Integrand size = 25, antiderivative size = 326 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {2 \left (b^2-a c\right ) \sqrt {d+e x}}{c^3}-\frac {2 (c d+b e) (d+e x)^{3/2}}{3 c^2 e^2}+\frac {2 (d+e x)^{5/2}}{5 c e^2}+\frac {\left (b^3-3 a b c-\sqrt {b^2-4 a c} \left (b^2-a c\right )\right ) \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{7/2} \sqrt {b^2-4 a c}}-\frac {\left (b^3-3 a b c+\sqrt {b^2-4 a c} \left (b^2-a c\right )\right ) \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{7/2} \sqrt {b^2-4 a c}} \]
-2/3*(b*e+c*d)*(e*x+d)^(3/2)/c^2/e^2+2/5*(e*x+d)^(5/2)/c/e^2+2*(-a*c+b^2)* (e*x+d)^(1/2)/c^3+1/2*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(- 4*a*c+b^2)^(1/2)))^(1/2))*(b^3-3*a*b*c-(-a*c+b^2)*(-4*a*c+b^2)^(1/2))*(2*c *d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/c^(7/2)*2^(1/2)/(-4*a*c+b^2)^(1/2)-1/2* arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/ 2))*(b^3-3*a*b*c+(-a*c+b^2)*(-4*a*c+b^2)^(1/2))*(2*c*d-e*(b+(-4*a*c+b^2)^( 1/2)))^(1/2)/c^(7/2)*2^(1/2)/(-4*a*c+b^2)^(1/2)
Time = 1.37 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.43 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2+c^2 \left (-2 d^2+d e x+3 e^2 x^2\right )-5 c e (3 a e+b (d+e x))\right )}{e^2}-\frac {15 \sqrt {2} \left (-b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 c \left (-\sqrt {b^2-4 a c} d+4 a e\right )+b^3 \left (c d+\sqrt {b^2-4 a c} e\right )-a b c \left (3 c d+2 \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )-b^2 c \left (\sqrt {b^2-4 a c} d+4 a e\right )+a b c \left (3 c d-2 \sqrt {b^2-4 a c} e\right )+b^3 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 c^{7/2}} \]
((2*Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2 + c^2*(-2*d^2 + d*e*x + 3*e^2*x^2) - 5*c*e*(3*a*e + b*(d + e*x))))/e^2 - (15*Sqrt[2]*(-(b^4*e) + a*c^2*(Sqrt[b ^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*ArcTan[(S qrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/( Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) - (15*Sqrt[2]* (b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*Sqrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqr t[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a* c])*e]))/(15*c^(7/2))
Time = 4.16 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1199, 2009}
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.
| \(\displaystyle \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {2 \int \left (\frac {(d+e x)^2}{c e}-\frac {(c d+b e) (d+e x)}{c^2 e}+\frac {\left (b^2-a c\right ) e}{c^3}-\frac {\left (b^2-a c\right ) \left (c d^2-b e d+a e^2\right )-\left (-e b^3+c d b^2+2 a c e b-a c^2 d\right ) (d+e x)}{c^3 e \left (\frac {c (d+e x)^2}{e^2}-\frac {(2 c d-b e) (d+e x)}{e^2}+a+\frac {d (c d-b e)}{e^2}\right )}\right )d\sqrt {d+e x}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {e \left (-\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {e \left (\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {e \left (b^2-a c\right ) \sqrt {d+e x}}{c^3}-\frac {(d+e x)^{3/2} (b e+c d)}{3 c^2 e}+\frac {(d+e x)^{5/2}}{5 c e}\right )}{e}\) |
(2*(((b^2 - a*c)*e*Sqrt[d + e*x])/c^3 - ((c*d + b*e)*(d + e*x)^(3/2))/(3*c ^2*e) + (d + e*x)^(5/2)/(5*c*e) - (e*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c* e - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^ 2 - 4*a*c])*e]])/(Sqrt[2]*c^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (e*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^ 4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[ c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(7/ 2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/e
3.6.26.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Time = 0.74 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.55
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (\left (\left (-\frac {1}{2} b^{3}+a b c \right ) e -\frac {c d \left (a c -b^{2}\right )}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{2} b^{4}\right ) e +\frac {3 d b \left (a c -\frac {b^{2}}{3}\right ) c}{2}\right )\right ) \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e^{2} \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\left (\left (\frac {1}{2} b^{3}-a b c \right ) e +\frac {c d \left (a c -b^{2}\right )}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{2} b^{4}\right ) e +\frac {3 d b \left (a c -\frac {b^{2}}{3}\right ) c}{2}\right )\right ) \sqrt {2}\, e^{2} \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (-\frac {c^{2} x^{2}}{5}+\left (\frac {b x}{3}+a \right ) c -b^{2}\right ) e^{2}+\frac {d \left (-\frac {c x}{5}+b \right ) c e}{3}+\frac {2 c^{2} d^{2}}{15}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\right )\right )}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c^{3} e^{2}}\) | \(504\) |
| risch | \(-\frac {2 \left (-3 c^{2} x^{2} e^{2}+5 b c \,e^{2} x -c^{2} d e x +15 a c \,e^{2}-15 b^{2} e^{2}+5 b c d e +2 c^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{2} c^{3}}+\frac {-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{2}}\) | \(552\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e^{2} \left (-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e^{2}}\) | \(559\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {5}{2}} c^{2}}{5}+\frac {b c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+a c \,e^{2} \sqrt {e x +d}-b^{2} e^{2} \sqrt {e x +d}\right )}{c^{3}}+\frac {8 e^{2} \left (-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c^{2}}}{e^{2}}\) | \(559\) |
-2/(-4*e^2*(a*c-1/4*b^2))^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))* c)^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((((-1/2*b^3+ a*b*c)*e-1/2*c*d*(a*c-b^2))*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*((a^2*c^2-2*a*b ^2*c+1/2*b^4)*e+3/2*d*b*(a*c-1/3*b^2)*c))*2^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c -1/4*b^2))^(1/2))*c)^(1/2)*e^2*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c* d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^ 2))^(1/2))*c)^(1/2)*((((1/2*b^3-a*b*c)*e+1/2*c*d*(a*c-b^2))*(-4*e^2*(a*c-1 /4*b^2))^(1/2)+e*((a^2*c^2-2*a*b^2*c+1/2*b^4)*e+3/2*d*b*(a*c-1/3*b^2)*c))* 2^(1/2)*e^2*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^ 2))^(1/2))*c)^(1/2))+((-1/5*c^2*x^2+(1/3*b*x+a)*c-b^2)*e^2+1/3*d*(-1/5*c*x +b)*c*e+2/15*c^2*d^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)*((b*e-2*c*d+(-4*e^2*(a* c-1/4*b^2))^(1/2))*c)^(1/2)*(e*x+d)^(1/2)))/c^3/e^2
Leaf count of result is larger than twice the leaf count of optimal. 4245 vs. \(2 (273) = 546\).
Time = 0.95 (sec) , antiderivative size = 4245, normalized size of antiderivative = 13.02 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \]
1/30*(15*sqrt(2)*c^3*e^2*sqrt(((b^6*c - 6*a*b^4*c^2 + 9*a^2*b^2*c^3 - 2*a^ 3*c^4)*d - (b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*e + (b^2*c^7 - 4*a*c^8)*sqrt(((b^10*c^2 - 8*a*b^8*c^3 + 22*a^2*b^6*c^4 - 24*a^3*b^4*c^5 + 9*a^4*b^2*c^6)*d^2 - 2*(b^11*c - 9*a*b^9*c^2 + 29*a^2*b^7*c^3 - 40*a^3*b ^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e + (b^12 - 10*a*b^10*c + 37*a^2* b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*e^2) /(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log(sqrt(2)*((b^9*c - 9*a*b^ 7*c^2 + 27*a^2*b^5*c^3 - 31*a^3*b^3*c^4 + 12*a^4*b*c^5)*d - (b^10 - 10*a*b ^8*c + 35*a^2*b^6*c^2 - 51*a^3*b^4*c^3 + 29*a^4*b^2*c^4 - 4*a^5*c^5)*e - ( b^5*c^7 - 7*a*b^3*c^8 + 12*a^2*b*c^9)*sqrt(((b^10*c^2 - 8*a*b^8*c^3 + 22*a ^2*b^6*c^4 - 24*a^3*b^4*c^5 + 9*a^4*b^2*c^6)*d^2 - 2*(b^11*c - 9*a*b^9*c^2 + 29*a^2*b^7*c^3 - 40*a^3*b^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e + ( b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12 *a^5*b^2*c^5 + a^6*c^6)*e^2)/(b^2*c^14 - 4*a*c^15)))*sqrt(((b^6*c - 6*a*b^ 4*c^2 + 9*a^2*b^2*c^3 - 2*a^3*c^4)*d - (b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*e + (b^2*c^7 - 4*a*c^8)*sqrt(((b^10*c^2 - 8*a*b^8*c^3 + 22*a ^2*b^6*c^4 - 24*a^3*b^4*c^5 + 9*a^4*b^2*c^6)*d^2 - 2*(b^11*c - 9*a*b^9*c^2 + 29*a^2*b^7*c^3 - 40*a^3*b^5*c^4 + 22*a^4*b^3*c^5 - 3*a^5*b*c^6)*d*e + ( b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12 *a^5*b^2*c^5 + a^6*c^6)*e^2)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8...
\[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{3} \sqrt {d + e x}}{a + b x + c x^{2}}\, dx \]
\[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d} x^{3}}{c x^{2} + b x + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (273) = 546\).
Time = 0.38 (sec) , antiderivative size = 1074, normalized size of antiderivative = 3.29 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \]
1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^4*c - 5*a*b^2*c^ 2 + 4*a^2*c^3)*d - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e)*c^2*e^2 - 2*((b^2*c^ 3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4*a*c)*d *e + (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^3*c^4 - 3*a*b*c^5)*d^2*e - (3*b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d*e^2 + (b^5*c^2 - 4*a*b^3*c^3 + 2* a^2*b*c^4)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2 *sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c^6*d*e^12 - b*c^5*e^13 + sqrt(-4*(c^6*d ^2*e^12 - b*c^5*d*e^13 + a*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13 )^2))/(c^6*e^12)))/((sqrt(b^2 - 4*a*c)*c^6*d^2 - sqrt(b^2 - 4*a*c)*b*c^5*d *e + sqrt(b^2 - 4*a*c)*a*c^5*e^2)*c^2*abs(e)) - 1/4*(sqrt(-4*c^2*d + 2*(b* c + sqrt(b^2 - 4*a*c)*c)*e)*((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e)*c^2*e^2 + 2*((b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a* c)*d^2 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c^2 - a^2*c^3) *sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*a bs(c)*abs(e) + (2*(b^3*c^4 - 3*a*b*c^5)*d^2*e - (3*b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d*e^2 + (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*e^3)*sqrt(-4*c^ 2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/s qrt(-(2*c^6*d*e^12 - b*c^5*e^13 - sqrt(-4*(c^6*d^2*e^12 - b*c^5*d*e^13 + a *c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13)^2))/(c^6*e^12)))/((sq...
Time = 14.18 (sec) , antiderivative size = 11143, normalized size of antiderivative = 34.18 \[ \int \frac {x^3 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \]
atan(((((8*(4*a^3*c^6*e^4 + a*b^4*c^4*e^4 - b^5*c^4*d*e^3 - 5*a^2*b^2*c^5* e^4 + 4*a^2*c^7*d^2*e^2 + b^4*c^5*d^2*e^2 + 5*a*b^3*c^5*d*e^3 - 4*a^2*b*c^ 6*d*e^3 - 5*a*b^2*c^6*d^2*e^2))/c^5 - (8*(d + e*x)^(1/2)*(-(b^9*e - 8*a^4* c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a ^3*b^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4*e + b^5*c*d*( -(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c ^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6 *a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^9 + b^4*c^7 - 8*a*b^ 2*c^8)))^(1/2)*(b^3*c^7*e^3 - 2*b^2*c^8*d*e^2 - 4*a*b*c^8*e^3 + 8*a*c^9*d* e^2))/c^5)*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c *d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c ^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d* (-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(1 6*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (8*(d + e*x)^(1/2)*(b^8*e^4 + 2*a^4*c^4*e^4 + 20*a^2*b^4*c^2*e^4 - 16*a^3*b^2*c^3*e^4 - 2*a^3*c^5*d^2*e ^2 + b^6*c^2*d^2*e^2 - 8*a*b^6*c*e^4 - 2*b^7*c*d*e^3 + 9*a^2*b^2*c^4*d^2*e ^2 + 14*a*b^5*c^2*d*e^3 + 14*a^3*b*c^4*d*e^3 - 6*a*b^4*c^3*d^2*e^2 - 28...