Optimal. Leaf size=34 \[ -\frac {a \left (a+b x^2\right )^5}{10 b^2}+\frac {\left (a+b x^2\right )^6}{12 b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45}
\begin {gather*} \frac {\left (a+b x^2\right )^6}{12 b^2}-\frac {a \left (a+b x^2\right )^5}{10 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac {\int x^3 \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac {\text {Subst}\left (\int x \left (a b+b^2 x\right )^4 \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {a \left (a b+b^2 x\right )^4}{b}+\frac {\left (a b+b^2 x\right )^5}{b^2}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a \left (a+b x^2\right )^5}{10 b^2}+\frac {\left (a+b x^2\right )^6}{12 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 56, normalized size = 1.65 \begin {gather*} \frac {a^4 x^4}{4}+\frac {2}{3} a^3 b x^6+\frac {3}{4} a^2 b^2 x^8+\frac {2}{5} a b^3 x^{10}+\frac {b^4 x^{12}}{12} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 47, normalized size = 1.38
method | result | size |
default | \(\frac {1}{12} b^{4} x^{12}+\frac {2}{5} a \,b^{3} x^{10}+\frac {3}{4} a^{2} b^{2} x^{8}+\frac {2}{3} a^{3} b \,x^{6}+\frac {1}{4} a^{4} x^{4}\) | \(47\) |
norman | \(\frac {1}{12} b^{4} x^{12}+\frac {2}{5} a \,b^{3} x^{10}+\frac {3}{4} a^{2} b^{2} x^{8}+\frac {2}{3} a^{3} b \,x^{6}+\frac {1}{4} a^{4} x^{4}\) | \(47\) |
risch | \(\frac {1}{12} b^{4} x^{12}+\frac {2}{5} a \,b^{3} x^{10}+\frac {3}{4} a^{2} b^{2} x^{8}+\frac {2}{3} a^{3} b \,x^{6}+\frac {1}{4} a^{4} x^{4}\) | \(47\) |
gosper | \(\frac {x^{4} \left (5 b^{4} x^{8}+24 a \,b^{3} x^{6}+45 a^{2} b^{2} x^{4}+40 a^{3} b \,x^{2}+15 a^{4}\right )}{60}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 46, normalized size = 1.35 \begin {gather*} \frac {1}{12} \, b^{4} x^{12} + \frac {2}{5} \, a b^{3} x^{10} + \frac {3}{4} \, a^{2} b^{2} x^{8} + \frac {2}{3} \, a^{3} b x^{6} + \frac {1}{4} \, a^{4} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 46, normalized size = 1.35 \begin {gather*} \frac {1}{12} \, b^{4} x^{12} + \frac {2}{5} \, a b^{3} x^{10} + \frac {3}{4} \, a^{2} b^{2} x^{8} + \frac {2}{3} \, a^{3} b x^{6} + \frac {1}{4} \, a^{4} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 53, normalized size = 1.56 \begin {gather*} \frac {a^{4} x^{4}}{4} + \frac {2 a^{3} b x^{6}}{3} + \frac {3 a^{2} b^{2} x^{8}}{4} + \frac {2 a b^{3} x^{10}}{5} + \frac {b^{4} x^{12}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.20, size = 46, normalized size = 1.35 \begin {gather*} \frac {1}{12} \, b^{4} x^{12} + \frac {2}{5} \, a b^{3} x^{10} + \frac {3}{4} \, a^{2} b^{2} x^{8} + \frac {2}{3} \, a^{3} b x^{6} + \frac {1}{4} \, a^{4} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 46, normalized size = 1.35 \begin {gather*} \frac {a^4\,x^4}{4}+\frac {2\,a^3\,b\,x^6}{3}+\frac {3\,a^2\,b^2\,x^8}{4}+\frac {2\,a\,b^3\,x^{10}}{5}+\frac {b^4\,x^{12}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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