Integrand size = 20, antiderivative size = 108 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{6 e^4}-\frac {c (3 B d-A e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \]
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Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {786} \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac {(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac {c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \]
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Rule 786
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^4}{e^3}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^5}{e^3}+\frac {c (-3 B d+A e) (d+e x)^6}{e^3}+\frac {B c (d+e x)^7}{e^3}\right ) \, dx \\ & = -\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{6 e^4}-\frac {c (3 B d-A e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=a A d^4 x+\frac {1}{2} a d^3 (B d+4 A e) x^2+\frac {1}{3} d^2 \left (A c d^2+4 a B d e+6 a A e^2\right ) x^3+\frac {1}{4} d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right ) x^4+\frac {1}{5} e \left (4 B c d^3+6 A c d^2 e+4 a B d e^2+a A e^3\right ) x^5+\frac {1}{6} e^2 \left (6 B c d^2+4 A c d e+a B e^2\right ) x^6+\frac {1}{7} c e^3 (4 B d+A e) x^7+\frac {1}{8} B c e^4 x^8 \]
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Time = 0.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.78
method | result | size |
norman | \(\frac {B \,e^{4} c \,x^{8}}{8}+\left (\frac {1}{7} A c \,e^{4}+\frac {4}{7} B c d \,e^{3}\right ) x^{7}+\left (\frac {2}{3} A c d \,e^{3}+\frac {1}{6} B \,e^{4} a +B c \,d^{2} e^{2}\right ) x^{6}+\left (\frac {1}{5} A a \,e^{4}+\frac {6}{5} A c \,d^{2} e^{2}+\frac {4}{5} B a d \,e^{3}+\frac {4}{5} B c \,d^{3} e \right ) x^{5}+\left (A a d \,e^{3}+A c \,d^{3} e +\frac {3}{2} B a \,d^{2} e^{2}+\frac {1}{4} B c \,d^{4}\right ) x^{4}+\left (2 A a \,d^{2} e^{2}+\frac {1}{3} A c \,d^{4}+\frac {4}{3} B a \,d^{3} e \right ) x^{3}+\left (2 A a \,d^{3} e +\frac {1}{2} B a \,d^{4}\right ) x^{2}+d^{4} A a x\) | \(192\) |
default | \(\frac {B \,e^{4} c \,x^{8}}{8}+\frac {\left (e^{4} A +4 e^{3} d B \right ) c \,x^{7}}{7}+\frac {\left (\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) c +B \,e^{4} a \right ) x^{6}}{6}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c +\left (e^{4} A +4 e^{3} d B \right ) a \right ) x^{5}}{5}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c +\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A c \,d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a \right ) x^{3}}{3}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a \,x^{2}}{2}+d^{4} A a x\) | \(199\) |
gosper | \(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {1}{6} x^{6} B \,e^{4} a +x^{6} B c \,d^{2} e^{2}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {4}{5} x^{5} B c \,d^{3} e +x^{4} A a d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {1}{4} x^{4} B c \,d^{4}+2 x^{3} A a \,d^{2} e^{2}+\frac {1}{3} x^{3} A c \,d^{4}+\frac {4}{3} x^{3} B a \,d^{3} e +2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} B a \,d^{4}+d^{4} A a x\) | \(216\) |
risch | \(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {1}{6} x^{6} B \,e^{4} a +x^{6} B c \,d^{2} e^{2}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {4}{5} x^{5} B c \,d^{3} e +x^{4} A a d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {1}{4} x^{4} B c \,d^{4}+2 x^{3} A a \,d^{2} e^{2}+\frac {1}{3} x^{3} A c \,d^{4}+\frac {4}{3} x^{3} B a \,d^{3} e +2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} B a \,d^{4}+d^{4} A a x\) | \(216\) |
parallelrisch | \(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {1}{6} x^{6} B \,e^{4} a +x^{6} B c \,d^{2} e^{2}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {4}{5} x^{5} B c \,d^{3} e +x^{4} A a d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {1}{4} x^{4} B c \,d^{4}+2 x^{3} A a \,d^{2} e^{2}+\frac {1}{3} x^{3} A c \,d^{4}+\frac {4}{3} x^{3} B a \,d^{3} e +2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} B a \,d^{4}+d^{4} A a x\) | \(216\) |
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Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{4} + 4 \, A a d^{3} e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (102) = 204\).
Time = 0.03 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.09 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=A a d^{4} x + \frac {B c e^{4} x^{8}}{8} + x^{7} \left (\frac {A c e^{4}}{7} + \frac {4 B c d e^{3}}{7}\right ) + x^{6} \cdot \left (\frac {2 A c d e^{3}}{3} + \frac {B a e^{4}}{6} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac {A a e^{4}}{5} + \frac {6 A c d^{2} e^{2}}{5} + \frac {4 B a d e^{3}}{5} + \frac {4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + A c d^{3} e + \frac {3 B a d^{2} e^{2}}{2} + \frac {B c d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a d^{2} e^{2} + \frac {A c d^{4}}{3} + \frac {4 B a d^{3} e}{3}\right ) + x^{2} \cdot \left (2 A a d^{3} e + \frac {B a d^{4}}{2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.80 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{4} + 4 \, A a d^{3} e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (100) = 200\).
Time = 0.26 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.99 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {4}{7} \, B c d e^{3} x^{7} + \frac {1}{7} \, A c e^{4} x^{7} + B c d^{2} e^{2} x^{6} + \frac {2}{3} \, A c d e^{3} x^{6} + \frac {1}{6} \, B a e^{4} x^{6} + \frac {4}{5} \, B c d^{3} e x^{5} + \frac {6}{5} \, A c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a d e^{3} x^{5} + \frac {1}{5} \, A a e^{4} x^{5} + \frac {1}{4} \, B c d^{4} x^{4} + A c d^{3} e x^{4} + \frac {3}{2} \, B a d^{2} e^{2} x^{4} + A a d e^{3} x^{4} + \frac {1}{3} \, A c d^{4} x^{3} + \frac {4}{3} \, B a d^{3} e x^{3} + 2 \, A a d^{2} e^{2} x^{3} + \frac {1}{2} \, B a d^{4} x^{2} + 2 \, A a d^{3} e x^{2} + A a d^{4} x \]
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Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.71 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx=x^3\,\left (\frac {A\,c\,d^4}{3}+\frac {4\,B\,a\,d^3\,e}{3}+2\,A\,a\,d^2\,e^2\right )+x^6\,\left (B\,c\,d^2\,e^2+\frac {2\,A\,c\,d\,e^3}{3}+\frac {B\,a\,e^4}{6}\right )+x^4\,\left (\frac {B\,c\,d^4}{4}+A\,c\,d^3\,e+\frac {3\,B\,a\,d^2\,e^2}{2}+A\,a\,d\,e^3\right )+x^5\,\left (\frac {4\,B\,c\,d^3\,e}{5}+\frac {6\,A\,c\,d^2\,e^2}{5}+\frac {4\,B\,a\,d\,e^3}{5}+\frac {A\,a\,e^4}{5}\right )+A\,a\,d^4\,x+\frac {B\,c\,e^4\,x^8}{8}+\frac {a\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c\,e^3\,x^7\,\left (A\,e+4\,B\,d\right )}{7} \]
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