Optimal. Leaf size=133 \[ -\frac {4 a^2 x^2 \left (a-b x^4\right )^{3/4}}{39 b^3}-\frac {10 a x^6 \left (a-b x^4\right )^{3/4}}{117 b^2}-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b}+\frac {8 a^{7/2} \sqrt [4]{1-\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a-b x^4}} \]
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Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {281, 327, 235,
234} \begin {gather*} \frac {8 a^{7/2} \sqrt [4]{1-\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 b^{7/2} \sqrt [4]{a-b x^4}}-\frac {4 a^2 x^2 \left (a-b x^4\right )^{3/4}}{39 b^3}-\frac {10 a x^6 \left (a-b x^4\right )^{3/4}}{117 b^2}-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 235
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^{13}}{\sqrt [4]{a-b x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{a-b x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b}+\frac {(5 a) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{a-b x^2}} \, dx,x,x^2\right )}{13 b}\\ &=-\frac {10 a x^6 \left (a-b x^4\right )^{3/4}}{117 b^2}-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b}+\frac {\left (10 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{a-b x^2}} \, dx,x,x^2\right )}{39 b^2}\\ &=-\frac {4 a^2 x^2 \left (a-b x^4\right )^{3/4}}{39 b^3}-\frac {10 a x^6 \left (a-b x^4\right )^{3/4}}{117 b^2}-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b}+\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-b x^2}} \, dx,x,x^2\right )}{39 b^3}\\ &=-\frac {4 a^2 x^2 \left (a-b x^4\right )^{3/4}}{39 b^3}-\frac {10 a x^6 \left (a-b x^4\right )^{3/4}}{117 b^2}-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b}+\frac {\left (4 a^3 \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 )}{39 b^3 \sqrt [4]{a-b x^4}}\\ &=-\frac {4 a^2 x^2 \left (a-b x^4\right )^{3/4}}{39 b^3}-\frac {10 a x^6 \left (a-b x^4\right )^{3/4}}{117 b^2}-\frac {x^{10} \left (a-b x^4\right )^{3/4}}{13 b}+\frac {8 a^{7/2} \sqrt [4]{1-\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 b^{7/2} \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 = 7.26, size = 91, normalized size = 0.68 \begin {gather*} \frac {x^2 \left (-12 a^3+2 a^2 b x^4+a b^2 x^8+9 b^3 x^{12}+12 a^3 \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 )}{117 b^3 \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 \[\int \frac {x^{13}}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
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 = 28, normalized size = 0.21 \begin {gather*} {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{13}}{b x^{4} - a}, 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 = 0.78, size = 29, normalized size = 0.22 \begin {gather*} \frac {x^{14} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{14 \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^{13}}{{\left (a-b\,x^4\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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