Integrand size = 29, antiderivative size = 431 \[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt {a+b x+c x^2}}{24 c^3 e^3}+\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}-\frac {g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2} e^4}+\frac {(e f-d g)^4 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^4 \sqrt {c d^2-b d e+a e^2}} \]
-1/16*g*(5*b^3*e^3*g^3-6*b*c*e^2*g^2*(2*a*e*g-b*d*g+4*b*e*f)-16*c^3*(-d^3* g^3+4*d^2*e*f*g^2-6*d*e^2*f^2*g+4*e^3*f^3)+8*c^2*e*g*(a*e*g*(-d*g+4*e*f)+b *(d^2*g^2-4*d*e*f*g+6*e^2*f^2)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+ a)^(1/2))/c^(7/2)/e^4+(-d*g+e*f)^4*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^4/(a*e^2-b*d*e+c*d^2)^(1/ 2)+1/24*g^2*(15*b^2*e^2*g^2-4*c*e*g*(4*a*e*g-7*b*d*g+18*b*e*f)+4*c^2*(11*d ^2*g^2-36*d*e*f*g+36*e^2*f^2))*(c*x^2+b*x+a)^(1/2)/c^3/e^3+1/12*g^3*(-5*b* e*g-14*c*d*g+24*c*e*f)*(e*x+d)*(c*x^2+b*x+a)^(1/2)/c^2/e^3+1/3*g^4*(e*x+d) ^2*(c*x^2+b*x+a)^(1/2)/c/e^3
Time = 1.98 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.88 \[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {2 e g^2 \sqrt {a+x (b+c x)} \left (15 b^2 e^2 g^2-2 c e g (8 a e g+b (36 e f-9 d g+5 e g x))+4 c^2 \left (6 d^2 g^2-3 d e g (8 f+g x)+2 e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )\right )}{c^3}+\frac {96 \sqrt {-c d^2+b d e-a e^2} (e f-d g)^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {3 g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{7/2}}}{48 e^4} \]
((2*e*g^2*Sqrt[a + x*(b + c*x)]*(15*b^2*e^2*g^2 - 2*c*e*g*(8*a*e*g + b*(36 *e*f - 9*d*g + 5*e*g*x)) + 4*c^2*(6*d^2*g^2 - 3*d*e*g*(8*f + g*x) + 2*e^2* (18*f^2 + 6*f*g*x + g^2*x^2))))/c^3 + (96*Sqrt[-(c*d^2) + b*d*e - a*e^2]*( e*f - d*g)^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c *d^2) + e*(b*d - a*e)]])/(c*d^2 + e*(-(b*d) + a*e)) + (3*g*(5*b^3*e^3*g^3 - 6*b*c*e^2*g^2*(4*b*e*f - b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6*d*e^2* f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*g) + b*(6*e ^2*f^2 - 4*d*e*f*g + d^2*g^2)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(7/2))/(48*e^4)
Time = 1.73 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1267, 27, 2184, 27, 2184, 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.
| \(\displaystyle \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle \frac {\int \frac {e^3 g^3 (24 c e f-14 c d g-5 b e g) x^3-e^2 g^2 \left (e (11 b d+4 a e) g^2-c \left (36 e^2 f^2-10 d^2 g^2\right )\right ) x^2-e g \left (d e (7 b d+8 a e) g^3-c \left (24 e^3 f^3-2 d^3 g^3\right )\right ) x+e \left (6 c e^3 f^4-d^2 (b d+4 a e) g^4\right )}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{3 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {e^3 g^3 (24 c e f-14 c d g-5 b e g) x^3-e^2 g^2 \left (e (11 b d+4 a e) g^2-c \left (36 e^2 f^2-10 d^2 g^2\right )\right ) x^2-e g \left (d e (7 b d+8 a e) g^3-c \left (24 e^3 f^3-2 d^3 g^3\right )\right ) x+e \left (6 c e^3 f^4-d^2 (b d+4 a e) g^4\right )}{(d+e x) \sqrt {c x^2+b x+a}}dx}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\frac {\int \frac {g^2 \left (4 \left (36 e^2 f^2-36 d e g f+11 d^2 g^2\right ) c^2-4 e g (18 b e f-7 b d g+4 a e g) c+15 b^2 e^2 g^2\right ) x^2 e^5+\left (24 c^2 e^3 f^4+5 b d e (b d+2 a e) g^4-2 c d g^3 (b d (12 e f-5 d g)+6 a e (4 e f-d g))\right ) e^4+2 g \left (5 b e^2 (2 b d+a e) g^3-c e (b d (48 e f-19 d g)+2 a e (12 e f+d g)) g^2+2 c^2 \left (24 e^3 f^3-12 d^2 e g^2 f+5 d^3 g^3\right )\right ) x e^4}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{2 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {g^2 \left (4 \left (36 e^2 f^2-36 d e g f+11 d^2 g^2\right ) c^2-4 e g (18 b e f-7 b d g+4 a e g) c+15 b^2 e^2 g^2\right ) x^2 e^5+\left (24 c^2 e^3 f^4+5 b d e (b d+2 a e) g^4-2 c d g^3 (b d (12 e f-5 d g)+6 a e (4 e f-d g))\right ) e^4+2 g \left (5 b e^2 (2 b d+a e) g^3-c e (b d (48 e f-19 d g)+2 a e (12 e f+d g)) g^2+2 c^2 \left (24 e^3 f^3-12 d^2 e g^2 f+5 d^3 g^3\right )\right ) x e^4}{(d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 e^6 \left (16 c^3 e^3 f^4-5 b^3 d e^2 g^4+6 b c d e g^3 (4 b e f-b d g+2 a e g)-8 c^2 d g^2 \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e g f+d^2 g^2\right )\right )-g \left (-16 \left (4 e^3 f^3-6 d e^2 g f^2+4 d^2 e g^2 f-d^3 g^3\right ) c^3+8 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e g f+d^2 g^2\right )\right ) c^2-6 b e^2 g^2 (4 b e f-b d g+2 a e g) c+5 b^3 e^3 g^3\right ) x\right )}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{c e^2}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 e^4 \int \frac {16 c^3 e^3 f^4-5 b^3 d e^2 g^4+6 b c d e g^3 (4 b e f-b d g+2 a e g)-8 c^2 d g^2 \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e g f+d^2 g^2\right )\right )-g \left (-16 \left (4 e^3 f^3-6 d e^2 g f^2+4 d^2 e g^2 f-d^3 g^3\right ) c^3+8 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e g f+d^2 g^2\right )\right ) c^2-6 b e^2 g^2 (4 b e f-b d g+2 a e g) c+5 b^3 e^3 g^3\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 c}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\frac {3 e^4 \left (\frac {16 c^3 (e f-d g)^4 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {g \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{2 c}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\frac {3 e^4 \left (\frac {16 c^3 (e f-d g)^4 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 g \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}\right )}{2 c}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 e^4 \left (\frac {16 c^3 (e f-d g)^4 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{\sqrt {c} e}\right )}{2 c}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\frac {3 e^4 \left (-\frac {32 c^3 (e f-d g)^4 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{\sqrt {c} e}\right )}{2 c}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 e^4 \left (\frac {16 c^3 (e f-d g)^4 \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}-\frac {g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{\sqrt {c} e}\right )}{2 c}+\frac {e^4 g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{c}}{4 c e^3}+\frac {e g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{2 c}}{6 c e^4}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}\) |
(g^4*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c*e^3) + ((e*g^3*(24*c*e*f - 14 *c*d*g - 5*b*e*g)*(d + e*x)*Sqrt[a + b*x + c*x^2])/(2*c) + ((e^4*g^2*(15*b ^2*e^2*g^2 - 4*c*e*g*(18*b*e*f - 7*b*d*g + 4*a*e*g) + 4*c^2*(36*e^2*f^2 - 36*d*e*f*g + 11*d^2*g^2))*Sqrt[a + b*x + c*x^2])/c + (3*e^4*(-((g*(5*b^3*e ^3*g^3 - 6*b*c*e^2*g^2*(4*b*e*f - b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6 *d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt [a + b*x + c*x^2])])/(Sqrt[c]*e)) + (16*c^3*(e*f - d*g)^4*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*c))/(4*c*e^3))/(6*c*e^4)
3.9.70.3.1 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_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
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]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 1.10 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(-\frac {g^{2} \left (-8 c^{2} e^{2} g^{2} x^{2}+10 b c \,e^{2} g^{2} x +12 c^{2} d e \,g^{2} x -48 c^{2} e^{2} f g x +16 a c \,e^{2} g^{2}-15 b^{2} e^{2} g^{2}-18 b c d e \,g^{2}+72 b c \,e^{2} f g -24 c^{2} d^{2} g^{2}+96 c^{2} d e f g -144 c^{2} e^{2} f^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{3} e^{3}}+\frac {-\frac {16 \left (g^{4} d^{4}-4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) c^{3} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {g \left (12 a b c \,e^{3} g^{3}+8 a \,c^{2} d \,e^{2} g^{3}-32 a \,c^{2} e^{3} f \,g^{2}-5 b^{3} e^{3} g^{3}-6 b^{2} c d \,e^{2} g^{3}+24 b^{2} c \,e^{3} f \,g^{2}-8 b \,c^{2} d^{2} e \,g^{3}+32 b \,c^{2} d \,e^{2} f \,g^{2}-48 b \,c^{2} e^{3} f^{2} g -16 c^{3} d^{3} g^{3}+64 c^{3} d^{2} e f \,g^{2}-96 c^{3} d \,e^{2} f^{2} g +64 c^{3} e^{3} f^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}}{16 c^{3} e^{3}}\) | \(556\) |
| default | \(-\frac {\left (g^{4} d^{4}-4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {g \left (\frac {d^{3} g^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {4 e^{3} f^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-e^{3} g^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )-\frac {4 d^{2} e f \,g^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {6 d \,e^{2} f^{2} g \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\left (d \,e^{2} g^{3}-4 e^{3} f \,g^{2}\right ) \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\left (-d^{2} e \,g^{3}+4 d \,e^{2} f \,g^{2}-6 e^{3} f^{2} g \right ) \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )\right )}{e^{4}}\) | \(755\) |
-1/24*g^2*(-8*c^2*e^2*g^2*x^2+10*b*c*e^2*g^2*x+12*c^2*d*e*g^2*x-48*c^2*e^2 *f*g*x+16*a*c*e^2*g^2-15*b^2*e^2*g^2-18*b*c*d*e*g^2+72*b*c*e^2*f*g-24*c^2* d^2*g^2+96*c^2*d*e*f*g-144*c^2*e^2*f^2)*(c*x^2+b*x+a)^(1/2)/c^3/e^3+1/16/c ^3/e^3*(-16*(d^4*g^4-4*d^3*e*f*g^3+6*d^2*e^2*f^2*g^2-4*d*e^3*f^3*g+e^4*f^4 )*c^3/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))+g*(12*a*b*c*e^3*g^ 3+8*a*c^2*d*e^2*g^3-32*a*c^2*e^3*f*g^2-5*b^3*e^3*g^3-6*b^2*c*d*e^2*g^3+24* b^2*c*e^3*f*g^2-8*b*c^2*d^2*e*g^3+32*b*c^2*d*e^2*f*g^2-48*b*c^2*e^3*f^2*g- 16*c^3*d^3*g^3+64*c^3*d^2*e*f*g^2-96*c^3*d*e^2*f^2*g+64*c^3*e^3*f^3)/e*ln( (1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2))
Timed out. \[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{4}}{\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as sume?` for
Exception generated. \[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^4}{\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]