Optimal. Leaf size=78 \[ -\frac {\sqrt [4]{a+b x^4}}{4 a x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 44, 65,
218, 212, 209} \begin {gather*} \frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {\sqrt [4]{a+b x^4}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^4\right )^{3/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 a x^4}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )}{16 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 a x^4}-\frac {3 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{4 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 a x^4}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 a^{3/2}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 a^{3/2}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 a x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 78, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{a+b x^4}}{4 a x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{5} \left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 94, normalized size = 1.21 \begin {gather*} -\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} b}{4 \, {\left ({\left (b x^{4} + a\right )} a - a^{2}\right )}} + \frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b \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 {3}{4}}}\right )}}{16 \, a} \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. 194 vs.
\(2 (58) = 116\).
time = 0.39, size = 194, normalized size = 2.49 \begin {gather*} -\frac {12 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{5} b \left (\frac {b^{4}}{a^{7}}\right )^{\frac {3}{4}} - \sqrt {a^{4} \sqrt {\frac {b^{4}}{a^{7}}} + \sqrt {b x^{4} + a} b^{2}} a^{5} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {3}{4}}}{b^{4}}\right ) - 3 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (3 \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b\right ) + 3 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b\right ) + 4 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, a x^{4}} \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 = 0.76, size = 39, normalized size = 0.50 \begin {gather*} - \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {3}{4}} x^{7} \Gamma \left (\frac {11}{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. 221 vs.
\(2 (58) = 116\).
time = 2.96, size = 221, normalized size = 2.83 \begin {gather*} \frac {\frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \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 {1}{4}} b^{2} \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} \left (-a\right )^{\frac {1}{4}} b^{2} \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 {3 \, \sqrt {2} b^{2} \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}} a} - \frac {8 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b}{a x^{4}}}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.29, size = 58, normalized size = 0.74 \begin {gather*} \frac {3\,b\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{7/4}}-\frac {{\left (b\,x^4+a\right )}^{1/4}}{4\,a\,x^4}+\frac {3\,b\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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