Integrand size = 17, antiderivative size = 209 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=a^4 d^3 x+\frac {1}{3} a^3 d \left (4 c d^2+3 a e^2\right ) x^3+\frac {1}{4} a^4 e^3 x^4+\frac {6}{5} a^2 c d \left (c d^2+2 a e^2\right ) x^5+\frac {2}{3} a^3 c e^3 x^6+\frac {2}{7} a c^2 d \left (2 c d^2+9 a e^2\right ) x^7+\frac {3}{4} a^2 c^2 e^3 x^8+\frac {1}{9} c^3 d \left (c d^2+12 a e^2\right ) x^9+\frac {2}{5} a c^3 e^3 x^{10}+\frac {3}{11} c^4 d e^2 x^{11}+\frac {1}{12} c^4 e^3 x^{12}+\frac {3 d^2 e \left (a+c x^2\right )^5}{10 c} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {710, 1824} \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=a^4 d^3 x+\frac {1}{4} a^4 e^3 x^4+\frac {1}{3} a^3 d x^3 \left (3 a e^2+4 c d^2\right )+\frac {2}{3} a^3 c e^3 x^6+\frac {3}{4} a^2 c^2 e^3 x^8+\frac {6}{5} a^2 c d x^5 \left (2 a e^2+c d^2\right )+\frac {1}{9} c^3 d x^9 \left (12 a e^2+c d^2\right )+\frac {2}{5} a c^3 e^3 x^{10}+\frac {2}{7} a c^2 d x^7 \left (9 a e^2+2 c d^2\right )+\frac {3 d^2 e \left (a+c x^2\right )^5}{10 c}+\frac {3}{11} c^4 d e^2 x^{11}+\frac {1}{12} c^4 e^3 x^{12} \]
[In]
[Out]
Rule 710
Rule 1824
Rubi steps \begin{align*} \text {integral}& = \frac {3 d^2 e \left (a+c x^2\right )^5}{10 c}+\int \left (a+c x^2\right )^4 \left (-3 d^2 e x+(d+e x)^3\right ) \, dx \\ & = \frac {3 d^2 e \left (a+c x^2\right )^5}{10 c}+\int \left (a^4 d^3+a^3 d \left (4 c d^2+3 a e^2\right ) x^2+a^4 e^3 x^3+6 a^2 c d \left (c d^2+2 a e^2\right ) x^4+4 a^3 c e^3 x^5+2 a c^2 d \left (2 c d^2+9 a e^2\right ) x^6+6 a^2 c^2 e^3 x^7+c^3 d \left (c d^2+12 a e^2\right ) x^8+4 a c^3 e^3 x^9+3 c^4 d e^2 x^{10}+c^4 e^3 x^{11}\right ) \, dx \\ & = a^4 d^3 x+\frac {1}{3} a^3 d \left (4 c d^2+3 a e^2\right ) x^3+\frac {1}{4} a^4 e^3 x^4+\frac {6}{5} a^2 c d \left (c d^2+2 a e^2\right ) x^5+\frac {2}{3} a^3 c e^3 x^6+\frac {2}{7} a c^2 d \left (2 c d^2+9 a e^2\right ) x^7+\frac {3}{4} a^2 c^2 e^3 x^8+\frac {1}{9} c^3 d \left (c d^2+12 a e^2\right ) x^9+\frac {2}{5} a c^3 e^3 x^{10}+\frac {3}{11} c^4 d e^2 x^{11}+\frac {1}{12} c^4 e^3 x^{12}+\frac {3 d^2 e \left (a+c x^2\right )^5}{10 c} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.94 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=\frac {x \left (3465 a^4 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+924 a^3 c x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+297 a^2 c^2 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+66 a c^3 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )+7 c^4 x^8 \left (220 d^3+594 d^2 e x+540 d e^2 x^2+165 e^3 x^3\right )\right )}{13860} \]
[In]
[Out]
Time = 2.18 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.15
method | result | size |
norman | \(\frac {c^{4} e^{3} x^{12}}{12}+\frac {3 c^{4} d \,e^{2} x^{11}}{11}+\left (\frac {2}{5} e^{3} c^{3} a +\frac {3}{10} d^{2} e \,c^{4}\right ) x^{10}+\left (\frac {4}{3} d \,e^{2} c^{3} a +\frac {1}{9} d^{3} c^{4}\right ) x^{9}+\left (\frac {3}{4} a^{2} c^{2} e^{3}+\frac {3}{2} d^{2} e \,c^{3} a \right ) x^{8}+\left (\frac {18}{7} a^{2} c^{2} d \,e^{2}+\frac {4}{7} d^{3} c^{3} a \right ) x^{7}+\left (\frac {2}{3} e^{3} c \,a^{3}+3 d^{2} e \,a^{2} c^{2}\right ) x^{6}+\left (\frac {12}{5} d \,e^{2} c \,a^{3}+\frac {6}{5} d^{3} a^{2} c^{2}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{4}+3 d^{2} e c \,a^{3}\right ) x^{4}+\left (d \,e^{2} a^{4}+\frac {4}{3} a^{3} c \,d^{3}\right ) x^{3}+\frac {3 d^{2} e \,a^{4} x^{2}}{2}+a^{4} d^{3} x\) | \(240\) |
default | \(\frac {c^{4} e^{3} x^{12}}{12}+\frac {3 c^{4} d \,e^{2} x^{11}}{11}+\frac {\left (4 e^{3} c^{3} a +3 d^{2} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (12 d \,e^{2} c^{3} a +d^{3} c^{4}\right ) x^{9}}{9}+\frac {\left (6 a^{2} c^{2} e^{3}+12 d^{2} e \,c^{3} a \right ) x^{8}}{8}+\frac {\left (18 a^{2} c^{2} d \,e^{2}+4 d^{3} c^{3} a \right ) x^{7}}{7}+\frac {\left (4 e^{3} c \,a^{3}+18 d^{2} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (12 d \,e^{2} c \,a^{3}+6 d^{3} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (e^{3} a^{4}+12 d^{2} e c \,a^{3}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{4}+4 a^{3} c \,d^{3}\right ) x^{3}}{3}+\frac {3 d^{2} e \,a^{4} x^{2}}{2}+a^{4} d^{3} x\) | \(247\) |
gosper | \(\frac {1}{12} c^{4} e^{3} x^{12}+\frac {3}{11} c^{4} d \,e^{2} x^{11}+\frac {2}{5} a \,c^{3} e^{3} x^{10}+\frac {3}{10} x^{10} d^{2} e \,c^{4}+\frac {4}{3} x^{9} d \,e^{2} c^{3} a +\frac {1}{9} x^{9} d^{3} c^{4}+\frac {3}{4} a^{2} c^{2} e^{3} x^{8}+\frac {3}{2} x^{8} d^{2} e \,c^{3} a +\frac {18}{7} x^{7} a^{2} c^{2} d \,e^{2}+\frac {4}{7} x^{7} d^{3} c^{3} a +\frac {2}{3} a^{3} c \,e^{3} x^{6}+3 x^{6} d^{2} e \,a^{2} c^{2}+\frac {12}{5} x^{5} d \,e^{2} c \,a^{3}+\frac {6}{5} x^{5} d^{3} a^{2} c^{2}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d^{2} e c \,a^{3}+x^{3} d \,e^{2} a^{4}+\frac {4}{3} a^{3} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+a^{4} d^{3} x\) | \(248\) |
risch | \(\frac {1}{12} c^{4} e^{3} x^{12}+\frac {3}{11} c^{4} d \,e^{2} x^{11}+\frac {2}{5} a \,c^{3} e^{3} x^{10}+\frac {3}{10} x^{10} d^{2} e \,c^{4}+\frac {4}{3} x^{9} d \,e^{2} c^{3} a +\frac {1}{9} x^{9} d^{3} c^{4}+\frac {3}{4} a^{2} c^{2} e^{3} x^{8}+\frac {3}{2} x^{8} d^{2} e \,c^{3} a +\frac {18}{7} x^{7} a^{2} c^{2} d \,e^{2}+\frac {4}{7} x^{7} d^{3} c^{3} a +\frac {2}{3} a^{3} c \,e^{3} x^{6}+3 x^{6} d^{2} e \,a^{2} c^{2}+\frac {12}{5} x^{5} d \,e^{2} c \,a^{3}+\frac {6}{5} x^{5} d^{3} a^{2} c^{2}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d^{2} e c \,a^{3}+x^{3} d \,e^{2} a^{4}+\frac {4}{3} a^{3} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+a^{4} d^{3} x\) | \(248\) |
parallelrisch | \(\frac {1}{12} c^{4} e^{3} x^{12}+\frac {3}{11} c^{4} d \,e^{2} x^{11}+\frac {2}{5} a \,c^{3} e^{3} x^{10}+\frac {3}{10} x^{10} d^{2} e \,c^{4}+\frac {4}{3} x^{9} d \,e^{2} c^{3} a +\frac {1}{9} x^{9} d^{3} c^{4}+\frac {3}{4} a^{2} c^{2} e^{3} x^{8}+\frac {3}{2} x^{8} d^{2} e \,c^{3} a +\frac {18}{7} x^{7} a^{2} c^{2} d \,e^{2}+\frac {4}{7} x^{7} d^{3} c^{3} a +\frac {2}{3} a^{3} c \,e^{3} x^{6}+3 x^{6} d^{2} e \,a^{2} c^{2}+\frac {12}{5} x^{5} d \,e^{2} c \,a^{3}+\frac {6}{5} x^{5} d^{3} a^{2} c^{2}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d^{2} e c \,a^{3}+x^{3} d \,e^{2} a^{4}+\frac {4}{3} a^{3} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+a^{4} d^{3} x\) | \(248\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=\frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {3}{11} \, c^{4} d e^{2} x^{11} + \frac {1}{10} \, {\left (3 \, c^{4} d^{2} e + 4 \, a c^{3} e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{3} + 12 \, a c^{3} d e^{2}\right )} x^{9} + \frac {3}{2} \, a^{4} d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, a c^{3} d^{2} e + a^{2} c^{2} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac {2}{7} \, {\left (2 \, a c^{3} d^{3} + 9 \, a^{2} c^{2} d e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (9 \, a^{2} c^{2} d^{2} e + 2 \, a^{3} c e^{3}\right )} x^{6} + \frac {6}{5} \, {\left (a^{2} c^{2} d^{3} + 2 \, a^{3} c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (12 \, a^{3} c d^{2} e + a^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2}\right )} x^{3} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=a^{4} d^{3} x + \frac {3 a^{4} d^{2} e x^{2}}{2} + \frac {3 c^{4} d e^{2} x^{11}}{11} + \frac {c^{4} e^{3} x^{12}}{12} + x^{10} \cdot \left (\frac {2 a c^{3} e^{3}}{5} + \frac {3 c^{4} d^{2} e}{10}\right ) + x^{9} \cdot \left (\frac {4 a c^{3} d e^{2}}{3} + \frac {c^{4} d^{3}}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c^{2} e^{3}}{4} + \frac {3 a c^{3} d^{2} e}{2}\right ) + x^{7} \cdot \left (\frac {18 a^{2} c^{2} d e^{2}}{7} + \frac {4 a c^{3} d^{3}}{7}\right ) + x^{6} \cdot \left (\frac {2 a^{3} c e^{3}}{3} + 3 a^{2} c^{2} d^{2} e\right ) + x^{5} \cdot \left (\frac {12 a^{3} c d e^{2}}{5} + \frac {6 a^{2} c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} c d^{2} e\right ) + x^{3} \left (a^{4} d e^{2} + \frac {4 a^{3} c d^{3}}{3}\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=\frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {3}{11} \, c^{4} d e^{2} x^{11} + \frac {1}{10} \, {\left (3 \, c^{4} d^{2} e + 4 \, a c^{3} e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{3} + 12 \, a c^{3} d e^{2}\right )} x^{9} + \frac {3}{2} \, a^{4} d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, a c^{3} d^{2} e + a^{2} c^{2} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac {2}{7} \, {\left (2 \, a c^{3} d^{3} + 9 \, a^{2} c^{2} d e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (9 \, a^{2} c^{2} d^{2} e + 2 \, a^{3} c e^{3}\right )} x^{6} + \frac {6}{5} \, {\left (a^{2} c^{2} d^{3} + 2 \, a^{3} c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (12 \, a^{3} c d^{2} e + a^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2}\right )} x^{3} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.18 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=\frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {3}{11} \, c^{4} d e^{2} x^{11} + \frac {3}{10} \, c^{4} d^{2} e x^{10} + \frac {2}{5} \, a c^{3} e^{3} x^{10} + \frac {1}{9} \, c^{4} d^{3} x^{9} + \frac {4}{3} \, a c^{3} d e^{2} x^{9} + \frac {3}{2} \, a c^{3} d^{2} e x^{8} + \frac {3}{4} \, a^{2} c^{2} e^{3} x^{8} + \frac {4}{7} \, a c^{3} d^{3} x^{7} + \frac {18}{7} \, a^{2} c^{2} d e^{2} x^{7} + 3 \, a^{2} c^{2} d^{2} e x^{6} + \frac {2}{3} \, a^{3} c e^{3} x^{6} + \frac {6}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {12}{5} \, a^{3} c d e^{2} x^{5} + 3 \, a^{3} c d^{2} e x^{4} + \frac {1}{4} \, a^{4} e^{3} x^{4} + \frac {4}{3} \, a^{3} c d^{3} x^{3} + a^{4} d e^{2} x^{3} + \frac {3}{2} \, a^{4} d^{2} e x^{2} + a^{4} d^{3} x \]
[In]
[Out]
Time = 9.49 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08 \[ \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx=x^3\,\left (a^4\,d\,e^2+\frac {4\,c\,a^3\,d^3}{3}\right )+x^4\,\left (\frac {a^4\,e^3}{4}+3\,c\,a^3\,d^2\,e\right )+x^9\,\left (\frac {c^4\,d^3}{9}+\frac {4\,a\,c^3\,d\,e^2}{3}\right )+x^{10}\,\left (\frac {3\,c^4\,d^2\,e}{10}+\frac {2\,a\,c^3\,e^3}{5}\right )+a^4\,d^3\,x+\frac {c^4\,e^3\,x^{12}}{12}+\frac {3\,a^4\,d^2\,e\,x^2}{2}+\frac {3\,c^4\,d\,e^2\,x^{11}}{11}+\frac {6\,a^2\,c\,d\,x^5\,\left (c\,d^2+2\,a\,e^2\right )}{5}+\frac {2\,a\,c^2\,d\,x^7\,\left (2\,c\,d^2+9\,a\,e^2\right )}{7}+\frac {3\,a\,c^2\,e\,x^8\,\left (2\,c\,d^2+a\,e^2\right )}{4}+\frac {a^2\,c\,e\,x^6\,\left (9\,c\,d^2+2\,a\,e^2\right )}{3} \]
[In]
[Out]