Integrand size = 17, antiderivative size = 70 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=a c^2 x+\frac {c (b c+2 a d) x^{1+n}}{1+n}+\frac {d (2 b c+a d) x^{1+2 n}}{1+2 n}+\frac {b d^2 x^{1+3 n}}{1+3 n} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {380} \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {c x^{n+1} (2 a d+b c)}{n+1}+\frac {d x^{2 n+1} (a d+2 b c)}{2 n+1}+a c^2 x+\frac {b d^2 x^{3 n+1}}{3 n+1} \]
[In]
[Out]
Rule 380
Rubi steps \begin{align*} \text {integral}& = \int \left (a c^2+c (b c+2 a d) x^n+d (2 b c+a d) x^{2 n}+b d^2 x^{3 n}\right ) \, dx \\ & = a c^2 x+\frac {c (b c+2 a d) x^{1+n}}{1+n}+\frac {d (2 b c+a d) x^{1+2 n}}{1+2 n}+\frac {b d^2 x^{1+3 n}}{1+3 n} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {b x \left (c+d x^n\right )^3-(b c-a d (1+3 n)) x \left (c^2+\frac {2 c d x^n}{1+n}+\frac {d^2 x^{2 n}}{1+2 n}\right )}{d+3 d n} \]
[In]
[Out]
Time = 3.93 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97
method | result | size |
risch | \(a \,c^{2} x +\frac {b \,d^{2} x \,x^{3 n}}{1+3 n}+\frac {c \left (2 a d +b c \right ) x \,x^{n}}{1+n}+\frac {d \left (a d +2 b c \right ) x \,x^{2 n}}{1+2 n}\) | \(68\) |
norman | \(a \,c^{2} x +\frac {b \,d^{2} x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {c \left (2 a d +b c \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {d \left (a d +2 b c \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}\) | \(74\) |
parallelrisch | \(\frac {2 x \,x^{3 n} b \,d^{2} n^{2}+3 x \,x^{3 n} b \,d^{2} n +3 x \,x^{2 n} a \,d^{2} n^{2}+6 x \,x^{2 n} b c d \,n^{2}+b \,d^{2} x \,x^{3 n}+4 x \,x^{2 n} a \,d^{2} n +8 x \,x^{2 n} b c d n +12 x \,x^{n} a c d \,n^{2}+6 x \,x^{n} b \,c^{2} n^{2}+6 x a \,c^{2} n^{3}+x \,x^{2 n} a \,d^{2}+2 x \,x^{2 n} b c d +10 x \,x^{n} a c d n +5 x \,x^{n} b \,c^{2} n +11 x a \,c^{2} n^{2}+2 x \,x^{n} a c d +x \,x^{n} b \,c^{2}+6 x a \,c^{2} n +a \,c^{2} x}{\left (1+3 n \right ) \left (1+n \right ) \left (1+2 n \right )}\) | \(235\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (70) = 140\).
Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.50 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {{\left (2 \, b d^{2} n^{2} + 3 \, b d^{2} n + b d^{2}\right )} x x^{3 \, n} + {\left (2 \, b c d + a d^{2} + 3 \, {\left (2 \, b c d + a d^{2}\right )} n^{2} + 4 \, {\left (2 \, b c d + a d^{2}\right )} n\right )} x x^{2 \, n} + {\left (b c^{2} + 2 \, a c d + 6 \, {\left (b c^{2} + 2 \, a c d\right )} n^{2} + 5 \, {\left (b c^{2} + 2 \, a c d\right )} n\right )} x x^{n} + {\left (6 \, a c^{2} n^{3} + 11 \, a c^{2} n^{2} + 6 \, a c^{2} n + a c^{2}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (63) = 126\).
Time = 0.36 (sec) , antiderivative size = 726, normalized size of antiderivative = 10.37 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=\begin {cases} a c^{2} x + 2 a c d \log {\left (x \right )} - \frac {a d^{2}}{x} + b c^{2} \log {\left (x \right )} - \frac {2 b c d}{x} - \frac {b d^{2}}{2 x^{2}} & \text {for}\: n = -1 \\a c^{2} x + 4 a c d \sqrt {x} + a d^{2} \log {\left (x \right )} + 2 b c^{2} \sqrt {x} + 2 b c d \log {\left (x \right )} - \frac {2 b d^{2}}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\a c^{2} x + 3 a c d x^{\frac {2}{3}} + 3 a d^{2} \sqrt [3]{x} + \frac {3 b c^{2} x^{\frac {2}{3}}}{2} + 6 b c d \sqrt [3]{x} + b d^{2} \log {\left (x \right )} & \text {for}\: n = - \frac {1}{3} \\\frac {6 a c^{2} n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {11 a c^{2} n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 a c^{2} n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a c^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {12 a c d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {10 a c d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 a c d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 a d^{2} n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {4 a d^{2} n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {a d^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 b c^{2} n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {5 b c^{2} n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b c^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {6 b c d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {8 b c d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 b c d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {2 b d^{2} n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {3 b d^{2} n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac {b d^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=a c^{2} x + \frac {b d^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {2 \, b c d x^{2 \, n + 1}}{2 \, n + 1} + \frac {a d^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c^{2} x^{n + 1}}{n + 1} + \frac {2 \, a c d x^{n + 1}}{n + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (70) = 140\).
Time = 0.28 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.31 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {6 \, a c^{2} n^{3} x + 2 \, b d^{2} n^{2} x x^{3 \, n} + 6 \, b c d n^{2} x x^{2 \, n} + 3 \, a d^{2} n^{2} x x^{2 \, n} + 6 \, b c^{2} n^{2} x x^{n} + 12 \, a c d n^{2} x x^{n} + 11 \, a c^{2} n^{2} x + 3 \, b d^{2} n x x^{3 \, n} + 8 \, b c d n x x^{2 \, n} + 4 \, a d^{2} n x x^{2 \, n} + 5 \, b c^{2} n x x^{n} + 10 \, a c d n x x^{n} + 6 \, a c^{2} n x + b d^{2} x x^{3 \, n} + 2 \, b c d x x^{2 \, n} + a d^{2} x x^{2 \, n} + b c^{2} x x^{n} + 2 \, a c d x x^{n} + a c^{2} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
[In]
[Out]
Time = 5.61 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^2 \, dx=a\,c^2\,x+\frac {x\,x^{2\,n}\,\left (a\,d^2+2\,b\,c\,d\right )}{2\,n+1}+\frac {x\,x^n\,\left (b\,c^2+2\,a\,d\,c\right )}{n+1}+\frac {b\,d^2\,x\,x^{3\,n}}{3\,n+1} \]
[In]
[Out]