Optimal. Leaf size=105 \[ -\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}}-\frac {7 a^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 b^{5/2} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 105, 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 = {288, 287, 342,
281, 202} \begin {gather*} -\frac {7 a^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 b^{5/2} \sqrt [4]{a+b x^4}}-\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 281
Rule 287
Rule 288
Rule 342
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac {x^7}{6 b \sqrt [4]{a+b x^4}}-\frac {(7 a) \int \frac {x^6}{\left (a+b x^4\right )^{5/4}} \, dx}{6 b}\\ &=-\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}}+\frac {\left (7 a^2\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{4 b^2}\\ &=-\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}}+\frac {\left (7 a^2 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{5/4} x^3} \, dx}{4 b^3 \sqrt [4]{a+b x^4}}\\ &=-\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}}-\frac {\left (7 a^2 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{4 b^3 \sqrt [4]{a+b x^4}}\\ &=-\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}}-\frac {\left (7 a^2 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{8 b^3 \sqrt [4]{a+b x^4}}\\ &=-\frac {7 a x^3}{12 b^2 \sqrt [4]{a+b x^4}}+\frac {x^7}{6 b \sqrt [4]{a+b x^4}}-\frac {7 a^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 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 = 7.91, size = 67, normalized size = 0.64 \begin {gather*} \frac {x^3 \left (-7 a+2 b x^4+7 a \sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (\frac {3}{4},\frac {5}{4};\frac {7}{4};-\frac {b x^4}{a}\right )\right )}{12 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 \[\int \frac {x^{10}}{\left (b \,x^{4}+a \right )^{\frac {5}{4}}}\, dx\]
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 = 35, normalized size = 0.33 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{10}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, 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 = 0.57, size = 37, normalized size = 0.35 \begin {gather*} \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{4}} \Gamma \left (\frac {15}{4}\right )} \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^{10}}{{\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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