Optimal. Leaf size=75 \[ \frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}-\frac {a x^2 (b c-a d)}{2 b^3}+\frac {x^4 (b c-a d)}{4 b^2}+\frac {d x^6}{6 b} \]
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Rubi [A] time = 0.09, antiderivative size = 75, 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} \begin {gather*} \frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}+\frac {x^4 (b c-a d)}{4 b^2}-\frac {a x^2 (b c-a d)}{2 b^3}+\frac {d x^6}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (c+d x)}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a (-b c+a d)}{b^3}+\frac {(b c-a d) x}{b^2}+\frac {d x^2}{b}-\frac {a^2 (-b c+a d)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a (b c-a d) x^2}{2 b^3}+\frac {(b c-a d) x^4}{4 b^2}+\frac {d x^6}{6 b}+\frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 0.95 \begin {gather*} \frac {b x^2 \left (6 a^2 d-3 a b \left (2 c+d x^2\right )+b^2 x^2 \left (3 c+2 d x^2\right )\right )+6 a^2 (b c-a d) \log \left (a+b x^2\right )}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.78, size = 75, normalized size = 1.00 \begin {gather*} \frac {2 \, b^{3} d x^{6} + 3 \, {\left (b^{3} c - a b^{2} d\right )} x^{4} - 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} + 6 \, {\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 77, normalized size = 1.03 \begin {gather*} \frac {2 \, b^{2} d x^{6} + 3 \, b^{2} c x^{4} - 3 \, a b d x^{4} - 6 \, a b c x^{2} + 6 \, a^{2} d x^{2}}{12 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 86, normalized size = 1.15 \begin {gather*} \frac {d \,x^{6}}{6 b}-\frac {a d \,x^{4}}{4 b^{2}}+\frac {c \,x^{4}}{4 b}+\frac {a^{2} d \,x^{2}}{2 b^{3}}-\frac {a c \,x^{2}}{2 b^{2}}-\frac {a^{3} d \ln \left (b \,x^{2}+a \right )}{2 b^{4}}+\frac {a^{2} c \ln \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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maxima [A] time = 1.02, size = 74, normalized size = 0.99 \begin {gather*} \frac {2 \, b^{2} d x^{6} + 3 \, {\left (b^{2} c - a b d\right )} x^{4} - 6 \, {\left (a b c - a^{2} d\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 76, normalized size = 1.01 \begin {gather*} x^4\,\left (\frac {c}{4\,b}-\frac {a\,d}{4\,b^2}\right )+\frac {d\,x^6}{6\,b}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d-a^2\,b\,c\right )}{2\,b^4}-\frac {a\,x^2\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 70, normalized size = 0.93 \begin {gather*} - \frac {a^{2} \left (a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{4}} + x^{4} \left (- \frac {a d}{4 b^{2}} + \frac {c}{4 b}\right ) + x^{2} \left (\frac {a^{2} d}{2 b^{3}} - \frac {a c}{2 b^{2}}\right ) + \frac {d x^{6}}{6 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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