Optimal. Leaf size=132 \[ a c^4 x+\frac {c^3 (b c+4 a d) x^{1+n}}{1+n}+\frac {2 c^2 d (2 b c+3 a d) x^{1+2 n}}{1+2 n}+\frac {2 c d^2 (3 b c+2 a d) x^{1+3 n}}{1+3 n}+\frac {d^3 (4 b c+a d) x^{1+4 n}}{1+4 n}+\frac {b d^4 x^{1+5 n}}{1+5 n} \]
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Rubi [A]
time = 0.08, antiderivative size = 132, 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^3 x^{n+1} (4 a d+b c)}{n+1}+\frac {2 c^2 d x^{2 n+1} (3 a d+2 b c)}{2 n+1}+\frac {d^3 x^{4 n+1} (a d+4 b c)}{4 n+1}+\frac {2 c d^2 x^{3 n+1} (2 a d+3 b c)}{3 n+1}+a c^4 x+\frac {b d^4 x^{5 n+1}}{5 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 )^4 \, dx &=\int \left (a c^4+c^3 (b c+4 a d) x^n+2 c^2 d (2 b c+3 a d) x^{2 n}+2 c d^2 (3 b c+2 a d) x^{3 n}+d^3 (4 b c+a d) x^{4 n}+b d^4 x^{5 n}\right ) \, dx\\ &=a c^4 x+\frac {c^3 (b c+4 a d) x^{1+n}}{1+n}+\frac {2 c^2 d (2 b c+3 a d) x^{1+2 n}}{1+2 n}+\frac {2 c d^2 (3 b c+2 a d) x^{1+3 n}}{1+3 n}+\frac {d^3 (4 b c+a d) x^{1+4 n}}{1+4 n}+\frac {b d^4 x^{1+5 n}}{1+5 n}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 123, normalized size = 0.93 \begin {gather*} x \left (a c^4+\frac {c^3 (b c+4 a d) x^n}{1+n}+\frac {2 c^2 d (2 b c+3 a d) x^{2 n}}{1+2 n}+\frac {2 c d^2 (3 b c+2 a d) x^{3 n}}{1+3 n}+\frac {d^3 (4 b c+a d) x^{4 n}}{1+4 n}+\frac {b d^4 x^{5 n}}{1+5 n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 128, normalized size = 0.97
method | result | size |
risch | \(a \,c^{4} x +\frac {b \,d^{4} x \,x^{5 n}}{1+5 n}+\frac {c^{3} \left (4 a d +b c \right ) x \,x^{n}}{1+n}+\frac {d^{3} \left (a d +4 b c \right ) x \,x^{4 n}}{1+4 n}+\frac {2 c \,d^{2} \left (2 a d +3 b c \right ) x \,x^{3 n}}{1+3 n}+\frac {2 c^{2} d \left (3 a d +2 b c \right ) x \,x^{2 n}}{1+2 n}\) | \(128\) |
norman | \(a \,c^{4} x +\frac {b \,d^{4} x \,{\mathrm e}^{5 n \ln \left (x \right )}}{1+5 n}+\frac {c^{3} \left (4 a d +b c \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {d^{3} \left (a d +4 b c \right ) x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}+\frac {2 c \,d^{2} \left (2 a d +3 b c \right ) x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {2 c^{2} d \left (3 a d +2 b c \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 186, normalized size = 1.41 \begin {gather*} a c^{4} x + \frac {b d^{4} x^{5 \, n + 1}}{5 \, n + 1} + \frac {4 \, b c d^{3} x^{4 \, n + 1}}{4 \, n + 1} + \frac {a d^{4} x^{4 \, n + 1}}{4 \, n + 1} + \frac {6 \, b c^{2} d^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {4 \, a c d^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {4 \, b c^{3} d x^{2 \, n + 1}}{2 \, n + 1} + \frac {6 \, a c^{2} d^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c^{4} x^{n + 1}}{n + 1} + \frac {4 \, a c^{3} 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. 527 vs.
\(2 (132) = 264\).
time = 2.60, size = 527, normalized size = 3.99 \begin {gather*} \frac {{\left (24 \, b d^{4} n^{4} + 50 \, b d^{4} n^{3} + 35 \, b d^{4} n^{2} + 10 \, b d^{4} n + b d^{4}\right )} x x^{5 \, n} + {\left (4 \, b c d^{3} + a d^{4} + 30 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n^{4} + 61 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n^{3} + 41 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n^{2} + 11 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n\right )} x x^{4 \, n} + 2 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3} + 40 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n^{4} + 78 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n^{3} + 49 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n^{2} + 12 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n\right )} x x^{3 \, n} + 2 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2} + 60 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n^{4} + 107 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n^{3} + 59 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n^{2} + 13 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (b c^{4} + 4 \, a c^{3} d + 120 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n^{4} + 154 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n^{3} + 71 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n^{2} + 14 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n\right )} x x^{n} + {\left (120 \, a c^{4} n^{5} + 274 \, a c^{4} n^{4} + 225 \, a c^{4} n^{3} + 85 \, a c^{4} n^{2} + 15 \, a c^{4} n + a c^{4}\right )} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, 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. 2744 vs.
\(2 (124) = 248\).
time = 0.82, size = 2744, normalized size = 20.79 \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. 740 vs.
\(2 (132) = 264\).
time = 1.81, size = 740, normalized size = 5.61 \begin {gather*} \frac {120 \, a c^{4} n^{5} x + 24 \, b d^{4} n^{4} x x^{5 \, n} + 120 \, b c d^{3} n^{4} x x^{4 \, n} + 30 \, a d^{4} n^{4} x x^{4 \, n} + 240 \, b c^{2} d^{2} n^{4} x x^{3 \, n} + 160 \, a c d^{3} n^{4} x x^{3 \, n} + 240 \, b c^{3} d n^{4} x x^{2 \, n} + 360 \, a c^{2} d^{2} n^{4} x x^{2 \, n} + 120 \, b c^{4} n^{4} x x^{n} + 480 \, a c^{3} d n^{4} x x^{n} + 274 \, a c^{4} n^{4} x + 50 \, b d^{4} n^{3} x x^{5 \, n} + 244 \, b c d^{3} n^{3} x x^{4 \, n} + 61 \, a d^{4} n^{3} x x^{4 \, n} + 468 \, b c^{2} d^{2} n^{3} x x^{3 \, n} + 312 \, a c d^{3} n^{3} x x^{3 \, n} + 428 \, b c^{3} d n^{3} x x^{2 \, n} + 642 \, a c^{2} d^{2} n^{3} x x^{2 \, n} + 154 \, b c^{4} n^{3} x x^{n} + 616 \, a c^{3} d n^{3} x x^{n} + 225 \, a c^{4} n^{3} x + 35 \, b d^{4} n^{2} x x^{5 \, n} + 164 \, b c d^{3} n^{2} x x^{4 \, n} + 41 \, a d^{4} n^{2} x x^{4 \, n} + 294 \, b c^{2} d^{2} n^{2} x x^{3 \, n} + 196 \, a c d^{3} n^{2} x x^{3 \, n} + 236 \, b c^{3} d n^{2} x x^{2 \, n} + 354 \, a c^{2} d^{2} n^{2} x x^{2 \, n} + 71 \, b c^{4} n^{2} x x^{n} + 284 \, a c^{3} d n^{2} x x^{n} + 85 \, a c^{4} n^{2} x + 10 \, b d^{4} n x x^{5 \, n} + 44 \, b c d^{3} n x x^{4 \, n} + 11 \, a d^{4} n x x^{4 \, n} + 72 \, b c^{2} d^{2} n x x^{3 \, n} + 48 \, a c d^{3} n x x^{3 \, n} + 52 \, b c^{3} d n x x^{2 \, n} + 78 \, a c^{2} d^{2} n x x^{2 \, n} + 14 \, b c^{4} n x x^{n} + 56 \, a c^{3} d n x x^{n} + 15 \, a c^{4} n x + b d^{4} x x^{5 \, n} + 4 \, b c d^{3} x x^{4 \, n} + a d^{4} x x^{4 \, n} + 6 \, b c^{2} d^{2} x x^{3 \, n} + 4 \, a c d^{3} x x^{3 \, n} + 4 \, b c^{3} d x x^{2 \, n} + 6 \, a c^{2} d^{2} x x^{2 \, n} + b c^{4} x x^{n} + 4 \, a c^{3} d x x^{n} + a c^{4} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.64, size = 131, normalized size = 0.99 \begin {gather*} a\,c^4\,x+\frac {x\,x^n\,\left (b\,c^4+4\,a\,d\,c^3\right )}{n+1}+\frac {x\,x^{4\,n}\,\left (a\,d^4+4\,b\,c\,d^3\right )}{4\,n+1}+\frac {b\,d^4\,x\,x^{5\,n}}{5\,n+1}+\frac {2\,c^2\,d\,x\,x^{2\,n}\,\left (3\,a\,d+2\,b\,c\right )}{2\,n+1}+\frac {2\,c\,d^2\,x\,x^{3\,n}\,\left (2\,a\,d+3\,b\,c\right )}{3\,n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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