Optimal. Leaf size=128 \[ -\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}+\frac {b^{3/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 331, 235,
233, 202} \begin {gather*} \frac {b^{3/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}-\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 281
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^7 \sqrt [4]{a+b x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}-\frac {b \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{8 a^2}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {\left (b^2 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{8 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}+\frac {\left (b^2 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{8 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}+\frac {b^{3/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 51, normalized size = 0.40 \begin {gather*} -\frac {\sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};-\frac {1}{2};-\frac {b x^4}{a}\right )}{6 x^6 \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{7} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.08, size = 25, normalized size = 0.20 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{11} + a x^{7}}, x\right ) \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 = 0.55, size = 32, normalized size = 0.25 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 \sqrt [4]{a} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^7\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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