Integrand size = 21, antiderivative size = 548 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 c d-7 b e}{5 b d^2 (c d-b e) \left (b x+c x^2\right )^{5/2}}+\frac {\left (\frac {4 c^2}{b^2}-\frac {7 e^2}{d^2}\right ) x}{3 d (c d-b e) \left (b x+c x^2\right )^{5/2}}-\frac {\left (32 c^3 d^3-14 b^2 c d e^2-21 b^3 e^3\right ) x^2}{3 b^3 d^4 (c d-b e) \left (b x+c x^2\right )^{5/2}}-\frac {c \left (64 c^4 d^4-64 b c^3 d^3 e-28 b^2 c^2 d^2 e^2+35 b^4 e^4\right ) x^3}{5 b^4 d^4 (c d-b e)^2 \left (b x+c x^2\right )^{5/2}}-\frac {e}{d (c d-b e) (d+e x) \left (b x+c x^2\right )^{5/2}}-\frac {c \left (256 c^5 d^5-512 b c^4 d^4 e+144 b^2 c^3 d^3 e^2+112 b^3 c^2 d^2 e^3+70 b^4 c d e^4-105 b^5 e^5\right ) x^2}{15 b^5 d^4 (c d-b e)^3 \left (b x+c x^2\right )^{3/2}}-\frac {c \left (512 c^6 d^6-1536 b c^5 d^5 e+1312 b^2 c^4 d^4 e^2-64 b^3 c^3 d^3 e^3-84 b^4 c^2 d^2 e^4-140 b^5 c d e^5+105 b^6 e^6\right ) x}{15 b^6 d^4 (c d-b e)^4 \sqrt {b x+c x^2}}+\frac {7 e^6 (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{d^{9/2} (c d-b e)^{9/2}} \] Output:
-1/5*(-7*b*e+2*c*d)/b/d^2/(-b*e+c*d)/(c*x^2+b*x)^(5/2)+1/3*(4*c^2/b^2-7*e^ 2/d^2)*x/d/(-b*e+c*d)/(c*x^2+b*x)^(5/2)-1/3*(-21*b^3*e^3-14*b^2*c*d*e^2+32 *c^3*d^3)*x^2/b^3/d^4/(-b*e+c*d)/(c*x^2+b*x)^(5/2)-1/5*c*(35*b^4*e^4-28*b^ 2*c^2*d^2*e^2-64*b*c^3*d^3*e+64*c^4*d^4)*x^3/b^4/d^4/(-b*e+c*d)^2/(c*x^2+b *x)^(5/2)-e/d/(-b*e+c*d)/(e*x+d)/(c*x^2+b*x)^(5/2)-1/15*c*(-105*b^5*e^5+70 *b^4*c*d*e^4+112*b^3*c^2*d^2*e^3+144*b^2*c^3*d^3*e^2-512*b*c^4*d^4*e+256*c ^5*d^5)*x^2/b^5/d^4/(-b*e+c*d)^3/(c*x^2+b*x)^(3/2)-1/15*c*(105*b^6*e^6-140 *b^5*c*d*e^5-84*b^4*c^2*d^2*e^4-64*b^3*c^3*d^3*e^3+1312*b^2*c^4*d^4*e^2-15 36*b*c^5*d^5*e+512*c^6*d^6)*x/b^6/d^4/(-b*e+c*d)^4/(c*x^2+b*x)^(1/2)+7*e^6 *(-b*e+2*c*d)*arctanh((-b*e+c*d)^(1/2)*x/d^(1/2)/(c*x^2+b*x)^(1/2))/d^(9/2 )/(-b*e+c*d)^(9/2)
Time = 5.02 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\frac {x \left (-\frac {\sqrt {d} (b+c x) \left (512 c^9 d^6 x^5 (d+e x)-256 b c^8 d^5 x^4 \left (-5 d^2+d e x+6 e^2 x^2\right )+32 b^2 c^7 d^4 x^3 \left (30 d^3-90 d^2 e x-79 d e^2 x^2+41 e^3 x^3\right )+b^9 e^4 \left (6 d^3-14 d^2 e x+70 d e^2 x^2+105 e^3 x^3\right )+16 b^3 c^6 d^3 x^2 \left (10 d^4-170 d^3 e x+25 d^2 e^2 x^2+201 d e^3 x^3-4 e^4 x^4\right )+b^8 c e^3 \left (-24 d^4+36 d^3 e x-140 d^2 e^2 x^2+70 d e^3 x^3+315 e^4 x^4\right )-4 b^4 c^5 d^2 x \left (5 d^5+125 d^4 e x-495 d^3 e^2 x^2-575 d^2 e^3 x^3+61 d e^4 x^4+21 e^5 x^5\right )+b^7 c^2 e^2 \left (36 d^5-4 d^4 e x+20 d^3 e^2 x^2-420 d^2 e^3 x^3-210 d e^4 x^4+315 e^5 x^5\right )-b^6 c^3 e \left (24 d^6+64 d^5 e x+80 d^4 e^2 x^2+160 d^3 e^3 x^3+560 d^2 e^4 x^4+350 d e^5 x^5-105 e^6 x^6\right )+2 b^5 c^4 d \left (3 d^6+33 d^5 e x+235 d^4 e^2 x^2+145 d^3 e^3 x^3-165 d^2 e^4 x^4-175 d e^5 x^5-70 e^6 x^6\right )\right )}{b^6 (c d-b e)^4 (d+e x)}-\frac {105 e^6 (2 c d-b e) x^{5/2} (b+c x)^{7/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{9/2}}\right )}{15 d^{9/2} (x (b+c x))^{7/2}} \] Input:
Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(7/2)),x]
Output:
(x*(-((Sqrt[d]*(b + c*x)*(512*c^9*d^6*x^5*(d + e*x) - 256*b*c^8*d^5*x^4*(- 5*d^2 + d*e*x + 6*e^2*x^2) + 32*b^2*c^7*d^4*x^3*(30*d^3 - 90*d^2*e*x - 79* d*e^2*x^2 + 41*e^3*x^3) + b^9*e^4*(6*d^3 - 14*d^2*e*x + 70*d*e^2*x^2 + 105 *e^3*x^3) + 16*b^3*c^6*d^3*x^2*(10*d^4 - 170*d^3*e*x + 25*d^2*e^2*x^2 + 20 1*d*e^3*x^3 - 4*e^4*x^4) + b^8*c*e^3*(-24*d^4 + 36*d^3*e*x - 140*d^2*e^2*x ^2 + 70*d*e^3*x^3 + 315*e^4*x^4) - 4*b^4*c^5*d^2*x*(5*d^5 + 125*d^4*e*x - 495*d^3*e^2*x^2 - 575*d^2*e^3*x^3 + 61*d*e^4*x^4 + 21*e^5*x^5) + b^7*c^2*e ^2*(36*d^5 - 4*d^4*e*x + 20*d^3*e^2*x^2 - 420*d^2*e^3*x^3 - 210*d*e^4*x^4 + 315*e^5*x^5) - b^6*c^3*e*(24*d^6 + 64*d^5*e*x + 80*d^4*e^2*x^2 + 160*d^3 *e^3*x^3 + 560*d^2*e^4*x^4 + 350*d*e^5*x^5 - 105*e^6*x^6) + 2*b^5*c^4*d*(3 *d^6 + 33*d^5*e*x + 235*d^4*e^2*x^2 + 145*d^3*e^3*x^3 - 165*d^2*e^4*x^4 - 175*d*e^5*x^5 - 70*e^6*x^6)))/(b^6*(c*d - b*e)^4*(d + e*x))) - (105*e^6*(2 *c*d - b*e)*x^(5/2)*(b + c*x)^(7/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + S qrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(9/2)))/(1 5*d^(9/2)*(x*(b + c*x))^(7/2))
Time = 1.75 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1165, 27, 1235, 27, 1235, 27, 1228, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{7/2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {(8 c d-7 b e) (2 c d+b e)+10 c e (2 c d-b e) x}{2 (d+e x)^2 \left (c x^2+b x\right )^{5/2}}dx}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(8 c d-7 b e) (2 c d+b e)+10 c e (2 c d-b e) x}{(d+e x)^2 \left (c x^2+b x\right )^{5/2}}dx}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {128 c^4 d^4-160 b c^3 e d^3-24 b^2 c^2 e^2 d^2+14 b^3 c e^3 d+35 b^4 e^4+6 c e (2 c d-b e) \left (16 c^2 d^2-16 b c e d-7 b^2 e^2\right ) x}{2 (d+e x)^2 \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {128 c^4 d^4-160 b c^3 e d^3-24 b^2 c^2 e^2 d^2+14 b^3 c e^3 d+35 b^4 e^4+6 c e (2 c d-b e) \left (16 c^2 d^2-16 b c e d-7 b^2 e^2\right ) x}{(d+e x)^2 \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {e \left (b \left (256 c^5 d^5-512 b c^4 e d^4+144 b^2 c^3 e^2 d^3+112 b^3 c^2 e^3 d^2+70 b^4 c e^4 d-105 b^5 e^5\right )+2 c (2 c d-b e) \left (128 c^4 d^4-256 b c^3 e d^3+72 b^2 c^2 e^2 d^2+56 b^3 c e^3 d+35 b^4 e^4\right ) x\right )}{2 (d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (35 b^4 e^4+56 b^3 c d e^3+72 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (35 b^4 e^4+14 b^3 c d e^3-24 b^2 c^2 d^2 e^2-160 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {e \int \frac {b \left (256 c^5 d^5-512 b c^4 e d^4+144 b^2 c^3 e^2 d^3+112 b^3 c^2 e^3 d^2+70 b^4 c e^4 d-105 b^5 e^5\right )+2 c (2 c d-b e) \left (128 c^4 d^4-256 b c^3 e d^3+72 b^2 c^2 e^2 d^2+56 b^3 c e^3 d+35 b^4 e^4\right ) x}{(d+e x)^2 \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (35 b^4 e^4+56 b^3 c d e^3+72 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (35 b^4 e^4+14 b^3 c d e^3-24 b^2 c^2 d^2 e^2-160 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {-\frac {-\frac {e \left (\frac {\sqrt {b x+c x^2} \left (105 b^6 e^6-140 b^5 c d e^5-84 b^4 c^2 d^2 e^4-64 b^3 c^3 d^3 e^3+1312 b^2 c^4 d^4 e^2-1536 b c^5 d^5 e+512 c^6 d^6\right )}{d (d+e x) (c d-b e)}-\frac {105 b^6 e^5 (2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (35 b^4 e^4+56 b^3 c d e^3+72 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (35 b^4 e^4+14 b^3 c d e^3-24 b^2 c^2 d^2 e^2-160 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-\frac {-\frac {e \left (\frac {105 b^6 e^5 (2 c d-b e) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}+\frac {\sqrt {b x+c x^2} \left (105 b^6 e^6-140 b^5 c d e^5-84 b^4 c^2 d^2 e^4-64 b^3 c^3 d^3 e^3+1312 b^2 c^4 d^4 e^2-1536 b c^5 d^5 e+512 c^6 d^6\right )}{d (d+e x) (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (35 b^4 e^4+56 b^3 c d e^3+72 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (35 b^4 e^4+14 b^3 c d e^3-24 b^2 c^2 d^2 e^2-160 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {-\frac {e \left (\frac {\sqrt {b x+c x^2} \left (105 b^6 e^6-140 b^5 c d e^5-84 b^4 c^2 d^2 e^4-64 b^3 c^3 d^3 e^3+1312 b^2 c^4 d^4 e^2-1536 b c^5 d^5 e+512 c^6 d^6\right )}{d (d+e x) (c d-b e)}-\frac {105 b^6 e^5 (2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}\right )}{b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (35 b^4 e^4+56 b^3 c d e^3+72 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )+b (c d-b e) \left (35 b^4 e^4+14 b^3 c d e^3-24 b^2 c^2 d^2 e^2-160 b c^3 d^3 e+128 c^4 d^4\right )\right )}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \left (c x (2 c d-b e) \left (-7 b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (8 c d-7 b e) (c d-b e) (b e+2 c d)\right )}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}}{5 b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{5 b^2 d \left (b x+c x^2\right )^{5/2} (d+e x) (c d-b e)}\) |
Input:
Int[1/((d + e*x)^2*(b*x + c*x^2)^(7/2)),x]
Output:
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(5*b^2*d*(c*d - b*e)*(d + e*x)*(b *x + c*x^2)^(5/2)) - ((-2*(b*(8*c*d - 7*b*e)*(c*d - b*e)*(2*c*d + b*e) + c *(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e - 7*b^2*e^2)*x))/(3*b^2*d*(c*d - b *e)*(d + e*x)*(b*x + c*x^2)^(3/2)) - ((-2*(b*(c*d - b*e)*(128*c^4*d^4 - 16 0*b*c^3*d^3*e - 24*b^2*c^2*d^2*e^2 + 14*b^3*c*d*e^3 + 35*b^4*e^4) + c*(2*c *d - b*e)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 72*b^2*c^2*d^2*e^2 + 56*b^3*c*d *e^3 + 35*b^4*e^4)*x))/(b^2*d*(c*d - b*e)*(d + e*x)*Sqrt[b*x + c*x^2]) - ( e*(((512*c^6*d^6 - 1536*b*c^5*d^5*e + 1312*b^2*c^4*d^4*e^2 - 64*b^3*c^3*d^ 3*e^3 - 84*b^4*c^2*d^2*e^4 - 140*b^5*c*d*e^5 + 105*b^6*e^6)*Sqrt[b*x + c*x ^2])/(d*(c*d - b*e)*(d + e*x)) - (105*b^6*e^5*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(3/ 2)*(c*d - b*e)^(3/2))))/(b^2*d*(c*d - b*e)))/(3*b^2*d*(c*d - b*e)))/(5*b^2 *d*(c*d - b*e))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Time = 1.12 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\frac {35 b^{6} e^{6} x^{2} \sqrt {x \left (c x +b \right )}\, \left (e x +d \right ) \left (c x +b \right )^{2} \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{2}+\sqrt {d \left (b e -c d \right )}\, \left (c^{4} \left (2 c x +b \right ) \left (\frac {128}{3} c^{4} x^{4}+\frac {256}{3} b \,c^{3} x^{3}+\frac {112}{3} b^{2} c^{2} x^{2}-\frac {16}{3} b^{3} c x +b^{4}\right ) d^{7}-4 e \,c^{3} \left (-\frac {64}{3} x^{6} c^{6}+\frac {32}{3} x^{5} b \,c^{5}+120 x^{4} b^{2} c^{4}+\frac {340}{3} b^{3} c^{3} x^{3}+\frac {125}{6} c^{2} x^{2} b^{4}-\frac {11}{4} x c \,b^{5}+b^{6}\right ) d^{6}+6 e^{2} c^{2} \left (-\frac {128}{3} x^{6} c^{6}-\frac {632}{9} x^{5} b \,c^{5}+\frac {100}{9} x^{4} b^{2} c^{4}+55 b^{3} c^{3} x^{3}+\frac {235}{18} c^{2} x^{2} b^{4}-\frac {16}{9} x c \,b^{5}+b^{6}\right ) b \,d^{5}-4 e^{3} \left (-\frac {164}{3} x^{6} c^{6}-134 x^{5} b \,c^{5}-\frac {575}{6} x^{4} b^{2} c^{4}-\frac {145}{12} b^{3} c^{3} x^{3}+\frac {10}{3} c^{2} x^{2} b^{4}+\frac {1}{6} x c \,b^{5}+b^{6}\right ) c \,b^{2} d^{4}+e^{4} \left (-\frac {32}{3} c^{2} x^{2}+2 c b x +b^{2}\right ) \left (c x +b \right )^{4} b^{3} d^{3}-\frac {7 b^{4} e^{5} x \left (6 c x +b \right ) \left (c x +b \right )^{4} d^{2}}{3}+\frac {35 b^{5} e^{6} x^{2} \left (-2 c x +b \right ) \left (c x +b \right )^{3} d}{3}+\frac {35 b^{6} e^{7} x^{3} \left (c x +b \right )^{3}}{2}\right )\right )}{5 \sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}\, x^{2} d^{4} \left (b e -c d \right )^{4} \left (c x +b \right )^{2} \left (e x +d \right ) b^{6}}\) | \(509\) |
| risch | \(-\frac {2 \left (c x +b \right ) \left (45 b^{2} e^{2} x^{2}+110 b c d e \,x^{2}+128 d^{2} c^{2} x^{2}-10 b^{2} d e x -19 x b c \,d^{2}+3 b^{2} d^{2}\right )}{15 b^{6} d^{4} \sqrt {x \left (c x +b \right )}\, x^{2}}-\frac {\frac {b^{2} c^{2} d^{4} \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{5 b \left (\frac {b}{c}+x \right )^{3}}+\frac {4 c \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{2} \left (\frac {b}{c}+x \right )}\right )}{5 b}\right )}{\left (b e -c d \right )^{2}}+\frac {e^{4} b^{5} d \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{\left (b e -c d \right )^{3}}+\frac {b \,c^{3} d^{4} \left (5 b e -3 c d \right ) \left (\frac {2 \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{2} \left (\frac {b}{c}+x \right )}\right )}{\left (b e -c d \right )^{3}}+\frac {6 c^{4} d^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{\left (b e -c d \right )^{4} b \left (\frac {b}{c}+x \right )}-\frac {3 e^{5} b^{5} \left (b e -2 c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right )^{4} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{b^{5} d^{4}}\) | \(811\) |
| default | \(\text {Expression too large to display}\) | \(1593\) |
Input:
int(1/(e*x+d)^2/(c*x^2+b*x)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/5/(x*(c*x+b))^(1/2)/(d*(b*e-c*d))^(1/2)*(-35/2*b^6*e^6*x^2*(x*(c*x+b))^ (1/2)*(e*x+d)*(c*x+b)^2*(b*e-2*c*d)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c *d))^(1/2))+(d*(b*e-c*d))^(1/2)*(c^4*(2*c*x+b)*(128/3*c^4*x^4+256/3*b*c^3* x^3+112/3*b^2*c^2*x^2-16/3*b^3*c*x+b^4)*d^7-4*e*c^3*(-64/3*x^6*c^6+32/3*x^ 5*b*c^5+120*x^4*b^2*c^4+340/3*b^3*c^3*x^3+125/6*c^2*x^2*b^4-11/4*x*c*b^5+b ^6)*d^6+6*e^2*c^2*(-128/3*x^6*c^6-632/9*x^5*b*c^5+100/9*x^4*b^2*c^4+55*b^3 *c^3*x^3+235/18*c^2*x^2*b^4-16/9*x*c*b^5+b^6)*b*d^5-4*e^3*(-164/3*x^6*c^6- 134*x^5*b*c^5-575/6*x^4*b^2*c^4-145/12*b^3*c^3*x^3+10/3*c^2*x^2*b^4+1/6*x* c*b^5+b^6)*c*b^2*d^4+e^4*(-32/3*c^2*x^2+2*c*b*x+b^2)*(c*x+b)^4*b^3*d^3-7/3 *b^4*e^5*x*(6*c*x+b)*(c*x+b)^4*d^2+35/3*b^5*e^6*x^2*(-2*c*x+b)*(c*x+b)^3*d +35/2*b^6*e^7*x^3*(c*x+b)^3))/x^2/d^4/(b*e-c*d)^4/(c*x+b)^2/(e*x+d)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 1408 vs. \(2 (512) = 1024\).
Time = 1.03 (sec) , antiderivative size = 2832, normalized size of antiderivative = 5.17 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(7/2),x, algorithm="fricas")
Output:
[-1/30*(105*((2*b^6*c^4*d*e^7 - b^7*c^3*e^8)*x^7 + (2*b^6*c^4*d^2*e^6 + 5* b^7*c^3*d*e^7 - 3*b^8*c^2*e^8)*x^6 + 3*(2*b^7*c^3*d^2*e^6 + b^8*c^2*d*e^7 - b^9*c*e^8)*x^5 + (6*b^8*c^2*d^2*e^6 - b^9*c*d*e^7 - b^10*e^8)*x^4 + (2*b ^9*c*d^2*e^6 - b^10*d*e^7)*x^3)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b* e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(6*b^5*c^5* d^9 - 30*b^6*c^4*d^8*e + 60*b^7*c^3*d^7*e^2 - 60*b^8*c^2*d^6*e^3 + 30*b^9* c*d^5*e^4 - 6*b^10*d^4*e^5 + (512*c^10*d^8*e - 2048*b*c^9*d^7*e^2 + 2848*b ^2*c^8*d^6*e^3 - 1376*b^3*c^7*d^5*e^4 - 20*b^4*c^6*d^4*e^5 - 56*b^5*c^5*d^ 3*e^6 + 245*b^6*c^4*d^2*e^7 - 105*b^7*c^3*d*e^8)*x^6 + (512*c^10*d^9 - 768 *b*c^9*d^8*e - 2272*b^2*c^8*d^7*e^2 + 5744*b^3*c^7*d^6*e^3 - 3460*b^4*c^6* d^5*e^4 - 106*b^5*c^5*d^4*e^5 + 665*b^7*c^3*d^2*e^7 - 315*b^8*c^2*d*e^8)*x ^5 + 5*(256*b*c^9*d^9 - 832*b^2*c^8*d^8*e + 656*b^3*c^7*d^7*e^2 + 380*b^4* c^6*d^6*e^3 - 526*b^5*c^5*d^5*e^4 - 46*b^6*c^4*d^4*e^5 + 70*b^7*c^3*d^3*e^ 6 + 105*b^8*c^2*d^2*e^7 - 63*b^9*c*d*e^8)*x^4 + 5*(192*b^2*c^8*d^9 - 736*b ^3*c^7*d^8*e + 940*b^4*c^6*d^7*e^2 - 338*b^5*c^5*d^6*e^3 - 90*b^6*c^4*d^5* e^4 - 52*b^7*c^3*d^4*e^5 + 98*b^8*c^2*d^3*e^6 + 7*b^9*c*d^2*e^7 - 21*b^10* d*e^8)*x^3 + 10*(16*b^3*c^7*d^9 - 66*b^4*c^6*d^8*e + 97*b^5*c^5*d^7*e^2 - 55*b^6*c^4*d^6*e^3 + 10*b^7*c^3*d^5*e^4 - 16*b^8*c^2*d^4*e^5 + 21*b^9*c*d^ 3*e^6 - 7*b^10*d^2*e^7)*x^2 - 2*(10*b^4*c^6*d^9 - 43*b^5*c^5*d^8*e + 65*b^ 6*c^4*d^7*e^2 - 30*b^7*c^3*d^6*e^3 - 20*b^8*c^2*d^5*e^4 + 25*b^9*c*d^4*...
\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(1/(e*x+d)**2/(c*x**2+b*x)**(7/2),x)
Output:
Integral(1/((x*(b + c*x))**(7/2)*(d + e*x)**2), x)
Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(7/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 2349 vs. \(2 (512) = 1024\).
Time = 0.52 (sec) , antiderivative size = 2349, normalized size of antiderivative = 4.29 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(7/2),x, algorithm="giac")
Output:
1/30*((210*b^6*c^(3/2)*d*e^9*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d* e)*sqrt(c)*abs(e))) - 105*b^7*sqrt(c)*e^10*log(abs(2*c*d*e - b*e^2 - 2*sqr t(c*d^2 - b*d*e)*sqrt(c)*abs(e))) + 1024*sqrt(c*d^2 - b*d*e)*c^7*d^6*e^2*a bs(e) - 3072*sqrt(c*d^2 - b*d*e)*b*c^6*d^5*e^3*abs(e) + 2624*sqrt(c*d^2 - b*d*e)*b^2*c^5*d^4*e^4*abs(e) - 128*sqrt(c*d^2 - b*d*e)*b^3*c^4*d^3*e^5*ab s(e) - 168*sqrt(c*d^2 - b*d*e)*b^4*c^3*d^2*e^6*abs(e) - 280*sqrt(c*d^2 - b *d*e)*b^5*c^2*d*e^7*abs(e) + 210*sqrt(c*d^2 - b*d*e)*b^6*c*e^8*abs(e))*sgn (1/(e*x + d))*sgn(e)/(sqrt(c*d^2 - b*d*e)*b^6*c^(9/2)*d^8*abs(e) - 4*sqrt( c*d^2 - b*d*e)*b^7*c^(7/2)*d^7*e*abs(e) + 6*sqrt(c*d^2 - b*d*e)*b^8*c^(5/2 )*d^6*e^2*abs(e) - 4*sqrt(c*d^2 - b*d*e)*b^9*c^(3/2)*d^5*e^3*abs(e) + sqrt (c*d^2 - b*d*e)*b^10*sqrt(c)*d^4*e^4*abs(e)) - 2*((512*c^9*d^6*e^19 - 1536 *b*c^8*d^5*e^20 + 1312*b^2*c^7*d^4*e^21 - 64*b^3*c^6*d^3*e^22 - 84*b^4*c^5 *d^2*e^23 - 140*b^5*c^4*d*e^24 + 105*b^6*c^3*e^25)/(b^6*c^4*d^8*e^17*sgn(1 /(e*x + d))*sgn(e) - 4*b^7*c^3*d^7*e^18*sgn(1/(e*x + d))*sgn(e) + 6*b^8*c^ 2*d^6*e^19*sgn(1/(e*x + d))*sgn(e) - 4*b^9*c*d^5*e^20*sgn(1/(e*x + d))*sgn (e) + b^10*d^4*e^21*sgn(1/(e*x + d))*sgn(e)) - (5*(512*c^9*d^7*e^20 - 1792 *b*c^8*d^6*e^21 + 2080*b^2*c^7*d^5*e^22 - 720*b^3*c^6*d^4*e^23 - 52*b^4*c^ 5*d^3*e^24 - 98*b^5*c^4*d^2*e^25 + 196*b^6*c^3*d*e^26 - 63*b^7*c^2*e^27)/( b^6*c^4*d^8*e^17*sgn(1/(e*x + d))*sgn(e) - 4*b^7*c^3*d^7*e^18*sgn(1/(e*x + d))*sgn(e) + 6*b^8*c^2*d^6*e^19*sgn(1/(e*x + d))*sgn(e) - 4*b^9*c*d^5*...
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{7/2}\,{\left (d+e\,x\right )}^2} \,d x \] Input:
int(1/((b*x + c*x^2)^(7/2)*(d + e*x)^2),x)
Output:
int(1/((b*x + c*x^2)^(7/2)*(d + e*x)^2), x)
\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (e x +d \right )^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}d x \] Input:
int(1/(e*x+d)^2/(c*x^2+b*x)^(7/2),x)
Output:
int(1/(e*x+d)^2/(c*x^2+b*x)^(7/2),x)