Optimal. Leaf size=45 \[ \frac {A b x^{3+m}}{3+m}+\frac {(b B+A c) x^{5+m}}{5+m}+\frac {B c x^{7+m}}{7+m} \]
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Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1598, 459}
\begin {gather*} \frac {x^{m+5} (A c+b B)}{m+5}+\frac {A b x^{m+3}}{m+3}+\frac {B c x^{m+7}}{m+7} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rule 1598
Rubi steps
\begin {align*} \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^{2+m} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^{2+m}+(b B+A c) x^{4+m}+B c x^{6+m}\right ) \, dx\\ &=\frac {A b x^{3+m}}{3+m}+\frac {(b B+A c) x^{5+m}}{5+m}+\frac {B c x^{7+m}}{7+m}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 42, normalized size = 0.93 \begin {gather*} x^{3+m} \left (\frac {A b}{3+m}+\frac {(b B+A c) x^2}{5+m}+\frac {B c x^4}{7+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 55, normalized size = 1.22
method | result | size |
norman | \(\frac {\left (A c +B b \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {A b \,x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {B c \,x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}\) | \(55\) |
gosper | \(\frac {x^{3+m} \left (B c \,m^{2} x^{4}+8 B c m \,x^{4}+A c \,m^{2} x^{2}+B b \,m^{2} x^{2}+15 B c \,x^{4}+10 A c m \,x^{2}+10 B b m \,x^{2}+A b \,m^{2}+21 A c \,x^{2}+21 b B \,x^{2}+12 A b m +35 A b \right )}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right )}\) | \(110\) |
risch | \(\frac {x^{m} \left (B c \,m^{2} x^{4}+8 B c m \,x^{4}+A c \,m^{2} x^{2}+B b \,m^{2} x^{2}+15 B c \,x^{4}+10 A c m \,x^{2}+10 B b m \,x^{2}+A b \,m^{2}+21 A c \,x^{2}+21 b B \,x^{2}+12 A b m +35 A b \right ) x^{3}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right )}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 53, normalized size = 1.18 \begin {gather*} \frac {B c x^{m + 7}}{m + 7} + \frac {B b x^{m + 5}}{m + 5} + \frac {A c x^{m + 5}}{m + 5} + \frac {A b x^{m + 3}}{m + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (45) = 90\).
time = 2.46, size = 94, normalized size = 2.09 \begin {gather*} \frac {{\left ({\left (B c m^{2} + 8 \, B c m + 15 \, B c\right )} x^{7} + {\left ({\left (B b + A c\right )} m^{2} + 21 \, B b + 21 \, A c + 10 \, {\left (B b + A c\right )} m\right )} x^{5} + {\left (A b m^{2} + 12 \, A b m + 35 \, A b\right )} x^{3}\right )} x^{m}}{m^{3} + 15 \, m^{2} + 71 \, m + 105} \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. 415 vs.
\(2 (37) = 74\).
time = 0.33, size = 415, normalized size = 9.22 \begin {gather*} \begin {cases} - \frac {A b}{4 x^{4}} - \frac {A c}{2 x^{2}} - \frac {B b}{2 x^{2}} + B c \log {\left (x \right )} & \text {for}\: m = -7 \\- \frac {A b}{2 x^{2}} + A c \log {\left (x \right )} + B b \log {\left (x \right )} + \frac {B c x^{2}}{2} & \text {for}\: m = -5 \\A b \log {\left (x \right )} + \frac {A c x^{2}}{2} + \frac {B b x^{2}}{2} + \frac {B c x^{4}}{4} & \text {for}\: m = -3 \\\frac {A b m^{2} x^{3} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {12 A b m x^{3} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {35 A b x^{3} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {A c m^{2} x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {10 A c m x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {21 A c x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {B b m^{2} x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {10 B b m x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {21 B b x^{5} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {B c m^{2} x^{7} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {8 B c m x^{7} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} + \frac {15 B c x^{7} x^{m}}{m^{3} + 15 m^{2} + 71 m + 105} & \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. 149 vs.
\(2 (45) = 90\).
time = 1.03, size = 149, normalized size = 3.31 \begin {gather*} \frac {B c m^{2} x^{7} x^{m} + 8 \, B c m x^{7} x^{m} + B b m^{2} x^{5} x^{m} + A c m^{2} x^{5} x^{m} + 15 \, B c x^{7} x^{m} + 10 \, B b m x^{5} x^{m} + 10 \, A c m x^{5} x^{m} + A b m^{2} x^{3} x^{m} + 21 \, B b x^{5} x^{m} + 21 \, A c x^{5} x^{m} + 12 \, A b m x^{3} x^{m} + 35 \, A b x^{3} x^{m}}{m^{3} + 15 \, m^{2} + 71 \, m + 105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 97, normalized size = 2.16 \begin {gather*} x^m\,\left (\frac {x^5\,\left (A\,c+B\,b\right )\,\left (m^2+10\,m+21\right )}{m^3+15\,m^2+71\,m+105}+\frac {A\,b\,x^3\,\left (m^2+12\,m+35\right )}{m^3+15\,m^2+71\,m+105}+\frac {B\,c\,x^7\,\left (m^2+8\,m+15\right )}{m^3+15\,m^2+71\,m+105}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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