Optimal. Leaf size=92 \[ -\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}-\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}+\frac {\sqrt [4]{a x^4+b} \left (7 a x^4-4 b\right )}{32 b^2 x^8} \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 51, 63, 212, 206, 203} \begin {gather*} -\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}-\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}+\frac {7 a \sqrt [4]{a x^4+b}}{32 b^2 x^4}-\frac {\sqrt [4]{a x^4+b}}{8 b x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^9 \left (b+a x^4\right )^{3/4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^3 (b+a x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [4]{b+a x^4}}{8 b x^8}-\frac {(7 a) \operatorname {Subst}\left (\int \frac {1}{x^2 (b+a x)^{3/4}} \, dx,x,x^4\right )}{32 b}\\ &=-\frac {\sqrt [4]{b+a x^4}}{8 b x^8}+\frac {7 a \sqrt [4]{b+a x^4}}{32 b^2 x^4}+\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^{3/4}} \, dx,x,x^4\right )}{128 b^2}\\ &=-\frac {\sqrt [4]{b+a x^4}}{8 b x^8}+\frac {7 a \sqrt [4]{b+a x^4}}{32 b^2 x^4}+\frac {(21 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^4}\right )}{32 b^2}\\ &=-\frac {\sqrt [4]{b+a x^4}}{8 b x^8}+\frac {7 a \sqrt [4]{b+a x^4}}{32 b^2 x^4}-\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )}{64 b^{5/2}}-\frac {\left (21 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )}{64 b^{5/2}}\\ &=-\frac {\sqrt [4]{b+a x^4}}{8 b x^8}+\frac {7 a \sqrt [4]{b+a x^4}}{32 b^2 x^4}-\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}-\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.40 \begin {gather*} -\frac {a^2 \sqrt [4]{a x^4+b} \, _2F_1\left (\frac {1}{4},3;\frac {5}{4};\frac {a x^4}{b}+1\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 92, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{b+a x^4} \left (-4 b+7 a x^4\right )}{32 b^2 x^8}-\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{64 b^{11/4}}-\frac {21 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{64 b^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 216, normalized size = 2.35 \begin {gather*} \frac {84 \, b^{2} x^{8} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{4} + 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^{4} + b} a^{4}} b^{8} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {3}{4}}}{a^{8}}\right ) - 21 \, b^{2} x^{8} \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^{4} + b\right )}^{\frac {1}{4}} a^{2}\right ) + 21 \, b^{2} x^{8} \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^{4} + b\right )}^{\frac {1}{4}} a^{2}\right ) + 4 \, {\left (7 \, a x^{4} - 4 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{128 \, b^{2} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 244, normalized size = 2.65 \begin {gather*} \frac {\frac {42 \, \sqrt {2} a^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {42 \, \sqrt {2} a^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {21 \, \sqrt {2} a^{3} \log \left (\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {21 \, \sqrt {2} a^{3} \left (-b\right )^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right )}{b^{3}} + \frac {8 \, {\left (7 \, {\left (a x^{4} + b\right )}^{\frac {5}{4}} a^{3} - 11 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} a^{3} b\right )}}{a^{2} b^{2} x^{8}}}{256 \, 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 \[\int \frac {1}{x^{9} \left (a \,x^{4}+b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 132, normalized size = 1.43 \begin {gather*} \frac {7 \, {\left (a x^{4} + b\right )}^{\frac {5}{4}} a^{2} - 11 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} a^{2} b}{32 \, {\left ({\left (a x^{4} + b\right )}^{2} b^{2} - 2 \, {\left (a x^{4} + b\right )} b^{3} + b^{4}\right )}} - \frac {21 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a^{2} \log \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}}\right )}}{128 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 82, normalized size = 0.89 \begin {gather*} \frac {7\,{\left (a\,x^4+b\right )}^{5/4}}{32\,b^2\,x^8}-\frac {11\,{\left (a\,x^4+b\right )}^{1/4}}{32\,b\,x^8}-\frac {21\,a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )}{64\,b^{11/4}}+\frac {a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}\,1{}\mathrm {i}}{b^{1/4}}\right )\,21{}\mathrm {i}}{64\,b^{11/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.60, size = 39, normalized size = 0.42 \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^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac {3}{4}} x^{11} \Gamma \left (\frac {15}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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