Optimal. Leaf size=61 \[ \frac {d (b c-a d) x^2}{2 b^2}+\frac {\left (c+d x^2\right )^2}{4 b}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, 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 = {455, 45}
\begin {gather*} \frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac {d x^2 (b c-a d)}{2 b^2}+\frac {\left (c+d x^2\right )^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rubi steps
\begin {align*} \int \frac {x \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {d (b c-a d) x^2}{2 b^2}+\frac {\left (c+d x^2\right )^2}{4 b}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.80 \begin {gather*} \frac {b d x^2 \left (4 b c-2 a d+b d x^2\right )+2 (b c-a d)^2 \log \left (a+b x^2\right )}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 63, normalized size = 1.03
method | result | size |
default | \(-\frac {d \left (-\frac {1}{2} b d \,x^{4}+a d \,x^{2}-2 c \,x^{2} b \right )}{2 b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\) | \(63\) |
norman | \(\frac {d^{2} x^{4}}{4 b}-\frac {d \left (a d -2 b c \right ) x^{2}}{2 b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\) | \(64\) |
risch | \(\frac {d^{2} x^{4}}{4 b}-\frac {a \,d^{2} x^{2}}{2 b^{2}}+\frac {c d \,x^{2}}{b}+\frac {a^{2} d^{2}}{4 b^{3}}-\frac {a c d}{b^{2}}+\frac {c^{2}}{b}+\frac {\ln \left (b \,x^{2}+a \right ) a^{2} d^{2}}{2 b^{3}}-\frac {\ln \left (b \,x^{2}+a \right ) a c d}{b^{2}}+\frac {\ln \left (b \,x^{2}+a \right ) c^{2}}{2 b}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 66, normalized size = 1.08 \begin {gather*} \frac {b d^{2} x^{4} + 2 \, {\left (2 \, b c d - a d^{2}\right )} x^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.27, size = 67, normalized size = 1.10 \begin {gather*} \frac {b^{2} d^{2} x^{4} + 2 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 49, normalized size = 0.80 \begin {gather*} x^{2} \left (- \frac {a d^{2}}{2 b^{2}} + \frac {c d}{b}\right ) + \frac {d^{2} x^{4}}{4 b} + \frac {\left (a d - b c\right )^{2} \log {\left (a + b x^{2} \right )}}{2 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 67, normalized size = 1.10 \begin {gather*} \frac {b d^{2} x^{4} + 4 \, b c d x^{2} - 2 \, a d^{2} x^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 68, normalized size = 1.11 \begin {gather*} \frac {d^2\,x^4}{4\,b}-x^2\,\left (\frac {a\,d^2}{2\,b^2}-\frac {c\,d}{b}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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