Optimal. Leaf size=60 \[ \frac {A c (e x)^{1+m}}{e (1+m)}+\frac {(B c+A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {B d (e x)^{5+m}}{e^5 (5+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459}
\begin {gather*} \frac {(e x)^{m+3} (A d+B c)}{e^3 (m+3)}+\frac {A c (e x)^{m+1}}{e (m+1)}+\frac {B d (e x)^{m+5}}{e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx &=\int \left (A c (e x)^m+\frac {(B c+A d) (e x)^{2+m}}{e^2}+\frac {B d (e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {A c (e x)^{1+m}}{e (1+m)}+\frac {(B c+A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {B d (e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 0.72 \begin {gather*} x (e x)^m \left (\frac {A c}{1+m}+\frac {(B c+A d) x^2}{3+m}+\frac {B d x^4}{5+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 59, normalized size = 0.98
method | result | size |
norman | \(\frac {\left (A d +B c \right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {A c x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {B d \,x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}\) | \(59\) |
gosper | \(\frac {x \left (B d \,m^{2} x^{4}+4 B d m \,x^{4}+A d \,m^{2} x^{2}+B c \,m^{2} x^{2}+3 B d \,x^{4}+6 A d m \,x^{2}+6 B c m \,x^{2}+A c \,m^{2}+5 A d \,x^{2}+5 B c \,x^{2}+8 A c m +15 A c \right ) \left (e x \right )^{m}}{\left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(111\) |
risch | \(\frac {x \left (B d \,m^{2} x^{4}+4 B d m \,x^{4}+A d \,m^{2} x^{2}+B c \,m^{2} x^{2}+3 B d \,x^{4}+6 A d m \,x^{2}+6 B c m \,x^{2}+A c \,m^{2}+5 A d \,x^{2}+5 B c \,x^{2}+8 A c m +15 A c \right ) \left (e x \right )^{m}}{\left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 73, normalized size = 1.22 \begin {gather*} \frac {B d x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {B c x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {A d x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {\left (x e\right )^{m + 1} A c e^{\left (-1\right )}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.45, size = 95, normalized size = 1.58 \begin {gather*} \frac {{\left ({\left (B d m^{2} + 4 \, B d m + 3 \, B d\right )} x^{5} + {\left ({\left (B c + A d\right )} m^{2} + 5 \, B c + 5 \, A d + 6 \, {\left (B c + A d\right )} m\right )} x^{3} + {\left (A c m^{2} + 8 \, A c m + 15 \, A c\right )} x\right )} \left (x e\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 439 vs.
\(2 (51) = 102\).
time = 0.25, size = 439, normalized size = 7.32 \begin {gather*} \begin {cases} \frac {- \frac {A c}{4 x^{4}} - \frac {A d}{2 x^{2}} - \frac {B c}{2 x^{2}} + B d \log {\left (x \right )}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {A c}{2 x^{2}} + A d \log {\left (x \right )} + B c \log {\left (x \right )} + \frac {B d x^{2}}{2}}{e^{3}} & \text {for}\: m = -3 \\\frac {A c \log {\left (x \right )} + \frac {A d x^{2}}{2} + \frac {B c x^{2}}{2} + \frac {B d x^{4}}{4}}{e} & \text {for}\: m = -1 \\\frac {A c m^{2} x \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 A c m x \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 A c x \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {A d m^{2} x^{3} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 A d m x^{3} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 A d x^{3} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B c m^{2} x^{3} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 B c m x^{3} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 B c x^{3} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B d m^{2} x^{5} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 B d m x^{5} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 B d x^{5} \left (e x\right )^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (60) = 120\).
time = 1.41, size = 167, normalized size = 2.78 \begin {gather*} \frac {B d m^{2} x^{5} x^{m} e^{m} + 4 \, B d m x^{5} x^{m} e^{m} + B c m^{2} x^{3} x^{m} e^{m} + A d m^{2} x^{3} x^{m} e^{m} + 3 \, B d x^{5} x^{m} e^{m} + 6 \, B c m x^{3} x^{m} e^{m} + 6 \, A d m x^{3} x^{m} e^{m} + A c m^{2} x x^{m} e^{m} + 5 \, B c x^{3} x^{m} e^{m} + 5 \, A d x^{3} x^{m} e^{m} + 8 \, A c m x x^{m} e^{m} + 15 \, A c x x^{m} e^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 97, normalized size = 1.62 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {x^3\,\left (A\,d+B\,c\right )\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}+\frac {B\,d\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {A\,c\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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