Optimal. Leaf size=16 \[ \frac {1}{28} x^{28} \left (b+c x^2\right )^{14} \]
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Rubi [A]
time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {457, 75}
\begin {gather*} \frac {1}{28} x^{28} \left (b+c x^2\right )^{14} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 457
Rubi steps
\begin {align*} \int x^{27} \left (b+c x^2\right )^{13} \left (b+2 c x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x^{13} (b+c x)^{13} (b+2 c x) \, dx,x,x^2\right )\\ &=\frac {1}{28} x^{28} \left (b+c x^2\right )^{14}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(16)=32\).
time = 0.00, size = 182, normalized size = 11.38 \begin {gather*} \frac {b^{14} x^{28}}{28}+\frac {1}{2} b^{13} c x^{30}+\frac {13}{4} b^{12} c^2 x^{32}+13 b^{11} c^3 x^{34}+\frac {143}{4} b^{10} c^4 x^{36}+\frac {143}{2} b^9 c^5 x^{38}+\frac {429}{4} b^8 c^6 x^{40}+\frac {858}{7} b^7 c^7 x^{42}+\frac {429}{4} b^6 c^8 x^{44}+\frac {143}{2} b^5 c^9 x^{46}+\frac {143}{4} b^4 c^{10} x^{48}+13 b^3 c^{11} x^{50}+\frac {13}{4} b^2 c^{12} x^{52}+\frac {1}{2} b c^{13} x^{54}+\frac {c^{14} x^{56}}{28} \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. \(156\) vs.
\(2(14)=28\).
time = 0.34, size = 157, normalized size = 9.81
method | result | size |
gosper | \(\frac {429}{4} b^{8} c^{6} x^{40}+\frac {858}{7} b^{7} c^{7} x^{42}+\frac {429}{4} b^{6} c^{8} x^{44}+\frac {143}{2} b^{5} c^{9} x^{46}+\frac {143}{4} b^{4} c^{10} x^{48}+13 b^{3} c^{11} x^{50}+\frac {13}{4} b^{2} c^{12} x^{52}+\frac {1}{2} b \,c^{13} x^{54}+\frac {1}{28} c^{14} x^{56}+\frac {1}{28} b^{14} x^{28}+\frac {1}{2} b^{13} c \,x^{30}+\frac {13}{4} b^{12} c^{2} x^{32}+13 b^{11} c^{3} x^{34}+\frac {143}{4} b^{10} c^{4} x^{36}+\frac {143}{2} b^{9} c^{5} x^{38}\) | \(157\) |
default | \(\frac {429}{4} b^{8} c^{6} x^{40}+\frac {858}{7} b^{7} c^{7} x^{42}+\frac {429}{4} b^{6} c^{8} x^{44}+\frac {143}{2} b^{5} c^{9} x^{46}+\frac {143}{4} b^{4} c^{10} x^{48}+13 b^{3} c^{11} x^{50}+\frac {13}{4} b^{2} c^{12} x^{52}+\frac {1}{2} b \,c^{13} x^{54}+\frac {1}{28} c^{14} x^{56}+\frac {1}{28} b^{14} x^{28}+\frac {1}{2} b^{13} c \,x^{30}+\frac {13}{4} b^{12} c^{2} x^{32}+13 b^{11} c^{3} x^{34}+\frac {143}{4} b^{10} c^{4} x^{36}+\frac {143}{2} b^{9} c^{5} x^{38}\) | \(157\) |
risch | \(\frac {429}{4} b^{8} c^{6} x^{40}+\frac {858}{7} b^{7} c^{7} x^{42}+\frac {429}{4} b^{6} c^{8} x^{44}+\frac {143}{2} b^{5} c^{9} x^{46}+\frac {143}{4} b^{4} c^{10} x^{48}+13 b^{3} c^{11} x^{50}+\frac {13}{4} b^{2} c^{12} x^{52}+\frac {1}{2} b \,c^{13} x^{54}+\frac {1}{28} c^{14} x^{56}+\frac {1}{28} b^{14} x^{28}+\frac {1}{2} b^{13} c \,x^{30}+\frac {13}{4} b^{12} c^{2} x^{32}+13 b^{11} c^{3} x^{34}+\frac {143}{4} b^{10} c^{4} x^{36}+\frac {143}{2} b^{9} c^{5} x^{38}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs.
\(2 (14) = 28\).
time = 0.34, size = 156, normalized size = 9.75 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac {143}{4} \, b^{4} c^{10} x^{48} + \frac {143}{2} \, b^{5} c^{9} x^{46} + \frac {429}{4} \, b^{6} c^{8} x^{44} + \frac {858}{7} \, b^{7} c^{7} x^{42} + \frac {429}{4} \, b^{8} c^{6} x^{40} + \frac {143}{2} \, b^{9} c^{5} x^{38} + \frac {143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac {13}{4} \, b^{12} c^{2} x^{32} + \frac {1}{2} \, b^{13} c x^{30} + \frac {1}{28} \, b^{14} x^{28} \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. 156 vs.
\(2 (14) = 28\).
time = 1.12, size = 156, normalized size = 9.75 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac {143}{4} \, b^{4} c^{10} x^{48} + \frac {143}{2} \, b^{5} c^{9} x^{46} + \frac {429}{4} \, b^{6} c^{8} x^{44} + \frac {858}{7} \, b^{7} c^{7} x^{42} + \frac {429}{4} \, b^{8} c^{6} x^{40} + \frac {143}{2} \, b^{9} c^{5} x^{38} + \frac {143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac {13}{4} \, b^{12} c^{2} x^{32} + \frac {1}{2} \, b^{13} c x^{30} + \frac {1}{28} \, b^{14} x^{28} \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. 182 vs.
\(2 (12) = 24\).
time = 0.03, size = 182, normalized size = 11.38 \begin {gather*} \frac {b^{14} x^{28}}{28} + \frac {b^{13} c x^{30}}{2} + \frac {13 b^{12} c^{2} x^{32}}{4} + 13 b^{11} c^{3} x^{34} + \frac {143 b^{10} c^{4} x^{36}}{4} + \frac {143 b^{9} c^{5} x^{38}}{2} + \frac {429 b^{8} c^{6} x^{40}}{4} + \frac {858 b^{7} c^{7} x^{42}}{7} + \frac {429 b^{6} c^{8} x^{44}}{4} + \frac {143 b^{5} c^{9} x^{46}}{2} + \frac {143 b^{4} c^{10} x^{48}}{4} + 13 b^{3} c^{11} x^{50} + \frac {13 b^{2} c^{12} x^{52}}{4} + \frac {b c^{13} x^{54}}{2} + \frac {c^{14} x^{56}}{28} \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. 156 vs.
\(2 (14) = 28\).
time = 0.59, size = 156, normalized size = 9.75 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac {143}{4} \, b^{4} c^{10} x^{48} + \frac {143}{2} \, b^{5} c^{9} x^{46} + \frac {429}{4} \, b^{6} c^{8} x^{44} + \frac {858}{7} \, b^{7} c^{7} x^{42} + \frac {429}{4} \, b^{8} c^{6} x^{40} + \frac {143}{2} \, b^{9} c^{5} x^{38} + \frac {143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac {13}{4} \, b^{12} c^{2} x^{32} + \frac {1}{2} \, b^{13} c x^{30} + \frac {1}{28} \, b^{14} x^{28} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.93, size = 156, normalized size = 9.75 \begin {gather*} \frac {b^{14}\,x^{28}}{28}+\frac {b^{13}\,c\,x^{30}}{2}+\frac {13\,b^{12}\,c^2\,x^{32}}{4}+13\,b^{11}\,c^3\,x^{34}+\frac {143\,b^{10}\,c^4\,x^{36}}{4}+\frac {143\,b^9\,c^5\,x^{38}}{2}+\frac {429\,b^8\,c^6\,x^{40}}{4}+\frac {858\,b^7\,c^7\,x^{42}}{7}+\frac {429\,b^6\,c^8\,x^{44}}{4}+\frac {143\,b^5\,c^9\,x^{46}}{2}+\frac {143\,b^4\,c^{10}\,x^{48}}{4}+13\,b^3\,c^{11}\,x^{50}+\frac {13\,b^2\,c^{12}\,x^{52}}{4}+\frac {b\,c^{13}\,x^{54}}{2}+\frac {c^{14}\,x^{56}}{28} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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