Optimal. Leaf size=98 \[ -\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac {5 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, 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 = {272, 43, 65,
218, 212, 209} \begin {gather*} -\frac {5 b^2 \text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x^9} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^{5/4}}{x^3} \, dx,x,x^4\right )\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac {1}{32} (5 b) \text {Subst}\left (\int \frac {\sqrt [4]{a+b x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac {1}{128} \left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}+\frac {1}{32} (5 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 \sqrt {a}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 \sqrt {a}}\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac {\left (a+b x^4\right )^{5/4}}{8 x^8}-\frac {5 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 89, normalized size = 0.91 \begin {gather*} \frac {\left (-4 a-9 b x^4\right ) \sqrt [4]{a+b x^4}}{32 x^8}-\frac {5 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{x^{9}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 120, normalized size = 1.22 \begin {gather*} -\frac {5 \, b^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{64 \, a^{\frac {3}{4}}} + \frac {5 \, b^{2} \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 )}{128 \, a^{\frac {3}{4}}} - \frac {9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2} - 5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{2}}{32 \, {\left ({\left (b x^{4} + a\right )}^{2} - 2 \, {\left (b x^{4} + a\right )} a + a^{2}\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. 200 vs.
\(2 (74) = 148\).
time = 0.39, size = 200, normalized size = 2.04 \begin {gather*} \frac {20 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \arctan \left (-\frac {\left (\frac {b^{8}}{a^{3}}\right )^{\frac {3}{4}} {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b^{2} - \left (\frac {b^{8}}{a^{3}}\right )^{\frac {3}{4}} \sqrt {\sqrt {b x^{4} + a} b^{4} + \sqrt {\frac {b^{8}}{a^{3}}} a^{2}} a^{2}}{b^{8}}\right ) - 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} a\right ) + 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} x^{8} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 5 \, \left (\frac {b^{8}}{a^{3}}\right )^{\frac {1}{4}} a\right ) - 4 \, {\left (9 \, b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, 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 = 1.25, size = 41, normalized size = 0.42 \begin {gather*} - \frac {b^{\frac {5}{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{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. 232 vs.
\(2 (74) = 148\).
time = 1.22, size = 232, normalized size = 2.37 \begin {gather*} \frac {\frac {10 \, \sqrt {2} 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 )}{\left (-a\right )^{\frac {3}{4}}} + \frac {10 \, \sqrt {2} 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 )}{\left (-a\right )^{\frac {3}{4}}} + \frac {5 \, \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 {3}{4}}} + \frac {5 \, \sqrt {2} \left (-a\right )^{\frac {1}{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} - \frac {8 \, {\left (9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3} - 5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{3}\right )}}{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 = 1.50, size = 77, normalized size = 0.79 \begin {gather*} \frac {5\,a\,{\left (b\,x^4+a\right )}^{1/4}}{32\,x^8}-\frac {5\,b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{3/4}}-\frac {9\,{\left (b\,x^4+a\right )}^{5/4}}{32\,x^8}+\frac {b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,5{}\mathrm {i}}{64\,a^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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