Optimal. Leaf size=58 \[ \frac {b x^{1+n} (d x)^m}{1+m+n}+\frac {c x^{1+2 n} (d x)^m}{1+m+2 n}+\frac {a (d x)^{1+m}}{d (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 20, 30}
\begin {gather*} \frac {a (d x)^{m+1}}{d (m+1)}+\frac {b x^{n+1} (d x)^m}{m+n+1}+\frac {c x^{2 n+1} (d x)^m}{m+2 n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 20
Rule 30
Rubi steps
\begin {align*} \int (d x)^m \left (a+b x^n+c x^{2 n}\right ) \, dx &=\int \left (a (d x)^m+b x^n (d x)^m+c x^{2 n} (d x)^m\right ) \, dx\\ &=\frac {a (d x)^{1+m}}{d (1+m)}+b \int x^n (d x)^m \, dx+c \int x^{2 n} (d x)^m \, dx\\ &=\frac {a (d x)^{1+m}}{d (1+m)}+\left (b x^{-m} (d x)^m\right ) \int x^{m+n} \, dx+\left (c x^{-m} (d x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac {b x^{1+n} (d x)^m}{1+m+n}+\frac {c x^{1+2 n} (d x)^m}{1+m+2 n}+\frac {a (d x)^{1+m}}{d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 41, normalized size = 0.71 \begin {gather*} x (d x)^m \left (\frac {a}{1+m}+x^n \left (\frac {b}{1+m+n}+\frac {c x^n}{1+m+2 n}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.05, size = 205, normalized size = 3.53
method | result | size |
risch | \(\frac {x \left (c \,m^{2} x^{2 n}+c m n \,x^{2 n}+b \,m^{2} x^{n}+2 b m n \,x^{n}+2 x^{2 n} c m +c \,x^{2 n} n +a \,m^{2}+3 a m n +2 a \,n^{2}+2 x^{n} b m +2 b \,x^{n} n +c \,x^{2 n}+2 a m +3 a n +b \,x^{n}+a \right ) {\mathrm e}^{\frac {m \left (i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right ) \mathrm {csgn}\left (i d \right )-i \pi \mathrm {csgn}\left (i d x \right )^{3}+i \pi \mathrm {csgn}\left (i d x \right )^{2} \mathrm {csgn}\left (i d \right )+2 \ln \left (x \right )+2 \ln \left (d \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right )}\) | \(205\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 65, normalized size = 1.12 \begin {gather*} \frac {c d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {b d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (d x\right )^{m + 1} 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. 142 vs.
\(2 (58) = 116\).
time = 0.36, size = 142, normalized size = 2.45 \begin {gather*} \frac {{\left (c m^{2} + 2 \, c m + {\left (c m + c\right )} n + c\right )} x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + {\left (b m^{2} + 2 \, b m + 2 \, {\left (b m + b\right )} n + b\right )} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + {\left (a m^{2} + 2 \, a n^{2} + 2 \, a m + 3 \, {\left (a m + a\right )} n + a\right )} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \, {\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \, {\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \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. 1091 vs.
\(2 (49) = 98\).
time = 18.64, size = 1091, normalized size = 18.81 \begin {gather*} \begin {cases} \frac {\left (a + b + c\right ) \log {\left (x \right )}}{d} & \text {for}\: m = -1 \wedge n = 0 \\\frac {a \log {\left (x \right )} + \frac {b x^{n}}{n} + \frac {c x^{2 n}}{2 n}}{d} & \text {for}\: m = -1 \\\frac {a \left (\begin {cases} - \frac {\left (d x\right )^{- 2 n}}{2 n} & \text {for}\: n \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{d} + \frac {b \left (\begin {cases} - \frac {x^{n} \left (d x\right )^{- 2 n}}{n} & \text {for}\: n \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{d} + \frac {c \left (\begin {cases} 0 & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\d^{- 2 n} \log {\left (x \right )} & \text {for}\: \left |{x}\right | < 1 \\- d^{- 2 n} \log {\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- d^{- 2 n} {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + d^{- 2 n} {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases}\right )}{d} & \text {for}\: m = - 2 n - 1 \\\frac {a \left (\begin {cases} - \frac {\left (d x\right )^{- n}}{n} & \text {for}\: n \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{d} + \frac {b \left (\begin {cases} 0 & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\d^{- n} \log {\left (x \right )} & \text {for}\: \left |{x}\right | < 1 \\- d^{- n} \log {\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- d^{- n} {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + d^{- n} {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases}\right )}{d} + \frac {c \left (\begin {cases} \frac {x^{2 n} \left (d x\right )^{- n}}{n} & \text {for}\: n \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}{d} & \text {for}\: m = - n - 1 \\\frac {a m^{2} x \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {3 a m n x \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {2 a m x \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {2 a n^{2} x \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {3 a n x \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {a x \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {b m^{2} x x^{n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {2 b m n x x^{n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {2 b m x x^{n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {2 b n x x^{n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {b x x^{n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {c m^{2} x x^{2 n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {c m n x x^{2 n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {2 c m x x^{2 n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {c n x x^{2 n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} + \frac {c x x^{2 n} \left (d x\right )^{m}}{m^{3} + 3 m^{2} n + 3 m^{2} + 2 m n^{2} + 6 m n + 3 m + 2 n^{2} + 3 n + 1} & \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. 557 vs.
\(2 (58) = 116\).
time = 3.74, size = 557, normalized size = 9.60 \begin {gather*} \frac {c m^{2} x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c m n x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + b m^{2} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c m^{2} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b m n x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c m n x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + a m^{2} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + b m^{2} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c m^{2} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \, a m n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b m n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c m n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, a n^{2} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, c m x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c n x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b m x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, c m x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b n x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c n x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, a m x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b m x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, c m x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \, a n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 2 \, b n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + b x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + a x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + b x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + c x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 83, normalized size = 1.43 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {a\,x}{m+1}+\frac {b\,x\,x^n\,\left (m+2\,n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}+\frac {c\,x\,x^{2\,n}\,\left (m+n+1\right )}{m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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