Optimal. Leaf size=36 \[ \frac {a}{8 b^2 \left (a-b x^2\right )^4}-\frac {1}{6 b^2 \left (a-b x^2\right )^3} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {266, 43} \[ \frac {a}{8 b^2 \left (a-b x^2\right )^4}-\frac {1}{6 b^2 \left (a-b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a-b x^2\right )^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a-b x)^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a}{b (a-b x)^5}-\frac {1}{b (a-b x)^4}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{8 b^2 \left (a-b x^2\right )^4}-\frac {1}{6 b^2 \left (a-b x^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.69 \[ -\frac {a-4 b x^2}{24 b^2 \left (a-b x^2\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 60, normalized size = 1.67 \[ \frac {4 \, b x^{2} - a}{24 \, {\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 39, normalized size = 1.08 \[ \frac {\frac {4}{{\left (b x^{2} - a\right )}^{3} b} + \frac {3 \, a}{{\left (b x^{2} - a\right )}^{4} b}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.97 \[ \frac {a}{8 \left (b \,x^{2}-a \right )^{4} b^{2}}+\frac {1}{6 \left (b \,x^{2}-a \right )^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 60, normalized size = 1.67 \[ \frac {4 \, b x^{2} - a}{24 \, {\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.60, size = 59, normalized size = 1.64 \[ -\frac {\frac {a}{24\,b^2}-\frac {x^2}{6\,b}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 60, normalized size = 1.67 \[ - \frac {a - 4 b x^{2}}{24 a^{4} b^{2} - 96 a^{3} b^{3} x^{2} + 144 a^{2} b^{4} x^{4} - 96 a b^{5} x^{6} + 24 b^{6} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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