Optimal. Leaf size=34 \[ \frac {\left (a+b x^2\right )^8}{16 b^2}-\frac {a \left (a+b x^2\right )^7}{14 b^2} \]
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Rubi [A] time = 0.05, 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, 266, 43} \begin {gather*} \frac {\left (a+b x^2\right )^8}{16 b^2}-\frac {a \left (a+b x^2\right )^7}{14 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x^3 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int x \left (a b+b^2 x\right )^6 \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a \left (a b+b^2 x\right )^6}{b}+\frac {\left (a b+b^2 x\right )^7}{b^2}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {a \left (a+b x^2\right )^7}{14 b^2}+\frac {\left (a+b x^2\right )^8}{16 b^2}\\ \end {align*}
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Mathematica [B] time = 0.00, size = 77, normalized size = 2.26 \begin {gather*} \frac {a^6 x^4}{4}+a^5 b x^6+\frac {15}{8} a^4 b^2 x^8+2 a^3 b^3 x^{10}+\frac {5}{4} a^2 b^4 x^{12}+\frac {3}{7} a b^5 x^{14}+\frac {b^6 x^{16}}{16} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.66, size = 67, normalized size = 1.97 \begin {gather*} \frac {1}{16} x^{16} b^{6} + \frac {3}{7} x^{14} b^{5} a + \frac {5}{4} x^{12} b^{4} a^{2} + 2 x^{10} b^{3} a^{3} + \frac {15}{8} x^{8} b^{2} a^{4} + x^{6} b a^{5} + \frac {1}{4} x^{4} a^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 67, normalized size = 1.97 \begin {gather*} \frac {1}{16} \, b^{6} x^{16} + \frac {3}{7} \, a b^{5} x^{14} + \frac {5}{4} \, a^{2} b^{4} x^{12} + 2 \, a^{3} b^{3} x^{10} + \frac {15}{8} \, a^{4} b^{2} x^{8} + a^{5} b x^{6} + \frac {1}{4} \, a^{6} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 68, normalized size = 2.00 \begin {gather*} \frac {1}{16} b^{6} x^{16}+\frac {3}{7} a \,b^{5} x^{14}+\frac {5}{4} a^{2} b^{4} x^{12}+2 a^{3} b^{3} x^{10}+\frac {15}{8} a^{4} b^{2} x^{8}+a^{5} b \,x^{6}+\frac {1}{4} a^{6} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 67, normalized size = 1.97 \begin {gather*} \frac {1}{16} \, b^{6} x^{16} + \frac {3}{7} \, a b^{5} x^{14} + \frac {5}{4} \, a^{2} b^{4} x^{12} + 2 \, a^{3} b^{3} x^{10} + \frac {15}{8} \, a^{4} b^{2} x^{8} + a^{5} b x^{6} + \frac {1}{4} \, a^{6} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 67, normalized size = 1.97 \begin {gather*} \frac {a^6\,x^4}{4}+a^5\,b\,x^6+\frac {15\,a^4\,b^2\,x^8}{8}+2\,a^3\,b^3\,x^{10}+\frac {5\,a^2\,b^4\,x^{12}}{4}+\frac {3\,a\,b^5\,x^{14}}{7}+\frac {b^6\,x^{16}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 75, normalized size = 2.21 \begin {gather*} \frac {a^{6} x^{4}}{4} + a^{5} b x^{6} + \frac {15 a^{4} b^{2} x^{8}}{8} + 2 a^{3} b^{3} x^{10} + \frac {5 a^{2} b^{4} x^{12}}{4} + \frac {3 a b^{5} x^{14}}{7} + \frac {b^{6} x^{16}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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