Integrand size = 13, antiderivative size = 65 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^4}{4}+\frac {3 a b^2 x^{2 (2+n)}}{2 (2+n)}+\frac {3 a^2 b x^{4+n}}{4+n}+\frac {b^3 x^{4+3 n}}{4+3 n} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^4}{4}+\frac {3 a^2 b x^{n+4}}{n+4}+\frac {3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac {b^3 x^{3 n+4}}{3 n+4} \]
[In]
[Out]
Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^3+b^3 x^{3 (1+n)}+3 a^2 b x^{3+n}+3 a b^2 x^{3+2 n}\right ) \, dx \\ & = \frac {a^3 x^4}{4}+\frac {3 a b^2 x^{2 (2+n)}}{2 (2+n)}+\frac {3 a^2 b x^{4+n}}{4+n}+\frac {b^3 x^{4+3 n}}{4+3 n} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {1}{4} x^4 \left (a^3+\frac {12 a^2 b x^n}{4+n}+\frac {6 a b^2 x^{2 n}}{2+n}+\frac {4 b^3 x^{3 n}}{4+3 n}\right ) \]
[In]
[Out]
Time = 3.70 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {a^{3} x^{4}}{4}+\frac {b^{3} x^{4} x^{3 n}}{4+3 n}+\frac {3 a \,b^{2} x^{4} x^{2 n}}{2 \left (2+n \right )}+\frac {3 a^{2} b \,x^{4} x^{n}}{4+n}\) | \(65\) |
parallelrisch | \(\frac {4 x^{4} x^{3 n} b^{3} n^{2}+24 x^{4} x^{3 n} b^{3} n +18 x^{4} x^{2 n} a \,b^{2} n^{2}+32 b^{3} x^{4} x^{3 n}+96 x^{4} x^{2 n} a \,b^{2} n +36 x^{4} x^{n} a^{2} b \,n^{2}+3 x^{4} a^{3} n^{3}+96 a \,b^{2} x^{4} x^{2 n}+120 x^{4} x^{n} a^{2} b n +22 x^{4} a^{3} n^{2}+96 x^{4} x^{n} a^{2} b +48 x^{4} a^{3} n +32 a^{3} x^{4}}{4 \left (4+3 n \right ) \left (2+n \right ) \left (4+n \right )}\) | \(189\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (61) = 122\).
Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.23 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {4 \, {\left (b^{3} n^{2} + 6 \, b^{3} n + 8 \, b^{3}\right )} x^{4} x^{3 \, n} + 6 \, {\left (3 \, a b^{2} n^{2} + 16 \, a b^{2} n + 16 \, a b^{2}\right )} x^{4} x^{2 \, n} + 12 \, {\left (3 \, a^{2} b n^{2} + 10 \, a^{2} b n + 8 \, a^{2} b\right )} x^{4} x^{n} + {\left (3 \, a^{3} n^{3} + 22 \, a^{3} n^{2} + 48 \, a^{3} n + 32 \, a^{3}\right )} x^{4}}{4 \, {\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (56) = 112\).
Time = 1.02 (sec) , antiderivative size = 507, normalized size of antiderivative = 7.80 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\begin {cases} \frac {a^{3} x^{4}}{4} + 3 a^{2} b \log {\left (x \right )} - \frac {3 a b^{2}}{4 x^{4}} - \frac {b^{3}}{8 x^{8}} & \text {for}\: n = -4 \\\frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{2}}{2} + 3 a b^{2} \log {\left (x \right )} - \frac {b^{3}}{2 x^{2}} & \text {for}\: n = -2 \\\frac {a^{3} x^{4}}{4} + \frac {9 a^{2} b x^{\frac {8}{3}}}{8} + \frac {9 a b^{2} x^{\frac {4}{3}}}{4} + b^{3} \log {\left (x \right )} & \text {for}\: n = - \frac {4}{3} \\\frac {3 a^{3} n^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {22 a^{3} n^{2} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {48 a^{3} n x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {32 a^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {36 a^{2} b n^{2} x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {120 a^{2} b n x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {96 a^{2} b x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {18 a b^{2} n^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {96 a b^{2} n x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {96 a b^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {4 b^{3} n^{2} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {24 b^{3} n x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {32 b^{3} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {1}{4} \, a^{3} x^{4} + \frac {b^{3} x^{3 \, n + 4}}{3 \, n + 4} + \frac {3 \, a b^{2} x^{2 \, n + 4}}{2 \, {\left (n + 2\right )}} + \frac {3 \, a^{2} b x^{n + 4}}{n + 4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (61) = 122\).
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.89 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {4 \, b^{3} n^{2} x^{4} x^{3 \, n} + 18 \, a b^{2} n^{2} x^{4} x^{2 \, n} + 36 \, a^{2} b n^{2} x^{4} x^{n} + 3 \, a^{3} n^{3} x^{4} + 24 \, b^{3} n x^{4} x^{3 \, n} + 96 \, a b^{2} n x^{4} x^{2 \, n} + 120 \, a^{2} b n x^{4} x^{n} + 22 \, a^{3} n^{2} x^{4} + 32 \, b^{3} x^{4} x^{3 \, n} + 96 \, a b^{2} x^{4} x^{2 \, n} + 96 \, a^{2} b x^{4} x^{n} + 48 \, a^{3} n x^{4} + 32 \, a^{3} x^{4}}{4 \, {\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \]
[In]
[Out]
Time = 6.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int x^3 \left (a+b x^n\right )^3 \, dx=\frac {a^3\,x^4}{4}+\frac {b^3\,x^{3\,n}\,x^4}{3\,n+4}+\frac {3\,a\,b^2\,x^{2\,n}\,x^4}{2\,n+4}+\frac {3\,a^2\,b\,x^n\,x^4}{n+4} \]
[In]
[Out]