Optimal. Leaf size=128 \[ \frac {4 a^2 x^2}{15 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b}-\frac {4 a^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.06, 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, 327, 235,
233, 202} \begin {gather*} -\frac {4 a^{5/2} \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 )}{15 b^{5/2} \sqrt [4]{a+b x^4}}+\frac {4 a^2 x^2}{15 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^9}{\sqrt [4]{a+b x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b}-\frac {a \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{3 b}\\ &=-\frac {2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{15 b^2}\\ &=-\frac {2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b}+\frac {\left (2 a^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 )}{15 b^2 \sqrt [4]{a+b x^4}}\\ &=\frac {4 a^2 x^2}{15 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b}-\frac {\left (2 a^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 )}{15 b^2 \sqrt [4]{a+b x^4}}\\ &=\frac {4 a^2 x^2}{15 b^2 \sqrt [4]{a+b x^4}}-\frac {2 a x^2 \left (a+b x^4\right )^{3/4}}{15 b^2}+\frac {x^6 \left (a+b x^4\right )^{3/4}}{9 b}-\frac {4 a^{5/2} \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 )}{15 b^{5/2} \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 = 6.65, size = 80, normalized size = 0.62 \begin {gather*} \frac {x^2 \left (-6 a^2-a b x^4+5 b^2 x^8+6 a^2 \sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^4}{a}\right )\right )}{45 b^2 \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 \[\int \frac {x^{9}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
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 = 0.07, size = 15, normalized size = 0.12 \begin {gather*} {\rm integral}\left (\frac {x^{9}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}}}, 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 = 0.54, size = 27, normalized size = 0.21 \begin {gather*} \frac {x^{10} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 \sqrt [4]{a}} \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 {x^9}{{\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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