Optimal. Leaf size=84 \[ a^3 c x+\frac {3 a^2 b c x^{1+n}}{1+n}+\frac {3 a b^2 c x^{1+2 n}}{1+2 n}+\frac {b^3 c x^{1+3 n}}{1+3 n}+\frac {d \left (a+b x^n\right )^4}{4 b n} \]
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Rubi [A]
time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1905, 250, 267}
\begin {gather*} a^3 c x+\frac {3 a^2 b c x^{n+1}}{n+1}+\frac {3 a b^2 c x^{2 n+1}}{2 n+1}+\frac {d \left (a+b x^n\right )^4}{4 b n}+\frac {b^3 c x^{3 n+1}}{3 n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 250
Rule 267
Rule 1905
Rubi steps
\begin {align*} \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right )^3 \, dx &=c \int \left (a+b x^n\right )^3 \, dx+d \int x^{-1+n} \left (a+b x^n\right )^3 \, dx\\ &=\frac {d \left (a+b x^n\right )^4}{4 b n}+c \int \left (a^3+3 a^2 b x^n+3 a b^2 x^{2 n}+b^3 x^{3 n}\right ) \, dx\\ &=a^3 c x+\frac {3 a^2 b c x^{1+n}}{1+n}+\frac {3 a b^2 c x^{1+2 n}}{1+2 n}+\frac {b^3 c x^{1+3 n}}{1+3 n}+\frac {d \left (a+b x^n\right )^4}{4 b n}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 162, normalized size = 1.93 \begin {gather*} \frac {4 a^3 \left (1+6 n+11 n^2+6 n^3\right ) \left (c n x+d x^n\right )+6 a^2 b \left (1+5 n+6 n^2\right ) x^n \left (2 c n x+d (1+n) x^n\right )+4 a b^2 \left (1+4 n+3 n^2\right ) x^{2 n} \left (3 c n x+d (1+2 n) x^n\right )+b^3 \left (1+3 n+2 n^2\right ) x^{3 n} \left (4 c n x+d (1+3 n) x^n\right )}{4 n (1+n) (1+2 n) (1+3 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 118, normalized size = 1.40
method | result | size |
risch | \(a^{3} c x +\frac {b^{3} d \,x^{4 n}}{4 n}+\frac {b^{2} \left (n b c x +3 a d n +a d \right ) x^{3 n}}{n \left (1+3 n \right )}+\frac {3 a b \left (2 n b c x +2 a d n +a d \right ) x^{2 n}}{2 n \left (1+2 n \right )}+\frac {a^{2} \left (3 n b c x +a d n +a d \right ) x^{n}}{n \left (1+n \right )}\) | \(118\) |
norman | \(a^{3} c x +\frac {a^{3} d \,{\mathrm e}^{n \ln \left (x \right )}}{n}+\frac {a \,b^{2} d \,{\mathrm e}^{3 n \ln \left (x \right )}}{n}+\frac {b^{3} c x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {b^{3} d \,{\mathrm e}^{4 n \ln \left (x \right )}}{4 n}+\frac {3 d \,a^{2} b \,{\mathrm e}^{2 n \ln \left (x \right )}}{2 n}+\frac {3 a c \,b^{2} x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {3 c \,a^{2} b x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 118, normalized size = 1.40 \begin {gather*} a^{3} c x + \frac {b^{3} d x^{4 \, n}}{4 \, n} + \frac {a b^{2} d x^{3 \, n}}{n} + \frac {3 \, a^{2} b d x^{2 \, n}}{2 \, n} + \frac {b^{3} c x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, a b^{2} c x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} b c x^{n + 1}}{n + 1} + \frac {a^{3} d x^{n}}{n} \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. 305 vs.
\(2 (82) = 164\).
time = 0.38, size = 305, normalized size = 3.63 \begin {gather*} \frac {4 \, {\left (6 \, a^{3} c n^{4} + 11 \, a^{3} c n^{3} + 6 \, a^{3} c n^{2} + a^{3} c n\right )} x + {\left (6 \, b^{3} d n^{3} + 11 \, b^{3} d n^{2} + 6 \, b^{3} d n + b^{3} d\right )} x^{4 \, n} + 4 \, {\left (6 \, a b^{2} d n^{3} + 11 \, a b^{2} d n^{2} + 6 \, a b^{2} d n + a b^{2} d + {\left (2 \, b^{3} c n^{3} + 3 \, b^{3} c n^{2} + b^{3} c n\right )} x\right )} x^{3 \, n} + 6 \, {\left (6 \, a^{2} b d n^{3} + 11 \, a^{2} b d n^{2} + 6 \, a^{2} b d n + a^{2} b d + 2 \, {\left (3 \, a b^{2} c n^{3} + 4 \, a b^{2} c n^{2} + a b^{2} c n\right )} x\right )} x^{2 \, n} + 4 \, {\left (6 \, a^{3} d n^{3} + 11 \, a^{3} d n^{2} + 6 \, a^{3} d n + a^{3} d + 3 \, {\left (6 \, a^{2} b c n^{3} + 5 \, a^{2} b c n^{2} + a^{2} b c n\right )} x\right )} x^{n}}{4 \, {\left (6 \, n^{4} + 11 \, n^{3} + 6 \, n^{2} + n\right )}} \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. 1251 vs.
\(2 (75) = 150\).
time = 1.00, size = 1251, normalized size = 14.89 \begin {gather*} \begin {cases} a^{3} c x - \frac {a^{3} d}{x} + 3 a^{2} b c \log {\left (x \right )} - \frac {3 a^{2} b d}{2 x^{2}} - \frac {3 a b^{2} c}{x} - \frac {a b^{2} d}{x^{3}} - \frac {b^{3} c}{2 x^{2}} - \frac {b^{3} d}{4 x^{4}} & \text {for}\: n = -1 \\a^{3} c x - \frac {2 a^{3} d}{\sqrt {x}} + 6 a^{2} b c \sqrt {x} - \frac {3 a^{2} b d}{x} + 3 a b^{2} c \log {\left (x \right )} - \frac {2 a b^{2} d}{x^{\frac {3}{2}}} - \frac {2 b^{3} c}{\sqrt {x}} - \frac {b^{3} d}{2 x^{2}} & \text {for}\: n = - \frac {1}{2} \\a^{3} c x - \frac {3 a^{3} d}{\sqrt [3]{x}} + \frac {9 a^{2} b c x^{\frac {2}{3}}}{2} - \frac {9 a^{2} b d}{2 x^{\frac {2}{3}}} + 9 a b^{2} c \sqrt [3]{x} - \frac {3 a b^{2} d}{x} + b^{3} c \log {\left (x \right )} - \frac {3 b^{3} d}{4 x^{\frac {4}{3}}} & \text {for}\: n = - \frac {1}{3} \\\left (a + b\right )^{3} \left (c x + d \log {\left (x \right )}\right ) & \text {for}\: n = 0 \\\frac {24 a^{3} c n^{4} x}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {44 a^{3} c n^{3} x}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {24 a^{3} c n^{2} x}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {4 a^{3} c n x}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {24 a^{3} d n^{3} x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {44 a^{3} d n^{2} x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {24 a^{3} d n x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {4 a^{3} d x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {72 a^{2} b c n^{3} x x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {60 a^{2} b c n^{2} x x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {12 a^{2} b c n x x^{n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {36 a^{2} b d n^{3} x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {66 a^{2} b d n^{2} x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {36 a^{2} b d n x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {6 a^{2} b d x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {36 a b^{2} c n^{3} x x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {48 a b^{2} c n^{2} x x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {12 a b^{2} c n x x^{2 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {24 a b^{2} d n^{3} x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {44 a b^{2} d n^{2} x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {24 a b^{2} d n x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {4 a b^{2} d x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {8 b^{3} c n^{3} x x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {12 b^{3} c n^{2} x x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {4 b^{3} c n x x^{3 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {6 b^{3} d n^{3} x^{4 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {11 b^{3} d n^{2} x^{4 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {6 b^{3} d n x^{4 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} + \frac {b^{3} d x^{4 n}}{24 n^{4} + 44 n^{3} + 24 n^{2} + 4 n} & \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. 392 vs.
\(2 (82) = 164\).
time = 0.82, size = 392, normalized size = 4.67 \begin {gather*} \frac {24 \, a^{3} c n^{4} x + 8 \, b^{3} c n^{3} x x^{3 \, n} + 36 \, a b^{2} c n^{3} x x^{2 \, n} + 72 \, a^{2} b c n^{3} x x^{n} + 44 \, a^{3} c n^{3} x + 6 \, b^{3} d n^{3} x^{4 \, n} + 24 \, a b^{2} d n^{3} x^{3 \, n} + 12 \, b^{3} c n^{2} x x^{3 \, n} + 36 \, a^{2} b d n^{3} x^{2 \, n} + 48 \, a b^{2} c n^{2} x x^{2 \, n} + 24 \, a^{3} d n^{3} x^{n} + 60 \, a^{2} b c n^{2} x x^{n} + 24 \, a^{3} c n^{2} x + 11 \, b^{3} d n^{2} x^{4 \, n} + 44 \, a b^{2} d n^{2} x^{3 \, n} + 4 \, b^{3} c n x x^{3 \, n} + 66 \, a^{2} b d n^{2} x^{2 \, n} + 12 \, a b^{2} c n x x^{2 \, n} + 44 \, a^{3} d n^{2} x^{n} + 12 \, a^{2} b c n x x^{n} + 4 \, a^{3} c n x + 6 \, b^{3} d n x^{4 \, n} + 24 \, a b^{2} d n x^{3 \, n} + 36 \, a^{2} b d n x^{2 \, n} + 24 \, a^{3} d n x^{n} + b^{3} d x^{4 \, n} + 4 \, a b^{2} d x^{3 \, n} + 6 \, a^{2} b d x^{2 \, n} + 4 \, a^{3} d x^{n}}{4 \, {\left (6 \, n^{4} + 11 \, n^{3} + 6 \, n^{2} + n\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.13, size = 115, normalized size = 1.37 \begin {gather*} a^3\,c\,x+\frac {a^3\,d\,x^n}{n}+\frac {b^3\,d\,x^{4\,n}}{4\,n}+\frac {b^3\,c\,x\,x^{3\,n}}{3\,n+1}+\frac {3\,a^2\,b\,d\,x^{2\,n}}{2\,n}+\frac {a\,b^2\,d\,x^{3\,n}}{n}+\frac {3\,a\,b^2\,c\,x\,x^{2\,n}}{2\,n+1}+\frac {3\,a^2\,b\,c\,x\,x^n}{n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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