Integrand size = 17, antiderivative size = 70 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=a c^3 x+\frac {1}{4} c^2 (b c+3 a d) x^4+\frac {3}{7} c d (b c+a d) x^7+\frac {1}{10} d^2 (3 b c+a d) x^{10}+\frac {1}{13} b d^3 x^{13} \]
[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} \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=\frac {1}{4} c^2 x^4 (3 a d+b c)+\frac {1}{10} d^2 x^{10} (a d+3 b c)+\frac {3}{7} c d x^7 (a d+b c)+a c^3 x+\frac {1}{13} b d^3 x^{13} \]
[In]
[Out]
Rule 380
Rubi steps \begin{align*} \text {integral}& = \int \left (a c^3+c^2 (b c+3 a d) x^3+3 c d (b c+a d) x^6+d^2 (3 b c+a d) x^9+b d^3 x^{12}\right ) \, dx \\ & = a c^3 x+\frac {1}{4} c^2 (b c+3 a d) x^4+\frac {3}{7} c d (b c+a d) x^7+\frac {1}{10} d^2 (3 b c+a d) x^{10}+\frac {1}{13} b d^3 x^{13} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=a c^3 x+\frac {1}{4} c^2 (b c+3 a d) x^4+\frac {3}{7} c d (b c+a d) x^7+\frac {1}{10} d^2 (3 b c+a d) x^{10}+\frac {1}{13} b d^3 x^{13} \]
[In]
[Out]
Time = 4.00 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {b \,d^{3} x^{13}}{13}+\left (\frac {1}{10} a \,d^{3}+\frac {3}{10} b c \,d^{2}\right ) x^{10}+\left (\frac {3}{7} a c \,d^{2}+\frac {3}{7} b \,c^{2} d \right ) x^{7}+\left (\frac {3}{4} a \,c^{2} d +\frac {1}{4} c^{3} b \right ) x^{4}+a \,c^{3} x\) | \(72\) |
default | \(\frac {b \,d^{3} x^{13}}{13}+\frac {\left (a \,d^{3}+3 b c \,d^{2}\right ) x^{10}}{10}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) x^{7}}{7}+\frac {\left (3 a \,c^{2} d +c^{3} b \right ) x^{4}}{4}+a \,c^{3} x\) | \(73\) |
gosper | \(\frac {1}{13} b \,d^{3} x^{13}+\frac {1}{10} x^{10} a \,d^{3}+\frac {3}{10} x^{10} b c \,d^{2}+\frac {3}{7} x^{7} a c \,d^{2}+\frac {3}{7} x^{7} b \,c^{2} d +\frac {3}{4} x^{4} a \,c^{2} d +\frac {1}{4} x^{4} c^{3} b +a \,c^{3} x\) | \(75\) |
risch | \(\frac {1}{13} b \,d^{3} x^{13}+\frac {1}{10} x^{10} a \,d^{3}+\frac {3}{10} x^{10} b c \,d^{2}+\frac {3}{7} x^{7} a c \,d^{2}+\frac {3}{7} x^{7} b \,c^{2} d +\frac {3}{4} x^{4} a \,c^{2} d +\frac {1}{4} x^{4} c^{3} b +a \,c^{3} x\) | \(75\) |
parallelrisch | \(\frac {1}{13} b \,d^{3} x^{13}+\frac {1}{10} x^{10} a \,d^{3}+\frac {3}{10} x^{10} b c \,d^{2}+\frac {3}{7} x^{7} a c \,d^{2}+\frac {3}{7} x^{7} b \,c^{2} d +\frac {3}{4} x^{4} a \,c^{2} d +\frac {1}{4} x^{4} c^{3} b +a \,c^{3} x\) | \(75\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=\frac {1}{13} \, b d^{3} x^{13} + \frac {1}{10} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{10} + \frac {3}{7} \, {\left (b c^{2} d + a c d^{2}\right )} x^{7} + a c^{3} x + \frac {1}{4} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{4} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.14 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=a c^{3} x + \frac {b d^{3} x^{13}}{13} + x^{10} \left (\frac {a d^{3}}{10} + \frac {3 b c d^{2}}{10}\right ) + x^{7} \cdot \left (\frac {3 a c d^{2}}{7} + \frac {3 b c^{2} d}{7}\right ) + x^{4} \cdot \left (\frac {3 a c^{2} d}{4} + \frac {b c^{3}}{4}\right ) \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=\frac {1}{13} \, b d^{3} x^{13} + \frac {1}{10} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{10} + \frac {3}{7} \, {\left (b c^{2} d + a c d^{2}\right )} x^{7} + a c^{3} x + \frac {1}{4} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{4} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=\frac {1}{13} \, b d^{3} x^{13} + \frac {3}{10} \, b c d^{2} x^{10} + \frac {1}{10} \, a d^{3} x^{10} + \frac {3}{7} \, b c^{2} d x^{7} + \frac {3}{7} \, a c d^{2} x^{7} + \frac {1}{4} \, b c^{3} x^{4} + \frac {3}{4} \, a c^{2} d x^{4} + a c^{3} x \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, dx=x^4\,\left (\frac {b\,c^3}{4}+\frac {3\,a\,d\,c^2}{4}\right )+x^{10}\,\left (\frac {a\,d^3}{10}+\frac {3\,b\,c\,d^2}{10}\right )+\frac {b\,d^3\,x^{13}}{13}+a\,c^3\,x+\frac {3\,c\,d\,x^7\,\left (a\,d+b\,c\right )}{7} \]
[In]
[Out]