Integrand size = 28, antiderivative size = 464 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=-\frac {15 \left (64 c^3 d^3-7 b^3 e^3+4 b c e^2 (14 b d-5 a e)-16 c^2 d e (7 b d-2 a e)-2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{32 e^6}-\frac {5 \left (8 c^2 d^2+b^2 e^2-c e (7 b d-2 a e)+c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{4 e^4 (d+e x)}+\frac {(3 c d-b e+c e x) \left (a+b x+c x^2\right )^{5/2}}{2 e^2 (d+e x)^2}+\frac {15 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 \sqrt {c} e^7}-\frac {15 (2 c d-b e) \sqrt {c d^2-b d e+a e^2} \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^7} \]
-5/4*(8*c^2*d^2+b^2*e^2-c*e*(-2*a*e+7*b*d)+c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+ a)^(3/2)/e^4/(e*x+d)+1/2*(c*e*x-b*e+3*c*d)*(c*x^2+b*x+a)^(5/2)/e^2/(e*x+d) ^2+15/64*(128*c^4*d^4+b^4*e^4-8*b^2*c*e^3*(-3*a*e+4*b*d)-128*c^3*d^2*e*(-a *e+2*b*d)+16*c^2*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2))*arctanh(1/2*(2*c*x+b) /c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^7/c^(1/2)-15/8*(-b*e+2*c*d)*(8*c^2*d^2+b^2 *e^2-4*c*e*(-a*e+2*b*d))*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))*(a*e^2-b*d*e+c*d^2)^(1/2)/e^7-15/32*( 64*c^3*d^3-7*b^3*e^3+4*b*c*e^2*(-5*a*e+14*b*d)-16*c^2*d*e*(-2*a*e+7*b*d)-2 *c*e*(16*c^2*d^2+3*b^2*e^2-4*c*e*(-a*e+4*b*d))*x)*(c*x^2+b*x+a)^(1/2)/e^6
Time = 11.41 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.15 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {-\frac {2 e \sqrt {a+x (b+c x)} \left (16 c^3 \left (60 d^5+90 d^4 e x+20 d^3 e^2 x^2-5 d^2 e^3 x^3+2 d e^4 x^4-e^5 x^5\right )+b e^3 \left (16 a^2 e^2+8 a b e (5 d+9 e x)-b^2 \left (105 d^2+170 d e x+49 e^2 x^2\right )\right )+2 c e^2 \left (16 a^2 e^2 (d+2 e x)-2 a b e \left (145 d^2+234 d e x+65 e^2 x^2\right )+b^2 \left (420 d^3+655 d^2 e x+166 d e^2 x^2-37 e^3 x^3\right )\right )-8 c^2 e \left (a e \left (-100 d^3-155 d^2 e x-38 d e^2 x^2+9 e^3 x^3\right )+b \left (210 d^4+320 d^3 e x+75 d^2 e^2 x^2-18 d e^3 x^3+7 e^4 x^4\right )\right )\right )}{(d+e x)^2}+\frac {15 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+120 (2 c d-b e) \left (8 c^2 d^2+b^2 e^2+4 c e (-2 b d+a e)\right ) \sqrt {c d^2+e (-b d+a e)} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{64 e^7} \]
((-2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*(60*d^5 + 90*d^4*e*x + 20*d^3*e^2*x^2 - 5*d^2*e^3*x^3 + 2*d*e^4*x^4 - e^5*x^5) + b*e^3*(16*a^2*e^2 + 8*a*b*e*(5 *d + 9*e*x) - b^2*(105*d^2 + 170*d*e*x + 49*e^2*x^2)) + 2*c*e^2*(16*a^2*e^ 2*(d + 2*e*x) - 2*a*b*e*(145*d^2 + 234*d*e*x + 65*e^2*x^2) + b^2*(420*d^3 + 655*d^2*e*x + 166*d*e^2*x^2 - 37*e^3*x^3)) - 8*c^2*e*(a*e*(-100*d^3 - 15 5*d^2*e*x - 38*d*e^2*x^2 + 9*e^3*x^3) + b*(210*d^4 + 320*d^3*e*x + 75*d^2* e^2*x^2 - 18*d*e^3*x^3 + 7*e^4*x^4))))/(d + e*x)^2 + (15*(128*c^4*d^4 + b^ 4*e^4 - 8*b^2*c*e^3*(4*b*d - 3*a*e) - 128*c^3*d^2*e*(2*b*d - a*e) + 16*c^2 *e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq rt[a + x*(b + c*x)])])/Sqrt[c] + 120*(2*c*d - b*e)*(8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*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)])])/(64*e^7)
Time = 1.19 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1230, 27, 1230, 27, 1231, 27, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \int \frac {4 \left (-e b^2+3 c d b-2 a c e+3 c (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^2}dx}{16 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \int \frac {\left (-e b^2+3 c d b-2 a c e+3 c (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^2}dx}{4 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}-\frac {\int -\frac {3 \left (-e^2 b^3+7 c d e b^2-4 c \left (2 c d^2+a e^2\right ) b+4 a c^2 d e-c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 e^2}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \int \frac {\left (-e^2 b^3+7 c d e b^2-4 c \left (2 c d^2+a e^2\right ) b+4 a c^2 d e-c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}-\frac {\int -\frac {c \left (7 d e^3 b^4-8 \left (a e^4+7 c d^2 e^2\right ) b^3+8 c d e \left (14 c d^2+11 a e^2\right ) b^2-32 c \left (2 c^2 d^4+5 a c e^2 d^2+a^2 e^4\right ) b+16 a c^2 d e \left (4 c d^2+3 a e^2\right )-\left (128 c^4 d^4-128 c^3 e (2 b d-a e) d^2+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b e d+a^2 e^2\right )\right ) x\right )}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {\int \frac {7 d e^3 b^4-8 \left (a e^4+7 c d^2 e^2\right ) b^3+8 c d e \left (14 c d^2+11 a e^2\right ) b^2-32 c \left (2 c^2 d^4+5 a c e^2 d^2+a^2 e^4\right ) b+16 a c^2 d e \left (4 c d^2+3 a e^2\right )-\left (128 c^4 d^4-128 c^3 e (2 b d-a e) d^2+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b e d+a^2 e^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{8 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {-\frac {16 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 (2 c d-b e) \sqrt {a e^2-b d e+c d^2} \left (4 a c e^2+b^2 e^2-8 b c d e+8 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{4 e^2}\right )}{2 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
((3*c*d - b*e + c*e*x)*(a + b*x + c*x^2)^(5/2))/(2*e^2*(d + e*x)^2) - (5*( ((8*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 2*a*e) + c*e*(2*c*d - b*e)*x)*(a + b* x + c*x^2)^(3/2))/(e^2*(d + e*x)) + (3*(((64*c^3*d^3 - 7*b^3*e^3 + 4*b*c*e ^2*(14*b*d - 5*a*e) - 16*c^2*d*e*(7*b*d - 2*a*e) - 2*c*e*(16*c^2*d^2 + 3*b ^2*e^2 - 4*c*e*(4*b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(4*e^2) + (-(((128 *c^4*d^4 + b^4*e^4 - 8*b^2*c*e^3*(4*b*d - 3*a*e) - 128*c^3*d^2*e*(2*b*d - a*e) + 16*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/ (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e)) + (8*(2*c*d - b*e)*Sqrt[c *d^2 - b*d*e + a*e^2]*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2 + 4*a*c*e^2)*ArcTan h[(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)/(8*e^2)))/(2*e^2)))/(4*e^2)
3.16.71.3.1 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)*(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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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. \(1702\) vs. \(2(432)=864\).
Time = 0.92 (sec) , antiderivative size = 1703, normalized size of antiderivative = 3.67
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1703\) |
| default | \(\text {Expression too large to display}\) | \(4474\) |
1/32*(16*c^3*e^3*x^3+56*b*c^2*e^3*x^2-64*c^3*d*e^2*x^2+72*a*c^2*e^3*x+74*b ^2*c*e^3*x-256*b*c^2*d*e^2*x+192*c^3*d^2*e*x+260*a*b*c*e^3-448*a*c^2*d*e^2 +49*b^3*e^3-480*b^2*c*d*e^2+1056*b*c^2*d^2*e-640*c^3*d^3)*(c*x^2+b*x+a)^(1 /2)/e^6+1/64/e^6*(15*(16*a^2*c^2*e^4+24*a*b^2*c*e^4-128*a*b*c^2*d*e^3+128* a*c^3*d^2*e^2+b^4*e^4-32*b^3*c*d*e^3+160*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+1 28*c^4*d^4)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-(576*a^2 *b*c*e^5-1152*a^2*c^2*d*e^4+192*a*b^3*e^5-2304*a*b^2*c*d*e^4+5760*a*b*c^2* d^2*e^3-3840*a*c^3*d^3*e^2-192*b^4*d*e^4+1920*b^3*c*d^2*e^3-5760*b^2*c^2*d ^3*e^2+6720*b*c^3*d^4*e-2688*c^4*d^5)/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))+(128*a^3*c*e^6+192*a^2*b^2*e^6-1152*a^2*b*c*d*e^5+1152*a^2*c ^2*d^2*e^4-384*a*b^3*d*e^5+2304*a*b^2*c*d^2*e^4-3840*a*b*c^2*d^3*e^3+1920* a*c^3*d^4*e^2+192*b^4*d^2*e^4-1280*b^3*c*d^3*e^3+2880*b^2*c^2*d^4*e^2-2688 *b*c^3*d^5*e+896*c^4*d^6)/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)))+1/e^4 *(64*a^3*b*e^7-128*a^3*c*d*e^6-192*a^2*b^2*d*e^6+576*a^2*b*c*d^2*e^5-38...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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. 2125 vs. \(2 (432) = 864\).
Time = 0.58 (sec) , antiderivative size = 2125, normalized size of antiderivative = 4.58 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
1/32*sqrt(c*x^2 + b*x + a)*(2*(4*(2*c^3*x/e^3 - (8*c^6*d*e^21 - 7*b*c^5*e^ 22)/(c^3*e^25))*x + (96*c^6*d^2*e^20 - 128*b*c^5*d*e^21 + 37*b^2*c^4*e^22 + 36*a*c^5*e^22)/(c^3*e^25))*x - (640*c^6*d^3*e^19 - 1056*b*c^5*d^2*e^20 + 480*b^2*c^4*d*e^21 + 448*a*c^5*d*e^21 - 49*b^3*c^3*e^22 - 260*a*b*c^4*e^2 2)/(c^3*e^25)) - 15/4*(16*c^4*d^5 - 40*b*c^3*d^4*e + 34*b^2*c^2*d^3*e^2 + 24*a*c^3*d^3*e^2 - 11*b^3*c*d^2*e^3 - 36*a*b*c^2*d^2*e^3 + b^4*d*e^4 + 14* a*b^2*c*d*e^4 + 8*a^2*c^2*d*e^4 - a*b^3*e^5 - 4*a^2*b*c*e^5)*arctan(-((sqr t(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2 ))/(sqrt(-c*d^2 + b*d*e - a*e^2)*e^7) - 15/64*(128*c^4*d^4 - 256*b*c^3*d^3 *e + 160*b^2*c^2*d^2*e^2 + 128*a*c^3*d^2*e^2 - 32*b^3*c*d*e^3 - 128*a*b*c^ 2*d*e^3 + b^4*e^4 + 24*a*b^2*c*e^4 + 16*a^2*c^2*e^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/(sqrt(c)*e^7) - 1/4*(96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^5*e - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e^2 + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d ^3*e^3 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*d^3*e^3 - 75*(sqr t(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c*d^2*e^4 - 180*(sqrt(c)*x - sqrt(c* x^2 + b*x + a))^3*a*b*c^2*d^2*e^4 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 3*b^4*d*e^5 + 78*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d*e^5 + 24* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d*e^5 - 9*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))^3*a*b^3*e^6 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]