Optimal. Leaf size=43 \[ \frac {a^2 x^{1+m}}{1+m}+\frac {2 a b x^{8+m}}{8+m}+\frac {b^2 x^{15+m}}{15+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\begin {gather*} \frac {a^2 x^{m+1}}{m+1}+\frac {2 a b x^{m+8}}{m+8}+\frac {b^2 x^{m+15}}{m+15} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int x^m \left (a+b x^7\right )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{7+m}+b^2 x^{14+m}\right ) \, dx\\ &=\frac {a^2 x^{1+m}}{1+m}+\frac {2 a b x^{8+m}}{8+m}+\frac {b^2 x^{15+m}}{15+m}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 40, normalized size = 0.93 \begin {gather*} x^{1+m} \left (\frac {a^2}{1+m}+\frac {2 a b x^7}{8+m}+\frac {b^2 x^{14}}{15+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs.
\(2(43)=86\).
time = 0.16, size = 92, normalized size = 2.14
method | result | size |
risch | \(\frac {x \left (b^{2} m^{2} x^{14}+9 m \,x^{14} b^{2}+8 b^{2} x^{14}+2 a b \,m^{2} x^{7}+32 m \,x^{7} a b +30 a b \,x^{7}+a^{2} m^{2}+23 m \,a^{2}+120 a^{2}\right ) x^{m}}{\left (1+m \right ) \left (8+m \right ) \left (15+m \right )}\) | \(92\) |
gosper | \(\frac {x^{1+m} \left (b^{2} m^{2} x^{14}+9 m \,x^{14} b^{2}+8 b^{2} x^{14}+2 a b \,m^{2} x^{7}+32 m \,x^{7} a b +30 a b \,x^{7}+a^{2} m^{2}+23 m \,a^{2}+120 a^{2}\right )}{\left (1+m \right ) \left (8+m \right ) \left (15+m \right )}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 43, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{m + 15}}{m + 15} + \frac {2 \, a b x^{m + 8}}{m + 8} + \frac {a^{2} x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 85, normalized size = 1.98 \begin {gather*} \frac {{\left ({\left (b^{2} m^{2} + 9 \, b^{2} m + 8 \, b^{2}\right )} x^{15} + 2 \, {\left (a b m^{2} + 16 \, a b m + 15 \, a b\right )} x^{8} + {\left (a^{2} m^{2} + 23 \, a^{2} m + 120 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \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. 313 vs.
\(2 (36) = 72\).
time = 0.88, size = 313, normalized size = 7.28 \begin {gather*} \begin {cases} - \frac {a^{2}}{14 x^{14}} - \frac {2 a b}{7 x^{7}} + b^{2} \log {\left (x \right )} & \text {for}\: m = -15 \\- \frac {a^{2}}{7 x^{7}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{7}}{7} & \text {for}\: m = -8 \\a^{2} \log {\left (x \right )} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{14}}{14} & \text {for}\: m = -1 \\\frac {a^{2} m^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {23 a^{2} m x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {120 a^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {2 a b m^{2} x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {32 a b m x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {30 a b x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {b^{2} m^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {9 b^{2} m x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac {8 b^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} & \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. 117 vs.
\(2 (43) = 86\).
time = 1.54, size = 117, normalized size = 2.72 \begin {gather*} \frac {b^{2} m^{2} x^{15} x^{m} + 9 \, b^{2} m x^{15} x^{m} + 8 \, b^{2} x^{15} x^{m} + 2 \, a b m^{2} x^{8} x^{m} + 32 \, a b m x^{8} x^{m} + 30 \, a b x^{8} x^{m} + a^{2} m^{2} x x^{m} + 23 \, a^{2} m x x^{m} + 120 \, a^{2} x x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 93, normalized size = 2.16 \begin {gather*} x^m\,\left (\frac {a^2\,x\,\left (m^2+23\,m+120\right )}{m^3+24\,m^2+143\,m+120}+\frac {b^2\,x^{15}\,\left (m^2+9\,m+8\right )}{m^3+24\,m^2+143\,m+120}+\frac {2\,a\,b\,x^8\,\left (m^2+16\,m+15\right )}{m^3+24\,m^2+143\,m+120}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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