Optimal. Leaf size=146 \[ -\frac {2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {4 a^{9/2} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.08, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 285, 327,
239, 237} \begin {gather*} \frac {4 a^{9/2} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac {2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 239
Rule 281
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^9 \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^4 \left (a+b x^2\right )^{5/4} \, dx,x,x^2\right )\\ &=\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {1}{6} a \text {Subst}\left (\int x^4 \sqrt [4]{a+b x^2} \, dx,x,x^2\right )\\ &=\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {1}{66} a^2 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}-\frac {a^3 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b}\\ &=-\frac {2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {\left (2 a^4\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{231 b^2}\\ &=-\frac {2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {\left (2 a^4 \left (1+\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{231 b^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac {a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac {1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac {1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac {4 a^{9/2} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.74, size = 81, normalized size = 0.55 \begin {gather*} \frac {x^2 \sqrt [4]{a+b x^4} \left (-\left (\left (6 a-11 b x^4\right ) \left (a+b x^4\right )^2\right )+\frac {6 a^3 \, _2F_1\left (-\frac {5}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^4}{a}\right )}{\sqrt [4]{1+\frac {b x^4}{a}}}\right )}{165 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{9} \left (b \,x^{4}+a \right )^{\frac {5}{4}}\, dx\]
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 = 23, normalized size = 0.16 \begin {gather*} {\rm integral}\left ({\left (b x^{13} + a x^{9}\right )} {\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 = 1.05, size = 29, normalized size = 0.20 \begin {gather*} \frac {a^{\frac {5}{4}} x^{10} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10} \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 x^9\,{\left (b\,x^4+a\right )}^{5/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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