Optimal. Leaf size=86 \[ \frac {x \left (1+b x^4\right )^{1+p}}{b (5+4 p)}-\frac {(1-b (5+4 p)) x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )}{b (5+4 p)}-\frac {2}{3} x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 0.92, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1221, 1218,
251, 371} \begin {gather*} x \left (1-\frac {1}{4 b p+5 b}\right ) \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )-\frac {2}{3} x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )+\frac {x \left (b x^4+1\right )^{p+1}}{b (4 p+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 371
Rule 1218
Rule 1221
Rubi steps
\begin {align*} \int \left (1-x^2\right )^2 \left (1+b x^4\right )^p \, dx &=\frac {x \left (1+b x^4\right )^{1+p}}{b (5+4 p)}+\frac {\int \left (-1+b (5+4 p)-2 b (5+4 p) x^2\right ) \left (1+b x^4\right )^p \, dx}{b (5+4 p)}\\ &=\frac {x \left (1+b x^4\right )^{1+p}}{b (5+4 p)}+\frac {\int \left ((-1+b (5+4 p)) \left (1+b x^4\right )^p-2 b (5+4 p) x^2 \left (1+b x^4\right )^p\right ) \, dx}{b (5+4 p)}\\ &=\frac {x \left (1+b x^4\right )^{1+p}}{b (5+4 p)}-2 \int x^2 \left (1+b x^4\right )^p \, dx+\left (1-\frac {1}{5 b+4 b p}\right ) \int \left (1+b x^4\right )^p \, dx\\ &=\frac {x \left (1+b x^4\right )^{1+p}}{b (5+4 p)}+\left (1-\frac {1}{5 b+4 b p}\right ) x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )-\frac {2}{3} x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 65, normalized size = 0.76 \begin {gather*} x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )-\frac {2}{3} x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )+\frac {1}{5} x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-b x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 56, normalized size = 0.65
method | result | size |
meijerg | \(\frac {x^{5} \hypergeom \left (\left [\frac {5}{4}, -p \right ], \left [\frac {9}{4}\right ], -b \,x^{4}\right )}{5}-\frac {2 x^{3} \hypergeom \left (\left [\frac {3}{4}, -p \right ], \left [\frac {7}{4}\right ], -b \,x^{4}\right )}{3}+x \hypergeom \left (\left [\frac {1}{4}, -p \right ], \left [\frac {5}{4}\right ], -b \,x^{4}\right )\) | \(56\) |
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 = 0.00, 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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Sympy [C] Result contains complex when optimal does not.
time = 30.55, size = 94, normalized size = 1.09 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{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 {\left (x^2-1\right )}^2\,{\left (b\,x^4+1\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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