Integrand size = 33, antiderivative size = 254 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x) (d+e x)^{11}}+\frac {2 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{10}}-\frac {2 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^9}+\frac {b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^8}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^5 (a+b x) (d+e x)^9}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{5 e^5 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^5 (a+b x) (d+e x)^{11}}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^5 (a+b x) (d+e x)^8} \]
[In]
[Out]
Rule 21
Rule 45
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^{12}} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^{12}} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{12}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{11}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^{10}}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^9}+\frac {b^4}{e^4 (d+e x)^8}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x) (d+e x)^{11}}+\frac {2 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{10}}-\frac {2 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^9}+\frac {b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x) (d+e x)^8}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+7 a b^3 e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )}{2310 e^5 (a+b x) (d+e x)^{11}} \]
[In]
[Out]
Time = 3.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{4} x^{4}}{7 e}-\frac {b^{3} \left (7 a e +b d \right ) x^{3}}{14 e^{2}}-\frac {b^{2} \left (28 e^{2} a^{2}+7 a b d e +b^{2} d^{2}\right ) x^{2}}{42 e^{3}}-\frac {b \left (84 a^{3} e^{3}+28 a^{2} b d \,e^{2}+7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{210 e^{4}}-\frac {210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}}{2310 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\) | \(187\) |
gosper | \(-\frac {\left (330 e^{4} x^{4} b^{4}+1155 x^{3} a \,b^{3} e^{4}+165 x^{3} b^{4} d \,e^{3}+1540 x^{2} a^{2} b^{2} e^{4}+385 x^{2} a \,b^{3} d \,e^{3}+55 x^{2} b^{4} d^{2} e^{2}+924 x \,a^{3} b \,e^{4}+308 x \,a^{2} b^{2} d \,e^{3}+77 x a \,b^{3} d^{2} e^{2}+11 x \,b^{4} d^{3} e +210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2310 e^{5} \left (e x +d \right )^{11} \left (b x +a \right )^{3}}\) | \(201\) |
default | \(-\frac {\left (330 e^{4} x^{4} b^{4}+1155 x^{3} a \,b^{3} e^{4}+165 x^{3} b^{4} d \,e^{3}+1540 x^{2} a^{2} b^{2} e^{4}+385 x^{2} a \,b^{3} d \,e^{3}+55 x^{2} b^{4} d^{2} e^{2}+924 x \,a^{3} b \,e^{4}+308 x \,a^{2} b^{2} d \,e^{3}+77 x a \,b^{3} d^{2} e^{2}+11 x \,b^{4} d^{3} e +210 e^{4} a^{4}+84 b d \,e^{3} a^{3}+28 b^{2} d^{2} e^{2} a^{2}+7 b^{3} d^{3} e a +b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2310 e^{5} \left (e x +d \right )^{11} \left (b x +a \right )^{3}}\) | \(201\) |
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=-\frac {330 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 7 \, a b^{3} d^{3} e + 28 \, a^{2} b^{2} d^{2} e^{2} + 84 \, a^{3} b d e^{3} + 210 \, a^{4} e^{4} + 165 \, {\left (b^{4} d e^{3} + 7 \, a b^{3} e^{4}\right )} x^{3} + 55 \, {\left (b^{4} d^{2} e^{2} + 7 \, a b^{3} d e^{3} + 28 \, a^{2} b^{2} e^{4}\right )} x^{2} + 11 \, {\left (b^{4} d^{3} e + 7 \, a b^{3} d^{2} e^{2} + 28 \, a^{2} b^{2} d e^{3} + 84 \, a^{3} b e^{4}\right )} x}{2310 \, {\left (e^{16} x^{11} + 11 \, d e^{15} x^{10} + 55 \, d^{2} e^{14} x^{9} + 165 \, d^{3} e^{13} x^{8} + 330 \, d^{4} e^{12} x^{7} + 462 \, d^{5} e^{11} x^{6} + 462 \, d^{6} e^{10} x^{5} + 330 \, d^{7} e^{9} x^{4} + 165 \, d^{8} e^{8} x^{3} + 55 \, d^{9} e^{7} x^{2} + 11 \, d^{10} e^{6} x + d^{11} e^{5}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (189) = 378\).
Time = 0.26 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\frac {b^{11} \mathrm {sgn}\left (b x + a\right )}{2310 \, {\left (b^{7} d^{7} e^{5} - 7 \, a b^{6} d^{6} e^{6} + 21 \, a^{2} b^{5} d^{5} e^{7} - 35 \, a^{3} b^{4} d^{4} e^{8} + 35 \, a^{4} b^{3} d^{3} e^{9} - 21 \, a^{5} b^{2} d^{2} e^{10} + 7 \, a^{6} b d e^{11} - a^{7} e^{12}\right )}} - \frac {330 \, b^{4} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{4} d e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1155 \, a b^{3} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{4} d^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 385 \, a b^{3} d e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1540 \, a^{2} b^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{4} d^{3} e x \mathrm {sgn}\left (b x + a\right ) + 77 \, a b^{3} d^{2} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 308 \, a^{2} b^{2} d e^{3} x \mathrm {sgn}\left (b x + a\right ) + 924 \, a^{3} b e^{4} x \mathrm {sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 28 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )}{2310 \, {\left (e x + d\right )}^{11} e^{5}} \]
[In]
[Out]
Time = 11.13 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{12}} \, dx=\frac {\left (\frac {-4\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-4\,a\,b^3\,d^2\,e+b^4\,d^3}{10\,e^5}+\frac {d\,\left (\frac {d\,\left (\frac {b^4\,d}{10\,e^3}-\frac {b^3\,\left (4\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^2\,\left (6\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {a^4}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {4\,a\,b^3}{11\,e}-\frac {b^4\,d}{11\,e^2}\right )}{e}-\frac {6\,a^2\,b^2}{11\,e}\right )}{e}+\frac {4\,a^3\,b}{11\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}-\frac {\left (\frac {6\,a^2\,b^2\,e^2-8\,a\,b^3\,d\,e+3\,b^4\,d^2}{9\,e^5}+\frac {d\,\left (\frac {b^4\,d}{9\,e^4}-\frac {2\,b^3\,\left (2\,a\,e-b\,d\right )}{9\,e^4}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {3\,b^4\,d-4\,a\,b^3\,e}{8\,e^5}+\frac {b^4\,d}{8\,e^5}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,e^5\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7} \]
[In]
[Out]