Optimal. Leaf size=54 \[ -\frac {a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac {x^2 (b c-a d)}{2 b^2}+\frac {d x^4}{4 b} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \[ \frac {x^2 (b c-a d)}{2 b^2}-\frac {a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac {d x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^3 \left (c+d x^2\right )}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (c+d x)}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b c-a d}{b^2}+\frac {d x}{b}+\frac {a (-b c+a d)}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d) x^2}{2 b^2}+\frac {d x^4}{4 b}-\frac {a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 0.87 \[ \frac {b x^2 \left (-2 a d+2 b c+b d x^2\right )+2 a (a d-b c) \log \left (a+b x^2\right )}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 51, normalized size = 0.94 \[ \frac {b^{2} d x^{4} + 2 \, {\left (b^{2} c - a b d\right )} x^{2} - 2 \, {\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 52, normalized size = 0.96 \[ \frac {b d x^{4} + 2 \, b c x^{2} - 2 \, a d x^{2}}{4 \, b^{2}} - \frac {{\left (a b c - a^{2} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 62, normalized size = 1.15 \[ \frac {d \,x^{4}}{4 b}-\frac {a d \,x^{2}}{2 b^{2}}+\frac {c \,x^{2}}{2 b}+\frac {a^{2} d \ln \left (b \,x^{2}+a \right )}{2 b^{3}}-\frac {a c \ln \left (b \,x^{2}+a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 50, normalized size = 0.93 \[ \frac {b d x^{4} + 2 \, {\left (b c - a d\right )} x^{2}}{4 \, b^{2}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 52, normalized size = 0.96 \[ x^2\,\left (\frac {c}{2\,b}-\frac {a\,d}{2\,b^2}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d-a\,b\,c\right )}{2\,b^3}+\frac {d\,x^4}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 46, normalized size = 0.85 \[ \frac {a \left (a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{3}} + x^{2} \left (- \frac {a d}{2 b^{2}} + \frac {c}{2 b}\right ) + \frac {d x^{4}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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