4}}{40\,b\,x^{10}}+\frac {7\,{\left (a\,x^5-b\right )}^{5/4}}{40\,b^2\,x^{10}}-\frac {21\,a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{80\,{\left (-b\right )}^{11/4}}+\frac {a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,21{}\mathrm {i}}{80\,{\left (-b\right )}^{11/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.88, size = 42, normalized size = 0.25 \begin {gather*} - \frac {\Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{5}}} \right )}}{5 a^{\frac {3}{4}} x^{\frac {55}{4}} \Gamma \left (\frac {15}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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