Optimal. Leaf size=108 \[ -\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )+\frac {(1-b (7+4 p)) x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )}{b (7+4 p)}+\frac {3}{5} x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-b x^4\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 103, normalized size of antiderivative = 0.95, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1221, 1907,
251, 371} \begin {gather*} x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )+\frac {3}{5} x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-b x^4\right )-x^3 \left (1-\frac {1}{4 b p+7 b}\right ) \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )-\frac {x^3 \left (b x^4+1\right )^{p+1}}{b (4 p+7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 371
Rule 1221
Rule 1907
Rubi steps
\begin {align*} \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx &=-\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+\frac {\int \left (1+b x^4\right )^p \left (b (7+4 p)+3 (1-b (7+4 p)) x^2+3 b (7+4 p) x^4\right ) \, dx}{b (7+4 p)}\\ &=-\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+\frac {\int \left (b (7+4 p) \left (1+b x^4\right )^p+3 (1-b (7+4 p)) x^2 \left (1+b x^4\right )^p+3 b (7+4 p) x^4 \left (1+b x^4\right )^p\right ) \, dx}{b (7+4 p)}\\ &=-\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+3 \int x^4 \left (1+b x^4\right )^p \, dx-\left (3 \left (1-\frac {1}{7 b+4 b p}\right )\right ) \int x^2 \left (1+b x^4\right )^p \, dx+\int \left (1+b x^4\right )^p \, dx\\ &=-\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )-\left (1-\frac {1}{7 b+4 b p}\right ) x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )+\frac {3}{5} x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-b x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 86, normalized size = 0.80 \begin {gather*} x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )-x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )+\frac {3}{5} x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-b x^4\right )-\frac {1}{7} x^7 \, _2F_1\left (\frac {7}{4},-p;\frac {11}{4};-b x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 75, normalized size = 0.69
method | result | size |
meijerg | \(-\frac {x^{7} \hypergeom \left (\left [\frac {7}{4}, -p \right ], \left [\frac {11}{4}\right ], -b \,x^{4}\right )}{7}+\frac {3 x^{5} \hypergeom \left (\left [\frac {5}{4}, -p \right ], \left [\frac {9}{4}\right ], -b \,x^{4}\right )}{5}-x^{3} \hypergeom \left (\left [\frac {3}{4}, -p \right ], \left [\frac {7}{4}\right ], -b \,x^{4}\right )+x \hypergeom \left (\left [\frac {1}{4}, -p \right ], \left [\frac {5}{4}\right ], -b \,x^{4}\right )\) | \(75\) |
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 = 55.55, size = 129, normalized size = 1.19 \begin {gather*} - \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, - p \\ \frac {11}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {3 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 {3 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 )}}{4 \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 )}^3\,{\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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