Integrand size = 28, antiderivative size = 309 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {3 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{4 e^4 (d+e x)}+\frac {(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}+\frac {3 \sqrt {c} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 e^5}-\frac {3 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^5 \sqrt {c d^2-b d e+a e^2}} \]
1/2*(2*c*e*x-b*e+4*c*d)*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)^2+3/4*(16*c^2*d^2+ 3*b^2*e^2-4*c*e*(-a*e+4*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^ (1/2))*c^(1/2)/e^5-3/8*(-b*e+2*c*d)*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*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))/e^5/(a*e^2-b*d*e+c*d^2)^(1/2)-3/4*(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)^(1/2)/e^4/(e*x+d)
Time = 11.47 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.76 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\frac {(2 c d-b e) (a+x (b+c x))^{5/2}}{(d+e x)^2}-\frac {\left (12 c^2 d^2+b^2 e^2+4 c e (-3 b d+2 a e)\right ) (a+x (b+c x))^{5/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)}-\frac {6 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (b^3 e^3+8 c^3 d^2 (-2 d+e x)+b c e^2 (-13 b d+8 a e+b e x)+4 c^2 e (b d (7 d-2 e x)+a e (-3 d+e x))\right )+2 e^3 (a+x (b+c x))^{3/2} \left (b^3 e^3-4 c^3 d^2 (4 d-3 e x)+b c e^2 (-15 b d+10 a e+b e x)+2 c^2 e (3 b d (5 d-2 e x)+2 a e (-3 d+2 e x))\right )+6 \sqrt {c} \left (16 c^2 d^2+3 b^2 e^2+4 c e (-4 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+3 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^{3/2} \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 e^5 \left (-c d^2+e (b d-a e)\right )}}{2 \left (c d^2+e (-b d+a e)\right )} \]
(((2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 - ((12*c^2*d^2 + b^2* e^2 + 4*c*e*(-3*b*d + 2*a*e))*(a + x*(b + c*x))^(5/2))/(2*(c*d^2 + e*(-(b* d) + a*e))*(d + e*x)) - (6*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c* x)]*(b^3*e^3 + 8*c^3*d^2*(-2*d + e*x) + b*c*e^2*(-13*b*d + 8*a*e + b*e*x) + 4*c^2*e*(b*d*(7*d - 2*e*x) + a*e*(-3*d + e*x))) + 2*e^3*(a + x*(b + c*x) )^(3/2)*(b^3*e^3 - 4*c^3*d^2*(4*d - 3*e*x) + b*c*e^2*(-15*b*d + 10*a*e + b *e*x) + 2*c^2*e*(3*b*d*(5*d - 2*e*x) + 2*a*e*(-3*d + 2*e*x))) + 6*Sqrt[c]* (16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e)) ^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 3*(2*c*d - b*e )*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*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)])])/(4*e^5*(-(c*d^2) + e*(b*d - a*e))))/( 2*(c*d^2 + e*(-(b*d) + a*e)))
Time = 0.69 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {2 \left (-e b^2+4 c d b-4 a c e+4 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {\left (-e b^2+4 c d b-4 a c e+4 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{4 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}-\frac {\int -\frac {-e^2 b^3+12 c d e b^2-4 c \left (4 c d^2+3 a e^2\right ) b+16 a c^2 d e-2 c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\int \frac {-e^2 b^3+12 c d e b^2-4 c \left (4 c d^2+3 a e^2\right ) b+16 a c^2 d e-2 c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 c \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {4 c \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {-\frac {2 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\) |
((4*c*d - b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) - (3 *(((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)*Sq rt[a + b*x + c*x^2])/(e^2*(d + e*x)) + ((-2*Sqrt[c]*(16*c^2*d^2 + 3*b^2*e^ 2 - 4*c*e*(4*b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x ^2])])/e + ((2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*A rcTanh[(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)
3.16.63.3.1 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. \(1316\) vs. \(2(281)=562\).
Time = 0.87 (sec) , antiderivative size = 1317, normalized size of antiderivative = 4.26
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1317\) |
| default | \(\text {Expression too large to display}\) | \(2961\) |
1/2*c*(2*c*e*x+7*b*e-12*c*d)*(c*x^2+b*x+a)^(1/2)/e^4+1/4/e^4*(3*c^(1/2)*(4 *a*c*e^2+3*b^2*e^2-16*b*c*d*e+16*c^2*d^2)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+ b*x+a)^(1/2))-(24*a*b*c*e^3-48*a*c^2*d*e^2+4*b^3*e^3-48*b^2*c*d*e^2+120*b* c^2*d^2*e-80*c^3*d^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))+(8 *a^2*c*e^4+8*a*b^2*e^4-48*a*b*c*d*e^3+48*a*c^2*d^2*e^2-8*b^3*d*e^3+48*b^2* c*d^2*e^2-80*b*c^2*d^3*e+40*c^3*d^4)/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*(4*a^2*b*e^5-8*a^2*c*d*e^4-8*a*b^2*d*e^4+24*a*b*c*d^2*e^3-16*a *c^2*d^3*e^2+4*b^3*d^2*e^3-16*b^2*c*d^3*e^2+20*b*c^2*d^4*e-8*c^3*d^5)*(-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*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...
Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (281) = 562\).
Time = 128.20 (sec) , antiderivative size = 3955, normalized size of antiderivative = 12.80 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
[1/16*(6*(16*c^3*d^6 - 32*b*c^2*d^5*e + (19*b^2*c + 20*a*c^2)*d^4*e^2 - (3 *b^3 + 20*a*b*c)*d^3*e^3 + (3*a*b^2 + 4*a^2*c)*d^2*e^4 + (16*c^3*d^4*e^2 - 32*b*c^2*d^3*e^3 + (19*b^2*c + 20*a*c^2)*d^2*e^4 - (3*b^3 + 20*a*b*c)*d*e ^5 + (3*a*b^2 + 4*a^2*c)*e^6)*x^2 + 2*(16*c^3*d^5*e - 32*b*c^2*d^4*e^2 + ( 19*b^2*c + 20*a*c^2)*d^3*e^3 - (3*b^3 + 20*a*b*c)*d^2*e^4 + (3*a*b^2 + 4*a ^2*c)*d*e^5)*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) - 3*(32*c^3*d^5 - 48*b*c^2*d^4*e + 6*( 3*b^2*c + 4*a*c^2)*d^3*e^2 - (b^3 + 12*a*b*c)*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5)*x^2 + 2*(32*c^3*d^4*e - 48*b*c^2*d^3*e^2 + 6*(3*b^2*c + 4*a*c^2)*d^2*e^3 - (b ^3 + 12*a*b*c)*d*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^ 2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^ 2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2* c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^ 2 + 2*d*e*x + d^2)) - 4*(48*c^3*d^5*e - 84*b*c^2*d^4*e^2 + 2*a^2*b*e^6 + 1 3*(3*b^2*c + 4*a*c^2)*d^3*e^3 - (3*b^3 + 38*a*b*c)*d^2*e^4 + (a*b^2 + 4*a^ 2*c)*d*e^5 - 4*(c^3*d^2*e^4 - b*c^2*d*e^5 + a*c^2*e^6)*x^3 + 2*(8*c^3*d^3* e^3 - 15*b*c^2*d^2*e^4 - 7*a*b*c*e^6 + (7*b^2*c + 8*a*c^2)*d*e^5)*x^2 + (7 2*c^3*d^4*e^2 - 128*b*c^2*d^3*e^3 + (61*b^2*c + 80*a*c^2)*d^2*e^4 - (5*b^3 + 64*a*b*c)*d*e^5 + (5*a*b^2 + 8*a^2*c)*e^6)*x)*sqrt(c*x^2 + b*x + a))...
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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. 1210 vs. \(2 (281) = 562\).
Time = 0.56 (sec) , antiderivative size = 1210, normalized size of antiderivative = 3.92 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
1/2*sqrt(c*x^2 + b*x + a)*(2*c^2*x/e^3 - (12*c^3*d*e^8 - 7*b*c^2*e^9)/(c*e ^12)) - 3/4*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 24*a*c^2*d*e^2 - b^3*e^3 - 12*a*b*c*e^3)*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 ^5) - 3/4*(16*c^3*d^2 - 16*b*c^2*d*e + 3*b^2*c*e^2 + 4*a*c^2*e^2)*log(abs( -2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/(sqrt(c)*e^5) - 1/4*( 64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^3*d^3*e - 96*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))^3*b*c^2*d^2*e^2 + 42*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^3*b^2*c*d*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^2*d*e^3 - 5* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c*e^4 + 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(7/ 2)*d^4 - 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(5/2)*d^3*e + 30*(s qrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(3/2)*d^2*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(5/2)*d^2*e^2 + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*sqrt(c)*d*e^3 + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b* c^(3/2)*d*e^3 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*sqrt(c)*e^4 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(3/2)*e^4 + 112*(sqrt(c) *x - sqrt(c*x^2 + b*x + a))*b*c^3*d^4 - 136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^2*d^3*e - 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^3*d^3*e + 42*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c*d^2*e^2 + 216*(sqrt(c)*x...
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \]