Optimal. Leaf size=128 \[ \frac {a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (1+p)}-\frac {a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{b^3 (3+2 p)}+\frac {\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (2+p)} \]
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Rubi [A]
time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1263, 784, 21,
45} \begin {gather*} \frac {\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (p+2)}-\frac {a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{b^3 (2 p+3)}+\frac {a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rule 1263
Rubi steps
\begin {align*} \int x^5 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx,x,x^2\right )\\ &=\frac {1}{2} \left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \text {Subst}\left (\int x^2 (a+b x) \left (a b+b^2 x\right )^{2 p} \, dx,x,x^2\right )\\ &=\frac {\left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \text {Subst}\left (\int x^2 \left (a b+b^2 x\right )^{1+2 p} \, dx,x,x^2\right )}{2 b}\\ &=\frac {\left (\left (b \left (a+b x^2\right )\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \text {Subst}\left (\int \left (\frac {a^2 \left (a b+b^2 x\right )^{1+2 p}}{b^2}-\frac {2 a \left (a b+b^2 x\right )^{2+2 p}}{b^3}+\frac {\left (a b+b^2 x\right )^{3+2 p}}{b^4}\right ) \, dx,x,x^2\right )}{2 b}\\ &=\frac {a^2 \left (a+b x^2\right )^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (1+p)}-\frac {a \left (a+b x^2\right )^3 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{b^3 (3+2 p)}+\frac {\left (a+b x^2\right )^4 \left (a^2+2 a b x^2+b^2 x^4\right )^p}{4 b^3 (2+p)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 68, normalized size = 0.53 \begin {gather*} \frac {\left (\left (a+b x^2\right )^2\right )^{1+p} \left (a^2-2 a b (1+p) x^2+b^2 \left (3+5 p+2 p^2\right ) x^4\right )}{4 b^3 (1+p) (2+p) (3+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 99, normalized size = 0.77
method | result | size |
gosper | \(\frac {\left (b \,x^{2}+a \right )^{2} \left (2 b^{2} p^{2} x^{4}+5 b^{2} p \,x^{4}+3 b^{2} x^{4}-2 a b p \,x^{2}-2 a b \,x^{2}+a^{2}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{4 b^{3} \left (2 p^{3}+9 p^{2}+13 p +6\right )}\) | \(99\) |
risch | \(\frac {\left (2 b^{4} p^{2} x^{8}+5 b^{4} p \,x^{8}+4 a \,b^{3} p^{2} x^{6}+3 b^{4} x^{8}+8 a \,b^{3} p \,x^{6}+2 a^{2} b^{2} p^{2} x^{4}+4 a \,b^{3} x^{6}+a^{2} b^{2} p \,x^{4}-2 a^{3} p \,x^{2} b +a^{4}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{p}}{4 \left (3+2 p \right ) \left (2+p \right ) \left (1+p \right ) b^{3}}\) | \(132\) |
norman | \(\frac {a \left (1+p \right ) x^{6} {\mathrm e}^{p \ln \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}}{2 p^{2}+7 p +6}+\frac {a^{4} {\mathrm e}^{p \ln \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}}{4 b^{3} \left (2 p^{3}+9 p^{2}+13 p +6\right )}+\frac {b \,x^{8} {\mathrm e}^{p \ln \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}}{8+4 p}-\frac {p \,a^{3} x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}}{2 b^{2} \left (2 p^{3}+9 p^{2}+13 p +6\right )}+\frac {p \left (1+2 p \right ) a^{2} x^{4} {\mathrm e}^{p \ln \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}}{4 b \left (2 p^{3}+9 p^{2}+13 p +6\right )}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 196, normalized size = 1.53 \begin {gather*} \frac {{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{6} + {\left (2 \, p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + a^{3}\right )} {\left (b x^{2} + a\right )}^{2 \, p} a}{2 \, {\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac {{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{8} + 2 \, {\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{6} - 3 \, {\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{4} + 6 \, a^{3} b p x^{2} - 3 \, a^{4}\right )} {\left (b x^{2} + a\right )}^{2 \, p}}{4 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 140, normalized size = 1.09 \begin {gather*} \frac {{\left ({\left (2 \, b^{4} p^{2} + 5 \, b^{4} p + 3 \, b^{4}\right )} x^{8} - 2 \, a^{3} b p x^{2} + 4 \, {\left (a b^{3} p^{2} + 2 \, a b^{3} p + a b^{3}\right )} x^{6} + {\left (2 \, a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{4} + a^{4}\right )} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{4 \, {\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {a x^{6} \left (a^{2}\right )^{p}}{6} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {2 a^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {3 a^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {4 a b x^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {4 a b x^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {4 a b x^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {2 b^{2} x^{4} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {2 b^{2} x^{4} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} & \text {for}\: p = -2 \\\int \frac {x^{5} \left (a + b x^{2}\right )}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx & \text {for}\: p = - \frac {3}{2} \\\frac {a^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {a^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{4}}{4 b} & \text {for}\: p = -1 \\\frac {a^{4} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} - \frac {2 a^{3} b p x^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {2 a^{2} b^{2} p^{2} x^{4} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {a^{2} b^{2} p x^{4} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {4 a b^{3} p^{2} x^{6} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {8 a b^{3} p x^{6} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {4 a b^{3} x^{6} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {2 b^{4} p^{2} x^{8} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {5 b^{4} p x^{8} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} + \frac {3 b^{4} x^{8} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}}{8 b^{3} p^{3} + 36 b^{3} p^{2} + 52 b^{3} p + 24 b^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 331 vs.
\(2 (124) = 248\).
time = 3.99, size = 331, normalized size = 2.59 \begin {gather*} \frac {2 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{4} p^{2} x^{8} + 5 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{4} p x^{8} + 4 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{3} p^{2} x^{6} + 3 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} b^{4} x^{8} + 8 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{3} p x^{6} + 2 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{2} b^{2} p^{2} x^{4} + 4 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a b^{3} x^{6} + {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{2} b^{2} p x^{4} - 2 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{3} b p x^{2} + {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} a^{4}}{4 \, {\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 169, normalized size = 1.32 \begin {gather*} {\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p\,\left (\frac {a^4}{4\,b^3\,\left (2\,p^3+9\,p^2+13\,p+6\right )}+\frac {a\,x^6\,{\left (p+1\right )}^2}{2\,p^3+9\,p^2+13\,p+6}+\frac {b\,x^8\,\left (2\,p^2+5\,p+3\right )}{4\,\left (2\,p^3+9\,p^2+13\,p+6\right )}-\frac {a^3\,p\,x^2}{2\,b^2\,\left (2\,p^3+9\,p^2+13\,p+6\right )}+\frac {a^2\,p\,x^4\,\left (2\,p+1\right )}{4\,b\,\left (2\,p^3+9\,p^2+13\,p+6\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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