Integrand size = 28, antiderivative size = 473 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {5 \left (64 c^3 d^3-b^3 e^3-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)+2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{8 e^6 (d+e x)}-\frac {5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac {(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac {5 \sqrt {c} (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^7}+\frac {5 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^7 \sqrt {c d^2-b d e+a e^2}} \]
-5/12*(16*c^2*d^2+b^2*e^2-4*c*e*(-a*e+3*b*d)+4*c*e*(-b*e+2*c*d)*x)*(c*x^2+ b*x+a)^(3/2)/e^4/(e*x+d)^2+1/3*(2*c*e*x-b*e+4*c*d)*(c*x^2+b*x+a)^(5/2)/e^2 /(e*x+d)^3-5/2*(-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*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/2)/e^7+5/16*(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*(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^7/(a*e^2-b*d*e+c*d^2)^(1/2)+5 /8*(64*c^3*d^3-b^3*e^3-16*c^2*d*e*(-2*a*e+5*b*d)+12*b*c*e^2*(-a*e+2*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/ (e*x+d)
Time = 11.86 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.13 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (16 c^3 \left (60 d^5+150 d^4 e x+110 d^3 e^2 x^2+15 d^2 e^3 x^3-3 d e^4 x^4+e^5 x^5\right )-b e^3 \left (8 a^2 e^2+2 a b e (5 d+13 e x)+b^2 \left (15 d^2+40 d e x+33 e^2 x^2\right )\right )+2 c e^2 \left (-4 a^2 e^2 (d+3 e x)-2 a b e \left (15 d^2+44 d e x+41 e^2 x^2\right )+b^2 \left (180 d^3+465 d^2 e x+361 d e^2 x^2+60 e^3 x^3\right )\right )+8 c^2 e \left (a e \left (40 d^3+105 d^2 e x+83 d e^2 x^2+14 e^3 x^3\right )-b \left (150 d^4+380 d^3 e x+285 d^2 e^2 x^2+42 d e^3 x^3-8 e^4 x^4\right )\right )\right )}{(d+e x)^3}-120 \sqrt {c} (2 c d-b e) \left (8 c^2 d^2+b^2 e^2+4 c e (-2 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {15 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d^2+e (-b d+a e)}}}{48 e^7} \]
((2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*(60*d^5 + 150*d^4*e*x + 110*d^3*e^2*x^ 2 + 15*d^2*e^3*x^3 - 3*d*e^4*x^4 + e^5*x^5) - b*e^3*(8*a^2*e^2 + 2*a*b*e*( 5*d + 13*e*x) + b^2*(15*d^2 + 40*d*e*x + 33*e^2*x^2)) + 2*c*e^2*(-4*a^2*e^ 2*(d + 3*e*x) - 2*a*b*e*(15*d^2 + 44*d*e*x + 41*e^2*x^2) + b^2*(180*d^3 + 465*d^2*e*x + 361*d*e^2*x^2 + 60*e^3*x^3)) + 8*c^2*e*(a*e*(40*d^3 + 105*d^ 2*e*x + 83*d*e^2*x^2 + 14*e^3*x^3) - b*(150*d^4 + 380*d^3*e*x + 285*d^2*e^ 2*x^2 + 42*d*e^3*x^3 - 8*e^4*x^4))))/(d + e*x)^3 - 120*Sqrt[c]*(2*c*d - b* e)*(8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqr t[c]*Sqrt[a + x*(b + c*x)])] - (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*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)])])/Sqrt[c*d^2 + e*(-(b*d) + a* e)])/(48*e^7)
Time = 1.04 (sec) , antiderivative size = 491, 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, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \int \frac {3 \left (-e b^2+4 c d b-4 a c e+4 c (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^3}dx}{18 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \int \frac {\left (-e b^2+4 c d b-4 a c e+4 c (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^3}dx}{6 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}-\frac {3 \int -\frac {2 \left (-e^2 b^3+12 c d e b^2-4 c \left (4 c d^2+3 a e^2\right ) b+16 a c^2 d e-2 c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \int \frac {\left (-e^2 b^3+12 c d e b^2-4 c \left (4 c d^2+3 a e^2\right ) b+16 a c^2 d e-2 c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (-\frac {\int -\frac {-e^3 b^4+24 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+32 c^2 d \left (2 c d^2+3 a e^2\right ) b-16 a c^2 e \left (4 c d^2+a e^2\right )+8 c (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (\frac {\int \frac {-e^3 b^4+24 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+32 c^2 d \left (2 c d^2+3 a e^2\right ) b-16 a c^2 e \left (4 c d^2+a e^2\right )+8 c (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 c (2 c d-b e) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (\frac {\frac {16 c (2 c d-b e) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {\left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )}{e}-\frac {\left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (\frac {\frac {2 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {8 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {3 \left (\frac {\frac {8 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )}{e}-\frac {\left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}\right )}{4 e^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^2 (d+e x)^2}\right )}{6 e^2}\) |
((4*c*d - b*e + 2*c*e*x)*(a + b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) - (5 *(((16*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - a*e) + 4*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) + (3*(-(((64*c^3*d^3 - b^3*e^3 - 16*c^2*d*e*(5*b*d - 2*a*e) + 12*b*c*e^2*(2*b*d - 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])/(e^2*(d + e *x))) + ((8*Sqrt[c]*(2*c*d - b*e)*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a* e))*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*(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*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)))/(4*e^2)))/(6*e^2)
3.16.72.3.1 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[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. \(2649\) vs. \(2(441)=882\).
Time = 0.91 (sec) , antiderivative size = 2650, normalized size of antiderivative = 5.60
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2650\) |
| default | \(\text {Expression too large to display}\) | \(7430\) |
1/3*c*(2*c^2*e^2*x^2+8*b*c*e^2*x-12*c^2*d*e*x+14*a*c*e^2+15*b^2*e^2-66*b*c *d*e+60*c^2*d^2)*(c*x^2+b*x+a)^(1/2)/e^6+1/2/e^6*(5*c^(1/2)*(4*a*b*c*e^3-8 *a*c^2*d*e^2+b^3*e^3-10*b^2*c*d*e^2+24*b*c^2*d^2*e-16*c^3*d^3)/e*ln((1/2*b +c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-(12*a^2*c^2*e^4+24*a*b^2*c*e^4-120*a*b* c^2*d*e^3+120*a*c^3*d^2*e^2+2*b^4*e^4-40*b^3*c*d*e^3+180*b^2*c^2*d^2*e^2-2 80*b*c^3*d^3*e+140*c^4*d^4)/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))+(18*a^2*b*c*e^5-36*a^2*c^2*d*e^4+6*a*b^3*e^5-72*a*b^2*c*d*e^4+180*a*b* c^2*d^2*e^3-120*a*c^3*d^3*e^2-6*b^4*d*e^4+60*b^3*c*d^2*e^3-180*b^2*c^2*d^3 *e^2+210*b*c^3*d^4*e-84*c^4*d^5)/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)) )+(4*a^3*c*e^6+6*a^2*b^2*e^6-36*a^2*b*c*d*e^5+36*a^2*c^2*d^2*e^4-12*a*b^3* d*e^5+72*a*b^2*c*d^2*e^4-120*a*b*c^2*d^3*e^3+60*a*c^3*d^4*e^2+6*b^4*d^2*e^ 4-40*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2-84*b*c^3*d^5*e+28*c^4*d^6)/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...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
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. 3027 vs. \(2 (441) = 882\).
Time = 4.99 (sec) , antiderivative size = 3027, normalized size of antiderivative = 6.40 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
1/3*sqrt(c*x^2 + b*x + a)*(2*x*(c^3*x/e^4 - 2*(3*c^5*d*e^17 - 2*b*c^4*e^18 )/(c^2*e^22)) + (60*c^5*d^2*e^16 - 66*b*c^4*d*e^17 + 15*b^2*c^3*e^18 + 14* a*c^4*e^18)/(c^2*e^22)) + 5/8*(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)*arctan(-((sqrt(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) + 5/2*(16*c^4*d^3 - 24*b*c^3*d^2*e + 10*b^2*c^2*d*e^2 + 8*a* c^3*d*e^2 - b^3*c*e^3 - 4*a*b*c^2*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/(sqrt(c)*e^7) + 1/24*(1440*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^5*c^4*d^4*e^2 - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5* b*c^3*d^3*e^3 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^2*d^2*e^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^3*d^2*e^4 - 480*(sqrt(c)* x - sqrt(c*x^2 + b*x + a))^5*b^3*c*d*e^5 - 960*(sqrt(c)*x - sqrt(c*x^2 + b *x + a))^5*a*b*c^2*d*e^5 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*e^ 6 + 216*(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 + 5184*(sqrt(c)*x - sqrt(c*x^2 + b* x + a))^4*c^(9/2)*d^5*e - 9360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^( 7/2)*d^4*e^2 + 5280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2)*d^3* e^3 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(7/2)*d^3*e^3 - 960*( sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c^(3/2)*d^2*e^4 - 480*(sqrt(c)...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]