Optimal. Leaf size=101 \[ -\frac {\left (a+b x^4\right )^{3/4}}{8 x^8}-\frac {3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {272, 43, 44, 65,
304, 209, 212} \begin {gather*} -\frac {3 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}-\frac {3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac {\left (a+b x^4\right )^{3/4}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x^9} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^{3/4}}{x^3} \, dx,x,x^4\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{8 x^8}+\frac {1}{32} (3 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{8 x^8}-\frac {3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )}{128 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{8 x^8}-\frac {3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac {(3 b) \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{32 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{8 x^8}-\frac {3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{8 x^8}-\frac {3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 92, normalized size = 0.91 \begin {gather*} \frac {\left (-4 a-3 b x^4\right ) \left (a+b x^4\right )^{3/4}}{32 a x^8}-\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{9}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 125, normalized size = 1.24 \begin {gather*} -\frac {3 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}\right )}}{128 \, a} - \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2} + {\left (b x^{4} + a\right )}^{\frac {3}{4}} a b^{2}}{32 \, {\left ({\left (b x^{4} + a\right )}^{2} a - 2 \, {\left (b x^{4} + a\right )} a^{2} + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (77) = 154\).
time = 0.38, size = 209, normalized size = 2.07 \begin {gather*} \frac {12 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a x^{8} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a b^{6} - \sqrt {\sqrt {b x^{4} + a} b^{12} + \sqrt {\frac {b^{8}}{a^{5}}} a^{3} b^{8}} \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a}{b^{8}}\right ) + 3 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a x^{8} \log \left (27 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{6} + 27 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4}\right ) - 3 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a x^{8} \log \left (27 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{6} - 27 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4}\right ) - 4 \, {\left (3 \, b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{128 \, a x^{8}} \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 = 1.32, size = 41, normalized size = 0.41 \begin {gather*} - \frac {b^{\frac {3}{4}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{5} \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs.
\(2 (77) = 154\).
time = 1.23, size = 243, normalized size = 2.41 \begin {gather*} \frac {\frac {6 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {6 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {3 \, \sqrt {2} b^{3} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {1}{4}} a} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{3} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{a^{2}} - \frac {8 \, {\left (3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3} + {\left (b x^{4} + a\right )}^{\frac {3}{4}} a b^{3}\right )}}{a b^{2} x^{8}}}{256 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.42, size = 79, normalized size = 0.78 \begin {gather*} -\frac {{\left (b\,x^4+a\right )}^{3/4}}{32\,x^8}-\frac {3\,b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{5/4}}-\frac {3\,{\left (b\,x^4+a\right )}^{7/4}}{32\,a\,x^8}-\frac {b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,3{}\mathrm {i}}{64\,a^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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