Integrand size = 28, antiderivative size = 462 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {\left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 b d-4 a e)+2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^4 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {4 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^5}+\frac {\left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^5 \left (c d^2-b d e+a e^2\right )^{3/2}} \]
-1/12*(16*c^2*d^3-b*e^2*(-4*a*e+b*d)-4*c*d*e*(-a*e+3*b*d)+3*e*(8*c^2*d^2+b ^2*e^2-4*c*e*(-a*e+2*b*d))*x)*(c*x^2+b*x+a)^(3/2)/e^2/(a*e^2-b*d*e+c*d^2)/ (e*x+d)^3-4*c^(3/2)*(-b*e+2*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+ a)^(1/2))/e^5+1/16*(128*c^4*d^4-b^4*e^4-8*b^2*c*e^3*(-3*a*e+2*b*d)-64*c^3* d^2*e*(-3*a*e+4*b*d)+48*c^2*e^2*(a^2*e^2-4*a*b*d*e+3*b^2*d^2))*arctanh(1/2 *(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2)) /e^5/(a*e^2-b*d*e+c*d^2)^(3/2)+1/8*(64*c^3*d^3+b^3*e^3+4*b*c*e^2*(-5*a*e+4 *b*d)-16*c^2*d*e*(-4*a*e+5*b*d)+2*c*e*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4* b*d))*x)*(c*x^2+b*x+a)^(1/2)/e^4/(a*e^2-b*d*e+c*d^2)/(e*x+d)
Time = 11.84 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.05 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (16 c^3 d^2 \left (12 d^3+30 d^2 e x+22 d e^2 x^2+3 e^3 x^3\right )+b e^3 \left (-8 a^2 e^2+2 a b e (d-7 e x)+b^2 \left (3 d^2+8 d e x-3 e^2 x^2\right )\right )+2 c e^2 \left (-4 a^2 e^2 (d+3 e x)-2 a b e \left (9 d^2+20 d e x+23 e^2 x^2\right )+b^2 d \left (24 d^2+63 d e x+55 e^2 x^2\right )\right )-8 c^2 e \left (-a e \left (20 d^3+51 d^2 e x+41 d e^2 x^2+6 e^3 x^3\right )+b d \left (30 d^3+76 d^2 e x+57 d e^2 x^2+6 e^3 x^3\right )\right )\right )}{\left (c d^2+e (-b d+a e)\right ) (d+e x)^3}-192 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {3 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2}}}{48 e^5} \]
((2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*d^2*(12*d^3 + 30*d^2*e*x + 22*d*e^2*x^ 2 + 3*e^3*x^3) + b*e^3*(-8*a^2*e^2 + 2*a*b*e*(d - 7*e*x) + b^2*(3*d^2 + 8* d*e*x - 3*e^2*x^2)) + 2*c*e^2*(-4*a^2*e^2*(d + 3*e*x) - 2*a*b*e*(9*d^2 + 2 0*d*e*x + 23*e^2*x^2) + b^2*d*(24*d^2 + 63*d*e*x + 55*e^2*x^2)) - 8*c^2*e* (-(a*e*(20*d^3 + 51*d^2*e*x + 41*d*e^2*x^2 + 6*e^3*x^3)) + b*d*(30*d^3 + 7 6*d^2*e*x + 57*d*e^2*x^2 + 6*e^3*x^3))))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^3) - 192*c^(3/2)*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - (3*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2* e^2))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(48*e^ 5)
Time = 0.99 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1229, 27, 1230, 25, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {\int \frac {\left (e^2 b^3+12 c d e b^2-4 c \left (4 c d^2+5 a e^2\right ) b+16 a c^2 d e-2 c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^2}dx}{4 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left (e^2 b^3+12 c d e b^2-4 c \left (4 c d^2+5 a e^2\right ) b+16 a c^2 d e-2 c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {-\frac {\int -\frac {e^3 b^4+16 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+64 c^2 d \left (c d^2+2 a e^2\right ) b-16 a c^2 e \left (4 c d^2+3 a e^2\right )+64 c^2 (2 c d-b e) \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {e^3 b^4+16 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+64 c^2 d \left (c d^2+2 a e^2\right ) b-16 a c^2 e \left (4 c d^2+3 a e^2\right )+64 c^2 (2 c d-b e) \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {\frac {\frac {64 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {\frac {\frac {128 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {64 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {\frac {\frac {2 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {64 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {64 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
-1/12*((16*c^2*d^3 - b*e^2*(b*d - 4*a*e) - 4*c*d*e*(3*b*d - a*e) + 3*e*(8* c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*(a + b*x + c*x^2)^(3/2))/(e^2* (c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (-(((64*c^3*d^3 + b^3*e^3 + 4*b*c*e ^2*(4*b*d - 5*a*e) - 16*c^2*d*e*(5*b*d - 4*a*e) + 2*c*e*(16*c^2*d^2 + b^2* e^2 - 4*c*e*(4*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x))) + ((64*c^(3/2)*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(b + 2*c*x)/(2* Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e - ((128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3 *(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^ 2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e*Sqrt[c*d^2 - b*d*e + a*e^2] ))/(2*e^2))/(8*e^2*(c*d^2 - b*d*e + a*e^2))
3.16.64.3.1 Defintions of rubi rules used
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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2272\) vs. \(2(436)=872\).
Time = 0.86 (sec) , antiderivative size = 2273, normalized size of antiderivative = 4.92
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2273\) |
| default | \(\text {Expression too large to display}\) | \(4950\) |
2*c^2/e^4*(c*x^2+b*x+a)^(1/2)+1/e^4*(4*c^(3/2)/e*(b*e-2*c*d)*ln((1/2*b+c*x )/c^(1/2)+(c*x^2+b*x+a)^(1/2))-4*c*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e ^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c* d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e* (x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+(6*a*b*c*e^3-12*a*c^2*d*e ^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/e^3*(-1/(a*e^2-b*d*e+ c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/ e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2) ^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e +c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/ e^2)^(1/2))/(x+d/e)))+(2*a^2*c*e^4+2*a*b^2*e^4-12*a*b*c*d*e^3+12*a*c^2*d^2 *e^2-2*b^3*d*e^3+12*b^2*c*d^2*e^2-20*b*c^2*d^3*e+10*c^3*d^4)/e^4*(-1/2/(a* e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b *d*e+c*d^2)/e^2)^(1/2)-3/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b* d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d ^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/ e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b *d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d ^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^ 2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 2889 vs. \(2 (436) = 872\).
Time = 4.45 (sec) , antiderivative size = 2889, normalized size of antiderivative = 6.25 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
1/8*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 + 192*a*c^3*d^2*e ^2 - 16*b^3*c*d*e^3 - 192*a*b*c^2*d*e^3 - b^4*e^4 + 24*a*b^2*c*e^4 + 48*a^ 2*c^2*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqr t(-c*d^2 + b*d*e - a*e^2))/((c*d^2*e^5 - b*d*e^6 + a*e^7)*sqrt(-c*d^2 + b* d*e - a*e^2)) + 2*sqrt(c*x^2 + b*x + a)*c^2/e^4 + 1/24*(576*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))^5*c^4*d^4*e^2 - 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^3*d^3*e^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^2*d ^2*e^4 + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^3*d^2*e^4 - 144*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c*d*e^5 - 576*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^5*a*b*c^2*d*e^5 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b ^4*e^6 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c*e^6 + 48*(sqrt( c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^2*e^6 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(9/2)*d^5*e - 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4* b*c^(7/2)*d^4*e^2 + 1584*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2) *d^3*e^3 + 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(7/2)*d^3*e^3 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c^(3/2)*d^2*e^4 - 576*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(5/2)*d^2*e^4 - 33*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))^4*b^4*sqrt(c)*d*e^5 - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(3/2)*d*e^5 - 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^ 2*c^(5/2)*d*e^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*sqrt(c...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \]