Optimal. Leaf size=29 \[ \frac {c x^2}{2 b}-\frac {a c \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 266, 43} \[ \frac {c x^2}{2 b}-\frac {a c \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx &=c \int \frac {x^3}{a+b x^2} \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {c x^2}{2 b}-\frac {a c \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 29, normalized size = 1.00 \[ c \left (\frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 24, normalized size = 0.83 \[ \frac {b c x^{2} - a c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 47, normalized size = 1.62 \[ \frac {\frac {a c \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b} + \frac {{\left (b x^{2} + a\right )} c}{b}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.90 \[ \frac {c \,x^{2}}{2 b}-\frac {a c \ln \left (b \,x^{2}+a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 25, normalized size = 0.86 \[ \frac {c x^{2}}{2 \, b} - \frac {a c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 23, normalized size = 0.79 \[ -\frac {c\,\left (a\,\ln \left (b\,x^2+a\right )-b\,x^2\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 22, normalized size = 0.76 \[ c \left (- \frac {a \log {\left (a + b x^{2} \right )}}{2 b^{2}} + \frac {x^{2}}{2 b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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