Integrand size = 28, antiderivative size = 202 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}+\frac {8 e^2 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {e^2 \left (64 c^2 d^2+15 b^2 e^2-2 c e (27 b d+8 a e)+10 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{3 c^3}+\frac {e (2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}} \]
1/2*e*(-b*e+2*c*d)*(8*c^2*d^2+5*b^2*e^2-4*c*e*(3*a*e+2*b*d))*arctanh(1/2*( 2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)-2*(e*x+d)^4/(c*x^2+b*x+a)^(1 /2)+8/3*e^2*(e*x+d)^2*(c*x^2+b*x+a)^(1/2)/c+1/3*e^2*(64*c^2*d^2+15*b^2*e^2 -2*c*e*(8*a*e+27*b*d)+10*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/c^3
Time = 2.92 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.22 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {15 b^2 e^4 (a+b x)+c^3 \left (-6 d^4-24 d^3 e x+36 d^2 e^2 x^2+12 d e^3 x^3+2 e^4 x^4\right )-c e^3 \left (16 a^2 e+b^2 x (54 d-5 e x)+a b (54 d+26 e x)\right )+2 c^2 e^2 \left (2 a \left (18 d^2+9 d e x-2 e^2 x^2\right )-b x \left (-36 d^2+9 d e x+e^2 x^2\right )\right )}{3 c^3 \sqrt {a+x (b+c x)}}+\frac {e (2 c d-b e) \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{7/2}} \]
(15*b^2*e^4*(a + b*x) + c^3*(-6*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 12*d*e ^3*x^3 + 2*e^4*x^4) - c*e^3*(16*a^2*e + b^2*x*(54*d - 5*e*x) + a*b*(54*d + 26*e*x)) + 2*c^2*e^2*(2*a*(18*d^2 + 9*d*e*x - 2*e^2*x^2) - b*x*(-36*d^2 + 9*d*e*x + e^2*x^2)))/(3*c^3*Sqrt[a + x*(b + c*x)]) + (e*(2*c*d - b*e)*(8* c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/c^(7/2)
Time = 0.42 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1222, 1166, 27, 1225, 1092, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle 8 e \int \frac {(d+e x)^3}{\sqrt {c x^2+b x+a}}dx-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle 8 e \left (\frac {\int \frac {(d+e x) \left (6 c d^2-e (b d+4 a e)+5 e (2 c d-b e) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {e (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}\right )-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 e \left (\frac {\int \frac {(d+e x) \left (6 c d^2-e (b d+4 a e)+5 e (2 c d-b e) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}+\frac {e (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}\right )-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle 8 e \left (\frac {\frac {3 (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}+\frac {e \sqrt {a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{4 c^2}}{6 c}+\frac {e (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}\right )-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle 8 e \left (\frac {\frac {3 (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}+\frac {e \sqrt {a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{4 c^2}}{6 c}+\frac {e (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}\right )-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 8 e \left (\frac {\frac {3 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{8 c^{5/2}}+\frac {e \sqrt {a+b x+c x^2} \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+10 c e x (2 c d-b e)+64 c^2 d^2\right )}{4 c^2}}{6 c}+\frac {e (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c}\right )-\frac {2 (d+e x)^4}{\sqrt {a+b x+c x^2}}\) |
(-2*(d + e*x)^4)/Sqrt[a + b*x + c*x^2] + 8*e*((e*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c) + ((e*(64*c^2*d^2 + 15*b^2*e^2 - 2*c*e*(27*b*d + 8*a*e) + 10*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*c^2) + (3*(2*c*d - b*e)* (8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqr t[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c))
3.16.81.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) Int[(d + e*x)^(m - 1)* (a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(639\) vs. \(2(182)=364\).
Time = 0.93 (sec) , antiderivative size = 640, normalized size of antiderivative = 3.17
| method | result | size |
| risch | \(-\frac {e^{2} \left (-2 c^{2} x^{2} e^{2}+4 b c \,e^{2} x -12 c^{2} d e x +10 a c \,e^{2}-9 b^{2} e^{2}+30 b c d e -36 c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{3 c^{3}}+\frac {-\frac {6 a \,b^{3} e^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {4 b \,d^{4} c^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {12 c \,e^{4} b \,a^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {16 a^{2} c^{2} d \,e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {20 a \,b^{2} c d \,e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {24 a b \,c^{2} d^{2} e^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (12 a b \,c^{2} e^{4}-24 a \,c^{3} d \,e^{3}-5 b^{3} c \,e^{4}+18 b^{2} c^{2} d \,e^{3}-24 b \,c^{3} d^{2} e^{2}+16 c^{4} d^{3} e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (4 a^{2} c^{2} e^{4}+4 a \,b^{2} c \,e^{4}-24 a \,c^{3} d^{2} e^{2}-3 b^{4} e^{4}+10 b^{3} c d \,e^{3}-12 b^{2} c^{2} d^{2} e^{2}+8 b \,c^{3} d^{3} e +4 c^{4} d^{4}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c^{3}}\) | \(640\) |
| default | \(\text {Expression too large to display}\) | \(1362\) |
-1/3*e^2*(-2*c^2*e^2*x^2+4*b*c*e^2*x-12*c^2*d*e*x+10*a*c*e^2-9*b^2*e^2+30* b*c*d*e-36*c^2*d^2)*(c*x^2+b*x+a)^(1/2)/c^3+1/2/c^3*(-6*a*b^3*e^4*(2*c*x+b )/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+4*b*d^4*c^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2 +b*x+a)^(1/2)+12*c*e^4*b*a^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-16* a^2*c^2*d*e^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+20*a*b^2*c*d*e^3*( 2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-24*a*b*c^2*d^2*e^2*(2*c*x+b)/(4*a *c-b^2)/(c*x^2+b*x+a)^(1/2)+(12*a*b*c^2*e^4-24*a*c^3*d*e^3-5*b^3*c*e^4+18* b^2*c^2*d*e^3-24*b*c^3*d^2*e^2+16*c^4*d^3*e)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2 *b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/ 2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(4*a^2*c^2*e^4+ 4*a*b^2*c*e^4-24*a*c^3*d^2*e^2-3*b^4*e^4+10*b^3*c*d*e^3-12*b^2*c^2*d^2*e^2 +8*b*c^3*d^3*e+4*c^4*d^4)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b ^2)/(c*x^2+b*x+a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (182) = 364\).
Time = 0.58 (sec) , antiderivative size = 965, normalized size of antiderivative = 4.78 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (16 \, a c^{3} d^{3} e - 24 \, a b c^{2} d^{2} e^{2} + 6 \, {\left (3 \, a b^{2} c - 4 \, a^{2} c^{2}\right )} d e^{3} - {\left (5 \, a b^{3} - 12 \, a^{2} b c\right )} e^{4} + {\left (16 \, c^{4} d^{3} e - 24 \, b c^{3} d^{2} e^{2} + 6 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{3} - {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} e^{4}\right )} x^{2} + {\left (16 \, b c^{3} d^{3} e - 24 \, b^{2} c^{2} d^{2} e^{2} + 6 \, {\left (3 \, b^{3} c - 4 \, a b c^{2}\right )} d e^{3} - {\left (5 \, b^{4} - 12 \, a b^{2} c\right )} e^{4}\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, c^{4} e^{4} x^{4} - 6 \, c^{4} d^{4} + 72 \, a c^{3} d^{2} e^{2} - 54 \, a b c^{2} d e^{3} + {\left (15 \, a b^{2} c - 16 \, a^{2} c^{2}\right )} e^{4} + 2 \, {\left (6 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + {\left (36 \, c^{4} d^{2} e^{2} - 18 \, b c^{3} d e^{3} + {\left (5 \, b^{2} c^{2} - 8 \, a c^{3}\right )} e^{4}\right )} x^{2} - {\left (24 \, c^{4} d^{3} e - 72 \, b c^{3} d^{2} e^{2} + 18 \, {\left (3 \, b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{3} - {\left (15 \, b^{3} c - 26 \, a b c^{2}\right )} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{12 \, {\left (c^{5} x^{2} + b c^{4} x + a c^{4}\right )}}, -\frac {3 \, {\left (16 \, a c^{3} d^{3} e - 24 \, a b c^{2} d^{2} e^{2} + 6 \, {\left (3 \, a b^{2} c - 4 \, a^{2} c^{2}\right )} d e^{3} - {\left (5 \, a b^{3} - 12 \, a^{2} b c\right )} e^{4} + {\left (16 \, c^{4} d^{3} e - 24 \, b c^{3} d^{2} e^{2} + 6 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{3} - {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} e^{4}\right )} x^{2} + {\left (16 \, b c^{3} d^{3} e - 24 \, b^{2} c^{2} d^{2} e^{2} + 6 \, {\left (3 \, b^{3} c - 4 \, a b c^{2}\right )} d e^{3} - {\left (5 \, b^{4} - 12 \, a b^{2} c\right )} e^{4}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (2 \, c^{4} e^{4} x^{4} - 6 \, c^{4} d^{4} + 72 \, a c^{3} d^{2} e^{2} - 54 \, a b c^{2} d e^{3} + {\left (15 \, a b^{2} c - 16 \, a^{2} c^{2}\right )} e^{4} + 2 \, {\left (6 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + {\left (36 \, c^{4} d^{2} e^{2} - 18 \, b c^{3} d e^{3} + {\left (5 \, b^{2} c^{2} - 8 \, a c^{3}\right )} e^{4}\right )} x^{2} - {\left (24 \, c^{4} d^{3} e - 72 \, b c^{3} d^{2} e^{2} + 18 \, {\left (3 \, b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{3} - {\left (15 \, b^{3} c - 26 \, a b c^{2}\right )} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{6 \, {\left (c^{5} x^{2} + b c^{4} x + a c^{4}\right )}}\right ] \]
[1/12*(3*(16*a*c^3*d^3*e - 24*a*b*c^2*d^2*e^2 + 6*(3*a*b^2*c - 4*a^2*c^2)* d*e^3 - (5*a*b^3 - 12*a^2*b*c)*e^4 + (16*c^4*d^3*e - 24*b*c^3*d^2*e^2 + 6* (3*b^2*c^2 - 4*a*c^3)*d*e^3 - (5*b^3*c - 12*a*b*c^2)*e^4)*x^2 + (16*b*c^3* d^3*e - 24*b^2*c^2*d^2*e^2 + 6*(3*b^3*c - 4*a*b*c^2)*d*e^3 - (5*b^4 - 12*a *b^2*c)*e^4)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b* x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(2*c^4*e^4*x^4 - 6*c^4*d^4 + 72*a* c^3*d^2*e^2 - 54*a*b*c^2*d*e^3 + (15*a*b^2*c - 16*a^2*c^2)*e^4 + 2*(6*c^4* d*e^3 - b*c^3*e^4)*x^3 + (36*c^4*d^2*e^2 - 18*b*c^3*d*e^3 + (5*b^2*c^2 - 8 *a*c^3)*e^4)*x^2 - (24*c^4*d^3*e - 72*b*c^3*d^2*e^2 + 18*(3*b^2*c^2 - 2*a* c^3)*d*e^3 - (15*b^3*c - 26*a*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c^5*x ^2 + b*c^4*x + a*c^4), -1/6*(3*(16*a*c^3*d^3*e - 24*a*b*c^2*d^2*e^2 + 6*(3 *a*b^2*c - 4*a^2*c^2)*d*e^3 - (5*a*b^3 - 12*a^2*b*c)*e^4 + (16*c^4*d^3*e - 24*b*c^3*d^2*e^2 + 6*(3*b^2*c^2 - 4*a*c^3)*d*e^3 - (5*b^3*c - 12*a*b*c^2) *e^4)*x^2 + (16*b*c^3*d^3*e - 24*b^2*c^2*d^2*e^2 + 6*(3*b^3*c - 4*a*b*c^2) *d*e^3 - (5*b^4 - 12*a*b^2*c)*e^4)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(2*c^4*e^4*x^4 - 6 *c^4*d^4 + 72*a*c^3*d^2*e^2 - 54*a*b*c^2*d*e^3 + (15*a*b^2*c - 16*a^2*c^2) *e^4 + 2*(6*c^4*d*e^3 - b*c^3*e^4)*x^3 + (36*c^4*d^2*e^2 - 18*b*c^3*d*e^3 + (5*b^2*c^2 - 8*a*c^3)*e^4)*x^2 - (24*c^4*d^3*e - 72*b*c^3*d^2*e^2 + 18*( 3*b^2*c^2 - 2*a*c^3)*d*e^3 - (15*b^3*c - 26*a*b*c^2)*e^4)*x)*sqrt(c*x^2...
\[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (b + 2 c x\right ) \left (d + e x\right )^{4}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (182) = 364\).
Time = 0.30 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.84 \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, {\left (\frac {{\left (b^{2} c^{3} e^{4} - 4 \, a c^{4} e^{4}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac {6 \, b^{2} c^{3} d e^{3} - 24 \, a c^{4} d e^{3} - b^{3} c^{2} e^{4} + 4 \, a b c^{3} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac {36 \, b^{2} c^{3} d^{2} e^{2} - 144 \, a c^{4} d^{2} e^{2} - 18 \, b^{3} c^{2} d e^{3} + 72 \, a b c^{3} d e^{3} + 5 \, b^{4} c e^{4} - 28 \, a b^{2} c^{2} e^{4} + 32 \, a^{2} c^{3} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {24 \, b^{2} c^{3} d^{3} e - 96 \, a c^{4} d^{3} e - 72 \, b^{3} c^{2} d^{2} e^{2} + 288 \, a b c^{3} d^{2} e^{2} + 54 \, b^{4} c d e^{3} - 252 \, a b^{2} c^{2} d e^{3} + 144 \, a^{2} c^{3} d e^{3} - 15 \, b^{5} e^{4} + 86 \, a b^{3} c e^{4} - 104 \, a^{2} b c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {6 \, b^{2} c^{3} d^{4} - 24 \, a c^{4} d^{4} - 72 \, a b^{2} c^{2} d^{2} e^{2} + 288 \, a^{2} c^{3} d^{2} e^{2} + 54 \, a b^{3} c d e^{3} - 216 \, a^{2} b c^{2} d e^{3} - 15 \, a b^{4} e^{4} + 76 \, a^{2} b^{2} c e^{4} - 64 \, a^{3} c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}}{3 \, \sqrt {c x^{2} + b x + a}} - \frac {{\left (16 \, c^{3} d^{3} e - 24 \, b c^{2} d^{2} e^{2} + 18 \, b^{2} c d e^{3} - 24 \, a c^{2} d e^{3} - 5 \, b^{3} e^{4} + 12 \, a b c e^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {7}{2}}} \]
1/3*(((2*((b^2*c^3*e^4 - 4*a*c^4*e^4)*x/(b^2*c^3 - 4*a*c^4) + (6*b^2*c^3*d *e^3 - 24*a*c^4*d*e^3 - b^3*c^2*e^4 + 4*a*b*c^3*e^4)/(b^2*c^3 - 4*a*c^4))* x + (36*b^2*c^3*d^2*e^2 - 144*a*c^4*d^2*e^2 - 18*b^3*c^2*d*e^3 + 72*a*b*c^ 3*d*e^3 + 5*b^4*c*e^4 - 28*a*b^2*c^2*e^4 + 32*a^2*c^3*e^4)/(b^2*c^3 - 4*a* c^4))*x - (24*b^2*c^3*d^3*e - 96*a*c^4*d^3*e - 72*b^3*c^2*d^2*e^2 + 288*a* b*c^3*d^2*e^2 + 54*b^4*c*d*e^3 - 252*a*b^2*c^2*d*e^3 + 144*a^2*c^3*d*e^3 - 15*b^5*e^4 + 86*a*b^3*c*e^4 - 104*a^2*b*c^2*e^4)/(b^2*c^3 - 4*a*c^4))*x - (6*b^2*c^3*d^4 - 24*a*c^4*d^4 - 72*a*b^2*c^2*d^2*e^2 + 288*a^2*c^3*d^2*e^ 2 + 54*a*b^3*c*d*e^3 - 216*a^2*b*c^2*d*e^3 - 15*a*b^4*e^4 + 76*a^2*b^2*c*e ^4 - 64*a^3*c^2*e^4)/(b^2*c^3 - 4*a*c^4))/sqrt(c*x^2 + b*x + a) - 1/2*(16* c^3*d^3*e - 24*b*c^2*d^2*e^2 + 18*b^2*c*d*e^3 - 24*a*c^2*d*e^3 - 5*b^3*e^4 + 12*a*b*c*e^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b ))/c^(7/2)
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]