Optimal. Leaf size=137 \[ \frac {a A (d x)^{1+m}}{d (1+m)}+\frac {a B (d x)^{2+m}}{d^2 (2+m)}+\frac {(A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {b B (d x)^{4+m}}{d^4 (4+m)}+\frac {(A c+b C) (d x)^{5+m}}{d^5 (5+m)}+\frac {B c (d x)^{6+m}}{d^6 (6+m)}+\frac {c C (d x)^{7+m}}{d^7 (7+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1642}
\begin {gather*} \frac {(d x)^{m+3} (a C+A b)}{d^3 (m+3)}+\frac {a A (d x)^{m+1}}{d (m+1)}+\frac {a B (d x)^{m+2}}{d^2 (m+2)}+\frac {(d x)^{m+5} (A c+b C)}{d^5 (m+5)}+\frac {b B (d x)^{m+4}}{d^4 (m+4)}+\frac {B c (d x)^{m+6}}{d^6 (m+6)}+\frac {c C (d x)^{m+7}}{d^7 (m+7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a A (d x)^m+\frac {a B (d x)^{1+m}}{d}+\frac {(A b+a C) (d x)^{2+m}}{d^2}+\frac {b B (d x)^{3+m}}{d^3}+\frac {(A c+b C) (d x)^{4+m}}{d^4}+\frac {B c (d x)^{5+m}}{d^5}+\frac {c C (d x)^{6+m}}{d^6}\right ) \, dx\\ &=\frac {a A (d x)^{1+m}}{d (1+m)}+\frac {a B (d x)^{2+m}}{d^2 (2+m)}+\frac {(A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {b B (d x)^{4+m}}{d^4 (4+m)}+\frac {(A c+b C) (d x)^{5+m}}{d^5 (5+m)}+\frac {B c (d x)^{6+m}}{d^6 (6+m)}+\frac {c C (d x)^{7+m}}{d^7 (7+m)}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 92, normalized size = 0.67 \begin {gather*} (d x)^m \left (\frac {a A x}{1+m}+\frac {a B x^2}{2+m}+\frac {(A b+a C) x^3}{3+m}+\frac {b B x^4}{4+m}+\frac {(A c+b C) x^5}{5+m}+\frac {B c x^6}{6+m}+\frac {c C x^7}{7+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 136, normalized size = 0.99
method | result | size |
norman | \(\frac {\left (A b +a C \right ) x^{3} {\mathrm e}^{m \ln \left (d x \right )}}{3+m}+\frac {\left (A c +b C \right ) x^{5} {\mathrm e}^{m \ln \left (d x \right )}}{5+m}+\frac {a A x \,{\mathrm e}^{m \ln \left (d x \right )}}{1+m}+\frac {a B \,x^{2} {\mathrm e}^{m \ln \left (d x \right )}}{2+m}+\frac {b B \,x^{4} {\mathrm e}^{m \ln \left (d x \right )}}{4+m}+\frac {c B \,x^{6} {\mathrm e}^{m \ln \left (d x \right )}}{6+m}+\frac {c C \,x^{7} {\mathrm e}^{m \ln \left (d x \right )}}{7+m}\) | \(136\) |
gosper | \(\frac {x \left (C c \,m^{6} x^{6}+B c \,m^{6} x^{5}+21 C c \,m^{5} x^{6}+A c \,m^{6} x^{4}+22 B c \,m^{5} x^{5}+C b \,m^{6} x^{4}+175 C c \,m^{4} x^{6}+23 A c \,m^{5} x^{4}+B b \,m^{6} x^{3}+190 B c \,m^{4} x^{5}+23 C b \,m^{5} x^{4}+735 C c \,m^{3} x^{6}+A b \,m^{6} x^{2}+207 A c \,m^{4} x^{4}+24 B b \,m^{5} x^{3}+820 B c \,m^{3} x^{5}+C a \,m^{6} x^{2}+207 C b \,m^{4} x^{4}+1624 C c \,m^{2} x^{6}+25 A b \,m^{5} x^{2}+925 A c \,m^{3} x^{4}+B a \,m^{6} x +226 B b \,m^{4} x^{3}+1849 B c \,m^{2} x^{5}+25 C a \,m^{5} x^{2}+925 C b \,m^{3} x^{4}+1764 C c m \,x^{6}+A a \,m^{6}+247 A b \,m^{4} x^{2}+2144 A c \,m^{2} x^{4}+26 B a \,m^{5} x +1056 B b \,m^{3} x^{3}+2038 B c m \,x^{5}+247 C a \,m^{4} x^{2}+2144 C b \,m^{2} x^{4}+720 c C \,x^{6}+27 A a \,m^{5}+1219 A b \,m^{3} x^{2}+2412 A c m \,x^{4}+270 B a \,m^{4} x +2545 B b \,m^{2} x^{3}+840 B c \,x^{5}+1219 C a \,m^{3} x^{2}+2412 C b m \,x^{4}+295 A a \,m^{4}+3112 A b \,m^{2} x^{2}+1008 A c \,x^{4}+1420 B a \,m^{3} x +2952 B b m \,x^{3}+3112 C a \,m^{2} x^{2}+1008 b C \,x^{4}+1665 A a \,m^{3}+3796 A b m \,x^{2}+3929 B a \,m^{2} x +1260 B b \,x^{3}+3796 C a m \,x^{2}+5104 A a \,m^{2}+1680 A b \,x^{2}+5274 B a m x +1680 C a \,x^{2}+8028 A a m +2520 a B x +5040 a A \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(585\) |
risch | \(\frac {x \left (C c \,m^{6} x^{6}+B c \,m^{6} x^{5}+21 C c \,m^{5} x^{6}+A c \,m^{6} x^{4}+22 B c \,m^{5} x^{5}+C b \,m^{6} x^{4}+175 C c \,m^{4} x^{6}+23 A c \,m^{5} x^{4}+B b \,m^{6} x^{3}+190 B c \,m^{4} x^{5}+23 C b \,m^{5} x^{4}+735 C c \,m^{3} x^{6}+A b \,m^{6} x^{2}+207 A c \,m^{4} x^{4}+24 B b \,m^{5} x^{3}+820 B c \,m^{3} x^{5}+C a \,m^{6} x^{2}+207 C b \,m^{4} x^{4}+1624 C c \,m^{2} x^{6}+25 A b \,m^{5} x^{2}+925 A c \,m^{3} x^{4}+B a \,m^{6} x +226 B b \,m^{4} x^{3}+1849 B c \,m^{2} x^{5}+25 C a \,m^{5} x^{2}+925 C b \,m^{3} x^{4}+1764 C c m \,x^{6}+A a \,m^{6}+247 A b \,m^{4} x^{2}+2144 A c \,m^{2} x^{4}+26 B a \,m^{5} x +1056 B b \,m^{3} x^{3}+2038 B c m \,x^{5}+247 C a \,m^{4} x^{2}+2144 C b \,m^{2} x^{4}+720 c C \,x^{6}+27 A a \,m^{5}+1219 A b \,m^{3} x^{2}+2412 A c m \,x^{4}+270 B a \,m^{4} x +2545 B b \,m^{2} x^{3}+840 B c \,x^{5}+1219 C a \,m^{3} x^{2}+2412 C b m \,x^{4}+295 A a \,m^{4}+3112 A b \,m^{2} x^{2}+1008 A c \,x^{4}+1420 B a \,m^{3} x +2952 B b m \,x^{3}+3112 C a \,m^{2} x^{2}+1008 b C \,x^{4}+1665 A a \,m^{3}+3796 A b m \,x^{2}+3929 B a \,m^{2} x +1260 B b \,x^{3}+3796 C a m \,x^{2}+5104 A a \,m^{2}+1680 A b \,x^{2}+5274 B a m x +1680 C a \,x^{2}+8028 A a m +2520 a B x +5040 a A \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(585\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 155, normalized size = 1.13 \begin {gather*} \frac {C c d^{m} x^{7} x^{m}}{m + 7} + \frac {B c d^{m} x^{6} x^{m}}{m + 6} + \frac {C b d^{m} x^{5} x^{m}}{m + 5} + \frac {A c d^{m} x^{5} x^{m}}{m + 5} + \frac {B b d^{m} x^{4} x^{m}}{m + 4} + \frac {C a d^{m} x^{3} x^{m}}{m + 3} + \frac {A b d^{m} x^{3} x^{m}}{m + 3} + \frac {B a d^{m} x^{2} x^{m}}{m + 2} + \frac {\left (d x\right )^{m + 1} A a}{d {\left (m + 1\right )}} \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. 444 vs.
\(2 (137) = 274\).
time = 0.38, size = 444, normalized size = 3.24 \begin {gather*} \frac {{\left ({\left (C c m^{6} + 21 \, C c m^{5} + 175 \, C c m^{4} + 735 \, C c m^{3} + 1624 \, C c m^{2} + 1764 \, C c m + 720 \, C c\right )} x^{7} + {\left (B c m^{6} + 22 \, B c m^{5} + 190 \, B c m^{4} + 820 \, B c m^{3} + 1849 \, B c m^{2} + 2038 \, B c m + 840 \, B c\right )} x^{6} + {\left ({\left (C b + A c\right )} m^{6} + 23 \, {\left (C b + A c\right )} m^{5} + 207 \, {\left (C b + A c\right )} m^{4} + 925 \, {\left (C b + A c\right )} m^{3} + 2144 \, {\left (C b + A c\right )} m^{2} + 1008 \, C b + 1008 \, A c + 2412 \, {\left (C b + A c\right )} m\right )} x^{5} + {\left (B b m^{6} + 24 \, B b m^{5} + 226 \, B b m^{4} + 1056 \, B b m^{3} + 2545 \, B b m^{2} + 2952 \, B b m + 1260 \, B b\right )} x^{4} + {\left ({\left (C a + A b\right )} m^{6} + 25 \, {\left (C a + A b\right )} m^{5} + 247 \, {\left (C a + A b\right )} m^{4} + 1219 \, {\left (C a + A b\right )} m^{3} + 3112 \, {\left (C a + A b\right )} m^{2} + 1680 \, C a + 1680 \, A b + 3796 \, {\left (C a + A b\right )} m\right )} x^{3} + {\left (B a m^{6} + 26 \, B a m^{5} + 270 \, B a m^{4} + 1420 \, B a m^{3} + 3929 \, B a m^{2} + 5274 \, B a m + 2520 \, B a\right )} x^{2} + {\left (A a m^{6} + 27 \, A a m^{5} + 295 \, A a m^{4} + 1665 \, A a m^{3} + 5104 \, A a m^{2} + 8028 \, A a m + 5040 \, A a\right )} x\right )} \left (d x\right )^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \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. 3628 vs.
\(2 (122) = 244\).
time = 0.58, size = 3628, normalized size = 26.48 \begin {gather*} \text {Too large to display} \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. 914 vs.
\(2 (137) = 274\).
time = 6.07, size = 914, normalized size = 6.67 \begin {gather*} \frac {\left (d x\right )^{m} C c m^{6} x^{7} + \left (d x\right )^{m} B c m^{6} x^{6} + 21 \, \left (d x\right )^{m} C c m^{5} x^{7} + \left (d x\right )^{m} C b m^{6} x^{5} + \left (d x\right )^{m} A c m^{6} x^{5} + 22 \, \left (d x\right )^{m} B c m^{5} x^{6} + 175 \, \left (d x\right )^{m} C c m^{4} x^{7} + \left (d x\right )^{m} B b m^{6} x^{4} + 23 \, \left (d x\right )^{m} C b m^{5} x^{5} + 23 \, \left (d x\right )^{m} A c m^{5} x^{5} + 190 \, \left (d x\right )^{m} B c m^{4} x^{6} + 735 \, \left (d x\right )^{m} C c m^{3} x^{7} + \left (d x\right )^{m} C a m^{6} x^{3} + \left (d x\right )^{m} A b m^{6} x^{3} + 24 \, \left (d x\right )^{m} B b m^{5} x^{4} + 207 \, \left (d x\right )^{m} C b m^{4} x^{5} + 207 \, \left (d x\right )^{m} A c m^{4} x^{5} + 820 \, \left (d x\right )^{m} B c m^{3} x^{6} + 1624 \, \left (d x\right )^{m} C c m^{2} x^{7} + \left (d x\right )^{m} B a m^{6} x^{2} + 25 \, \left (d x\right )^{m} C a m^{5} x^{3} + 25 \, \left (d x\right )^{m} A b m^{5} x^{3} + 226 \, \left (d x\right )^{m} B b m^{4} x^{4} + 925 \, \left (d x\right )^{m} C b m^{3} x^{5} + 925 \, \left (d x\right )^{m} A c m^{3} x^{5} + 1849 \, \left (d x\right )^{m} B c m^{2} x^{6} + 1764 \, \left (d x\right )^{m} C c m x^{7} + \left (d x\right )^{m} A a m^{6} x + 26 \, \left (d x\right )^{m} B a m^{5} x^{2} + 247 \, \left (d x\right )^{m} C a m^{4} x^{3} + 247 \, \left (d x\right )^{m} A b m^{4} x^{3} + 1056 \, \left (d x\right )^{m} B b m^{3} x^{4} + 2144 \, \left (d x\right )^{m} C b m^{2} x^{5} + 2144 \, \left (d x\right )^{m} A c m^{2} x^{5} + 2038 \, \left (d x\right )^{m} B c m x^{6} + 720 \, \left (d x\right )^{m} C c x^{7} + 27 \, \left (d x\right )^{m} A a m^{5} x + 270 \, \left (d x\right )^{m} B a m^{4} x^{2} + 1219 \, \left (d x\right )^{m} C a m^{3} x^{3} + 1219 \, \left (d x\right )^{m} A b m^{3} x^{3} + 2545 \, \left (d x\right )^{m} B b m^{2} x^{4} + 2412 \, \left (d x\right )^{m} C b m x^{5} + 2412 \, \left (d x\right )^{m} A c m x^{5} + 840 \, \left (d x\right )^{m} B c x^{6} + 295 \, \left (d x\right )^{m} A a m^{4} x + 1420 \, \left (d x\right )^{m} B a m^{3} x^{2} + 3112 \, \left (d x\right )^{m} C a m^{2} x^{3} + 3112 \, \left (d x\right )^{m} A b m^{2} x^{3} + 2952 \, \left (d x\right )^{m} B b m x^{4} + 1008 \, \left (d x\right )^{m} C b x^{5} + 1008 \, \left (d x\right )^{m} A c x^{5} + 1665 \, \left (d x\right )^{m} A a m^{3} x + 3929 \, \left (d x\right )^{m} B a m^{2} x^{2} + 3796 \, \left (d x\right )^{m} C a m x^{3} + 3796 \, \left (d x\right )^{m} A b m x^{3} + 1260 \, \left (d x\right )^{m} B b x^{4} + 5104 \, \left (d x\right )^{m} A a m^{2} x + 5274 \, \left (d x\right )^{m} B a m x^{2} + 1680 \, \left (d x\right )^{m} C a x^{3} + 1680 \, \left (d x\right )^{m} A b x^{3} + 8028 \, \left (d x\right )^{m} A a m x + 2520 \, \left (d x\right )^{m} B a x^{2} + 5040 \, \left (d x\right )^{m} A a x}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.07, size = 527, normalized size = 3.85 \begin {gather*} \frac {x^3\,{\left (d\,x\right )}^m\,\left (A\,b+C\,a\right )\,\left (m^6+25\,m^5+247\,m^4+1219\,m^3+3112\,m^2+3796\,m+1680\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {x^5\,{\left (d\,x\right )}^m\,\left (A\,c+C\,b\right )\,\left (m^6+23\,m^5+207\,m^4+925\,m^3+2144\,m^2+2412\,m+1008\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {A\,a\,x\,{\left (d\,x\right )}^m\,\left (m^6+27\,m^5+295\,m^4+1665\,m^3+5104\,m^2+8028\,m+5040\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {B\,a\,x^2\,{\left (d\,x\right )}^m\,\left (m^6+26\,m^5+270\,m^4+1420\,m^3+3929\,m^2+5274\,m+2520\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {B\,b\,x^4\,{\left (d\,x\right )}^m\,\left (m^6+24\,m^5+226\,m^4+1056\,m^3+2545\,m^2+2952\,m+1260\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {B\,c\,x^6\,{\left (d\,x\right )}^m\,\left (m^6+22\,m^5+190\,m^4+820\,m^3+1849\,m^2+2038\,m+840\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {C\,c\,x^7\,{\left (d\,x\right )}^m\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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