Optimal. Leaf size=62 \[ a d x+\frac {(b d+a e) x^{1+n}}{1+n}+\frac {(c d+b e) x^{1+2 n}}{1+2 n}+\frac {c e x^{1+3 n}}{1+3 n} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1421}
\begin {gather*} \frac {x^{n+1} (a e+b d)}{n+1}+a d x+\frac {x^{2 n+1} (b e+c d)}{2 n+1}+\frac {c e x^{3 n+1}}{3 n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 1421
Rubi steps
\begin {align*} \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right ) \, dx &=\int \left (a d+(b d+a e) x^n+(c d+b e) x^{2 n}+c e x^{3 n}\right ) \, dx\\ &=a d x+\frac {(b d+a e) x^{1+n}}{1+n}+\frac {(c d+b e) x^{1+2 n}}{1+2 n}+\frac {c e x^{1+3 n}}{1+3 n}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 57, normalized size = 0.92 \begin {gather*} x \left (a d+\frac {(b d+a e) x^n}{1+n}+\frac {(c d+b e) x^{2 n}}{1+2 n}+\frac {c e x^{3 n}}{1+3 n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 60, normalized size = 0.97
method | result | size |
risch | \(a d x +\frac {\left (a e +b d \right ) x \,x^{n}}{1+n}+\frac {\left (e b +c d \right ) x \,x^{2 n}}{1+2 n}+\frac {c e x \,x^{3 n}}{1+3 n}\) | \(60\) |
norman | \(a d x +\frac {\left (a e +b d \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {\left (e b +c d \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {c e x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 82, normalized size = 1.32 \begin {gather*} a d x + \frac {c e x^{3 \, n + 1}}{3 \, n + 1} + \frac {c d x^{2 \, n + 1}}{2 \, n + 1} + \frac {b e x^{2 \, n + 1}}{2 \, n + 1} + \frac {b d x^{n + 1}}{n + 1} + \frac {a e x^{n + 1}}{n + 1} \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. 145 vs.
\(2 (65) = 130\).
time = 0.35, size = 145, normalized size = 2.34 \begin {gather*} \frac {{\left (2 \, c n^{2} + 3 \, c n + c\right )} x x^{3 \, n} e + {\left (6 \, a d n^{3} + 11 \, a d n^{2} + 6 \, a d n + a d\right )} x + {\left ({\left (3 \, b n^{2} + 4 \, b n + b\right )} x e + {\left (3 \, c d n^{2} + 4 \, c d n + c d\right )} x\right )} x^{2 \, n} + {\left ({\left (6 \, a n^{2} + 5 \, a n + a\right )} x e + {\left (6 \, b d n^{2} + 5 \, b d n + b d\right )} x\right )} x^{n}}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 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. 656 vs.
\(2 (53) = 106\).
time = 0.36, size = 656, normalized size = 10.58 \begin {gather*} \begin {cases} a d x + a e \log {\left (x \right )} + b d \log {\left (x \right )} - \frac {b e}{x} - \frac {c d}{x} - \frac {c e}{2 x^{2}} & \text {for}\: n = -1 \\a d x + 2 a e \sqrt {x} + 2 b d \sqrt {x} + b e \log {\left (x \right )} + c d \log {\left (x \right )} - \frac {2 c e}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\a d x + \frac {3 a e x^{\frac {2}{3}}}{2} + \frac {3 b d x^{\frac {2}{3}}}{2} + 3 b e \sqrt [3]{x} + 3 c d \sqrt [3]{x} + c e \log {\left (x \right )} & \text {for}\: n = - \frac {1}{3} \\\frac {6 a d n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {11 a d n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a d n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a d x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a e n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {5 a e n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a e x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 b d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {5 b d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b e n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {4 b e n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b e x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 c d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {4 c d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {c d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 c e n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 c e n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {c e x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 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. 207 vs.
\(2 (65) = 130\).
time = 4.17, size = 207, normalized size = 3.34 \begin {gather*} \frac {6 \, a d n^{3} x + 3 \, c d n^{2} x x^{2 \, n} + 6 \, b d n^{2} x x^{n} + 2 \, c n^{2} x x^{3 \, n} e + 3 \, b n^{2} x x^{2 \, n} e + 6 \, a n^{2} x x^{n} e + 11 \, a d n^{2} x + 4 \, c d n x x^{2 \, n} + 5 \, b d n x x^{n} + 3 \, c n x x^{3 \, n} e + 4 \, b n x x^{2 \, n} e + 5 \, a n x x^{n} e + 6 \, a d n x + c d x x^{2 \, n} + b d x x^{n} + c x x^{3 \, n} e + b x x^{2 \, n} e + a x x^{n} e + a d x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.66, size = 59, normalized size = 0.95 \begin {gather*} a\,d\,x+\frac {x\,x^{2\,n}\,\left (b\,e+c\,d\right )}{2\,n+1}+\frac {x\,x^n\,\left (a\,e+b\,d\right )}{n+1}+\frac {c\,e\,x\,x^{3\,n}}{3\,n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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