Optimal. Leaf size=99 \[ a c^3 x+\frac {c^2 (b c+3 a d) x^{1+n}}{1+n}+\frac {3 c d (b c+a d) x^{1+2 n}}{1+2 n}+\frac {d^2 (3 b c+a d) x^{1+3 n}}{1+3 n}+\frac {b d^3 x^{1+4 n}}{1+4 n} \]
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Rubi [A]
time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {380}
\begin {gather*} \frac {c^2 x^{n+1} (3 a d+b c)}{n+1}+\frac {d^2 x^{3 n+1} (a d+3 b c)}{3 n+1}+\frac {3 c d x^{2 n+1} (a d+b c)}{2 n+1}+a c^3 x+\frac {b d^3 x^{4 n+1}}{4 n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin {align*} \int \left (a+b x^n\right ) \left (c+d x^n\right )^3 \, dx &=\int \left (a c^3+c^2 (b c+3 a d) x^n+3 c d (b c+a d) x^{2 n}+d^2 (3 b c+a d) x^{3 n}+b d^3 x^{4 n}\right ) \, dx\\ &=a c^3 x+\frac {c^2 (b c+3 a d) x^{1+n}}{1+n}+\frac {3 c d (b c+a d) x^{1+2 n}}{1+2 n}+\frac {d^2 (3 b c+a d) x^{1+3 n}}{1+3 n}+\frac {b d^3 x^{1+4 n}}{1+4 n}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 92, normalized size = 0.93 \begin {gather*} x \left (a c^3+\frac {c^2 (b c+3 a d) x^n}{1+n}+\frac {3 c d (b c+a d) x^{2 n}}{1+2 n}+\frac {d^2 (3 b c+a d) x^{3 n}}{1+3 n}+\frac {b d^3 x^{4 n}}{1+4 n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 96, normalized size = 0.97
method | result | size |
risch | \(a \,c^{3} x +\frac {b \,d^{3} x \,x^{4 n}}{1+4 n}+\frac {c^{2} \left (3 a d +b c \right ) x \,x^{n}}{1+n}+\frac {d^{2} \left (a d +3 b c \right ) x \,x^{3 n}}{1+3 n}+\frac {3 c d \left (a d +b c \right ) x \,x^{2 n}}{1+2 n}\) | \(96\) |
norman | \(a \,c^{3} x +\frac {b \,d^{3} x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}+\frac {c^{2} \left (3 a d +b c \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {d^{2} \left (a d +3 b c \right ) x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {3 c d \left (a d +b c \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 140, normalized size = 1.41 \begin {gather*} a c^{3} x + \frac {b d^{3} x^{4 \, n + 1}}{4 \, n + 1} + \frac {3 \, b c d^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {a d^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, b c^{2} d x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a c d^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c^{3} x^{n + 1}}{n + 1} + \frac {3 \, a c^{2} d 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. 319 vs.
\(2 (99) = 198\).
time = 2.63, size = 319, normalized size = 3.22 \begin {gather*} \frac {{\left (6 \, b d^{3} n^{3} + 11 \, b d^{3} n^{2} + 6 \, b d^{3} n + b d^{3}\right )} x x^{4 \, n} + {\left (3 \, b c d^{2} + a d^{3} + 8 \, {\left (3 \, b c d^{2} + a d^{3}\right )} n^{3} + 14 \, {\left (3 \, b c d^{2} + a d^{3}\right )} n^{2} + 7 \, {\left (3 \, b c d^{2} + a d^{3}\right )} n\right )} x x^{3 \, n} + 3 \, {\left (b c^{2} d + a c d^{2} + 12 \, {\left (b c^{2} d + a c d^{2}\right )} n^{3} + 19 \, {\left (b c^{2} d + a c d^{2}\right )} n^{2} + 8 \, {\left (b c^{2} d + a c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (b c^{3} + 3 \, a c^{2} d + 24 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} n^{3} + 26 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} n^{2} + 9 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} n\right )} x x^{n} + {\left (24 \, a c^{3} n^{4} + 50 \, a c^{3} n^{3} + 35 \, a c^{3} n^{2} + 10 \, a c^{3} n + a c^{3}\right )} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, 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. 1540 vs.
\(2 (90) = 180\).
time = 1.21, size = 1540, normalized size = 15.56 \begin {gather*} \begin {cases} a c^{3} x + 3 a c^{2} d \log {\left (x \right )} - \frac {3 a c d^{2}}{x} - \frac {a d^{3}}{2 x^{2}} + b c^{3} \log {\left (x \right )} - \frac {3 b c^{2} d}{x} - \frac {3 b c d^{2}}{2 x^{2}} - \frac {b d^{3}}{3 x^{3}} & \text {for}\: n = -1 \\a c^{3} x + 6 a c^{2} d \sqrt {x} + 3 a c d^{2} \log {\left (x \right )} - \frac {2 a d^{3}}{\sqrt {x}} + 2 b c^{3} \sqrt {x} + 3 b c^{2} d \log {\left (x \right )} - \frac {6 b c d^{2}}{\sqrt {x}} - \frac {b d^{3}}{x} & \text {for}\: n = - \frac {1}{2} \\a c^{3} x + \frac {9 a c^{2} d x^{\frac {2}{3}}}{2} + 9 a c d^{2} \sqrt [3]{x} + a d^{3} \log {\left (x \right )} + \frac {3 b c^{3} x^{\frac {2}{3}}}{2} + 9 b c^{2} d \sqrt [3]{x} + 3 b c d^{2} \log {\left (x \right )} - \frac {3 b d^{3}}{\sqrt [3]{x}} & \text {for}\: n = - \frac {1}{3} \\a c^{3} x + 4 a c^{2} d x^{\frac {3}{4}} + 6 a c d^{2} \sqrt {x} + 4 a d^{3} \sqrt [4]{x} + \frac {4 b c^{3} x^{\frac {3}{4}}}{3} + 6 b c^{2} d \sqrt {x} + 12 b c d^{2} \sqrt [4]{x} + b d^{3} \log {\left (x \right )} & \text {for}\: n = - \frac {1}{4} \\\frac {24 a c^{3} n^{4} x}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {50 a c^{3} n^{3} x}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {35 a c^{3} n^{2} x}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {10 a c^{3} n x}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {a c^{3} x}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {72 a c^{2} d n^{3} x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {78 a c^{2} d n^{2} x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {27 a c^{2} d n x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {3 a c^{2} d x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {36 a c d^{2} n^{3} x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {57 a c d^{2} n^{2} x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {24 a c d^{2} n x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {3 a c d^{2} x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {8 a d^{3} n^{3} x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {14 a d^{3} n^{2} x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {7 a d^{3} n x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {a d^{3} x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {24 b c^{3} n^{3} x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {26 b c^{3} n^{2} x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {9 b c^{3} n x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {b c^{3} x x^{n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {36 b c^{2} d n^{3} x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {57 b c^{2} d n^{2} x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {24 b c^{2} d n x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {3 b c^{2} d x x^{2 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {24 b c d^{2} n^{3} x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {42 b c d^{2} n^{2} x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {21 b c d^{2} n x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {3 b c d^{2} x x^{3 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {6 b d^{3} n^{3} x x^{4 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {11 b d^{3} n^{2} x x^{4 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {6 b d^{3} n x x^{4 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 n + 1} + \frac {b d^{3} x x^{4 n}}{24 n^{4} + 50 n^{3} + 35 n^{2} + 10 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. 450 vs.
\(2 (99) = 198\).
time = 0.91, size = 450, normalized size = 4.55 \begin {gather*} \frac {24 \, a c^{3} n^{4} x + 6 \, b d^{3} n^{3} x x^{4 \, n} + 24 \, b c d^{2} n^{3} x x^{3 \, n} + 8 \, a d^{3} n^{3} x x^{3 \, n} + 36 \, b c^{2} d n^{3} x x^{2 \, n} + 36 \, a c d^{2} n^{3} x x^{2 \, n} + 24 \, b c^{3} n^{3} x x^{n} + 72 \, a c^{2} d n^{3} x x^{n} + 50 \, a c^{3} n^{3} x + 11 \, b d^{3} n^{2} x x^{4 \, n} + 42 \, b c d^{2} n^{2} x x^{3 \, n} + 14 \, a d^{3} n^{2} x x^{3 \, n} + 57 \, b c^{2} d n^{2} x x^{2 \, n} + 57 \, a c d^{2} n^{2} x x^{2 \, n} + 26 \, b c^{3} n^{2} x x^{n} + 78 \, a c^{2} d n^{2} x x^{n} + 35 \, a c^{3} n^{2} x + 6 \, b d^{3} n x x^{4 \, n} + 21 \, b c d^{2} n x x^{3 \, n} + 7 \, a d^{3} n x x^{3 \, n} + 24 \, b c^{2} d n x x^{2 \, n} + 24 \, a c d^{2} n x x^{2 \, n} + 9 \, b c^{3} n x x^{n} + 27 \, a c^{2} d n x x^{n} + 10 \, a c^{3} n x + b d^{3} x x^{4 \, n} + 3 \, b c d^{2} x x^{3 \, n} + a d^{3} x x^{3 \, n} + 3 \, b c^{2} d x x^{2 \, n} + 3 \, a c d^{2} x x^{2 \, n} + b c^{3} x x^{n} + 3 \, a c^{2} d x x^{n} + a c^{3} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.56, size = 99, normalized size = 1.00 \begin {gather*} a\,c^3\,x+\frac {x\,x^n\,\left (b\,c^3+3\,a\,d\,c^2\right )}{n+1}+\frac {x\,x^{3\,n}\,\left (a\,d^3+3\,b\,c\,d^2\right )}{3\,n+1}+\frac {b\,d^3\,x\,x^{4\,n}}{4\,n+1}+\frac {3\,c\,d\,x\,x^{2\,n}\,\left (a\,d+b\,c\right )}{2\,n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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