Integrand size = 22, antiderivative size = 362 \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\frac {\sqrt {c+b x+a x^2} \left (15 b^6 c+28 a b^4 c^2+400 a^2 b^2 c^3-960 a^3 c^4+15 b^7 x+146 a b^5 c x+200 a^2 b^3 c^2 x-480 a^3 b c^3 x+118 a b^6 x^2-64 a^2 b^4 c x^2+160 a^3 b^2 c^2 x^2+136 a^2 b^5 x^3-80 a^3 b^3 c x^3+48 a^3 b^4 x^4\right )}{192 a b^5 (c+b x)}+\frac {5 \left (-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right ) \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}\right )}{2 b^6}-\frac {5 \left (-b^8+8 a b^6 c-64 a^3 b^2 c^3+128 a^4 c^4\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{128 a^{3/2} b^6}-\frac {5 \left (-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right ) \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+b x+a x^2}\right )}{2 b^6} \]
1/192*(a*x^2+b*x+c)^(1/2)*(48*a^3*b^4*x^4-80*a^3*b^3*c*x^3+136*a^2*b^5*x^3 +160*a^3*b^2*c^2*x^2-64*a^2*b^4*c*x^2+118*a*b^6*x^2-480*a^3*b*c^3*x+200*a^ 2*b^3*c^2*x+146*a*b^5*c*x+15*b^7*x-960*a^3*c^4+400*a^2*b^2*c^3+28*a*b^4*c^ 2+15*b^6*c)/a/b^5/(b*x+c)+5/2*(-a^(3/2)*b^2*c^3+2*a^(5/2)*c^4)*ln(-x*a^(1/ 2)+(a*x^2+b*x+c)^(1/2))/b^6-5/128*(128*a^4*c^4-64*a^3*b^2*c^3+8*a*b^6*c-b^ 8)*ln(b+2*a*x-2*a^(1/2)*(a*x^2+b*x+c)^(1/2))/a^(3/2)/b^6-5/2*(-a^(3/2)*b^2 *c^3+2*a^(5/2)*c^4)*ln(a^(1/2)*(b*x+2*c)-b*(a*x^2+b*x+c)^(1/2))/b^6
Time = 1.64 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\frac {\frac {2 \sqrt {a} b \sqrt {c+x (b+a x)} \left (-960 a^3 c^4+15 b^7 x-480 a^3 b c^3 x-40 a^2 b^3 c x \left (-5 c+2 a x^2\right )+80 a^2 b^2 c^2 \left (5 c+2 a x^2\right )+2 a b^5 x \left (73 c+68 a x^2\right )+b^6 \left (15 c+118 a x^2\right )+4 a b^4 \left (7 c^2-16 a c x^2+12 a^2 x^4\right )\right )}{c+b x}-15 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right ) \text {arctanh}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+x (b+a x)}}\right )+960 a^3 c^3 \left (-b^2+2 a c\right ) \log \left (-\sqrt {a} x+\sqrt {c+x (b+a x)}\right )-960 a^3 c^3 \left (-b^2+2 a c\right ) \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+x (b+a x)}\right )}{384 a^{3/2} b^6} \]
((2*Sqrt[a]*b*Sqrt[c + x*(b + a*x)]*(-960*a^3*c^4 + 15*b^7*x - 480*a^3*b*c ^3*x - 40*a^2*b^3*c*x*(-5*c + 2*a*x^2) + 80*a^2*b^2*c^2*(5*c + 2*a*x^2) + 2*a*b^5*x*(73*c + 68*a*x^2) + b^6*(15*c + 118*a*x^2) + 4*a*b^4*(7*c^2 - 16 *a*c*x^2 + 12*a^2*x^4)))/(c + b*x) - 15*(b^8 - 8*a*b^6*c + 64*a^3*b^2*c^3 - 128*a^4*c^4)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])] + 96 0*a^3*c^3*(-b^2 + 2*a*c)*Log[-(Sqrt[a]*x) + Sqrt[c + x*(b + a*x)]] - 960*a ^3*c^3*(-b^2 + 2*a*c)*Log[Sqrt[a]*(2*c + b*x) - b*Sqrt[c + x*(b + a*x)]])/ (384*a^(3/2)*b^6)
Time = 0.59 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1161, 1231, 25, 27, 1231, 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.
| \(\displaystyle \int \frac {\left (a x^2+b x+c\right )^{5/2}}{(b x+c)^2} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 a x) \left (a x^2+b x+c\right )^{3/2}}{c+b x}dx}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}-\frac {\int -\frac {a \left (b c \left (b^2+4 a c\right )+\left (b^4-4 a c b^2+16 a^2 c^2\right ) x\right ) \sqrt {a x^2+b x+c}}{c+b x}dx}{8 a b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {a \left (b c \left (b^2+4 a c\right )+\left (b^4-4 a c b^2+16 a^2 c^2\right ) x\right ) \sqrt {a x^2+b x+c}}{c+b x}dx}{8 a b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {\left (b c \left (b^2+4 a c\right )+\left (b^4-4 a c b^2+16 a^2 c^2\right ) x\right ) \sqrt {a x^2+b x+c}}{c+b x}dx}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\int \frac {c \left (b^2-8 a c\right ) b^5+\left (b^8-8 a c b^6+64 a^3 c^3 b^2-128 a^4 c^4\right ) x}{2 (c+b x) \sqrt {a x^2+b x+c}}dx}{4 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\int \frac {c \left (b^2-8 a c\right ) b^5+\left (b^8-8 a c b^6+64 a^3 c^3 b^2-128 a^4 c^4\right ) x}{(c+b x) \sqrt {a x^2+b x+c}}dx}{8 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {\left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \int \frac {1}{\sqrt {a x^2+b x+c}}dx}{b}-\frac {64 a^3 c^4 \left (b^2-2 a c\right ) \int \frac {1}{(c+b x) \sqrt {a x^2+b x+c}}dx}{b}}{8 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {2 \left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \int \frac {1}{4 a-\frac {(b+2 a x)^2}{a x^2+b x+c}}d\frac {b+2 a x}{\sqrt {a x^2+b x+c}}}{b}-\frac {64 a^3 c^4 \left (b^2-2 a c\right ) \int \frac {1}{(c+b x) \sqrt {a x^2+b x+c}}dx}{b}}{8 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {\left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} b}-\frac {64 a^3 c^4 \left (b^2-2 a c\right ) \int \frac {1}{(c+b x) \sqrt {a x^2+b x+c}}dx}{b}}{8 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {128 a^3 c^4 \left (b^2-2 a c\right ) \int \frac {1}{4 a c^2-\frac {\left (b c+\left (b^2-2 a c\right ) x\right )^2}{a x^2+b x+c}}d\frac {b c+\left (b^2-2 a c\right ) x}{\sqrt {a x^2+b x+c}}}{b}+\frac {\left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} b}}{8 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {64 a^{5/2} c^3 \left (b^2-2 a c\right ) \text {arctanh}\left (\frac {x \left (b^2-2 a c\right )+b c}{2 \sqrt {a} c \sqrt {a x^2+b x+c}}\right )}{b}+\frac {\left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} b}}{8 a b^2}}{8 b^2}+\frac {\left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{12 b^2}\right )}{2 b}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)}\) |
-((c + b*x + a*x^2)^(5/2)/(b*(c + b*x))) + (5*(((7*b^2 - 8*a*c + 6*a*b*x)* (c + b*x + a*x^2)^(3/2))/(12*b^2) + (((b^6 - 4*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 + 2*a*b*(b^4 - 4*a*b^2*c + 16*a^2*c^2)*x)*Sqrt[c + b*x + a*x^2] )/(4*a*b^2) - (((b^8 - 8*a*b^6*c + 64*a^3*b^2*c^3 - 128*a^4*c^4)*ArcTanh[( b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + b*x + a*x^2])])/(Sqrt[a]*b) + (64*a^(5/2)*c ^3*(b^2 - 2*a*c)*ArcTanh[(b*c + (b^2 - 2*a*c)*x)/(2*Sqrt[a]*c*Sqrt[c + b*x + a*x^2])])/b)/(8*a*b^2))/(8*b^2)))/(2*b)
3.30.59.3.1 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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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]
Time = 0.17 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.44
| method | result | size |
| risch | \(\frac {\left (48 a^{3} x^{3} b^{3}-128 a^{3} b^{2} c \,x^{2}+136 a^{2} b^{4} x^{2}+288 a^{3} b \,c^{2} x -200 a^{2} b^{3} c x +118 a \,b^{5} x -768 a^{3} c^{3}+400 a^{2} b^{2} c^{2}+28 a \,b^{4} c +15 b^{6}\right ) \sqrt {a \,x^{2}+b x +c}}{192 a \,b^{5}}+\frac {\frac {128 a^{4} c^{6} \left (-\frac {b^{2} \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{a \,c^{2} \left (x +\frac {c}{b}\right )}-\frac {\left (2 a c -b^{2}\right ) b \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{2 a \,c^{2} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\right )}{b^{3}}+\frac {384 a^{3} c^{4} \left (2 a c -b^{2}\right ) \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{2} \sqrt {\frac {a \,c^{2}}{b^{2}}}}+\frac {5 \left (128 a^{4} c^{4}-64 a^{3} b^{2} c^{3}+8 a \,b^{6} c -b^{8}\right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{b \sqrt {a}}}{128 a \,b^{5}}\) | \(523\) |
| default | \(\text {Expression too large to display}\) | \(1378\) |
1/192/a*(48*a^3*b^3*x^3-128*a^3*b^2*c*x^2+136*a^2*b^4*x^2+288*a^3*b*c^2*x- 200*a^2*b^3*c*x+118*a*b^5*x-768*a^3*c^3+400*a^2*b^2*c^2+28*a*b^4*c+15*b^6) *(a*x^2+b*x+c)^(1/2)/b^5+1/128/a/b^5*(128/b^3*a^4*c^6*(-1/a/c^2*b^2/(x+c/b )*((x+c/b)^2*a-(2*a*c-b^2)/b*(x+c/b)+a*c^2/b^2)^(1/2)-1/2*(2*a*c-b^2)*b/a/ c^2/(a*c^2/b^2)^(1/2)*ln((2*a*c^2/b^2-(2*a*c-b^2)/b*(x+c/b)+2*(a*c^2/b^2)^ (1/2)*((x+c/b)^2*a-(2*a*c-b^2)/b*(x+c/b)+a*c^2/b^2)^(1/2))/(x+c/b)))+384*a ^3/b^2*c^4*(2*a*c-b^2)/(a*c^2/b^2)^(1/2)*ln((2*a*c^2/b^2-(2*a*c-b^2)/b*(x+ c/b)+2*(a*c^2/b^2)^(1/2)*((x+c/b)^2*a-(2*a*c-b^2)/b*(x+c/b)+a*c^2/b^2)^(1/ 2))/(x+c/b))+5*(128*a^4*c^4-64*a^3*b^2*c^3+8*a*b^6*c-b^8)/b*ln((1/2*b+a*x) /a^(1/2)+(a*x^2+b*x+c)^(1/2))/a^(1/2))
Time = 14.61 (sec) , antiderivative size = 854, normalized size of antiderivative = 2.36 \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\left [-\frac {15 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 64 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5} + {\left (b^{9} - 8 \, a b^{7} c + 64 \, a^{3} b^{3} c^{3} - 128 \, a^{4} b c^{4}\right )} x\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - 4 \, \sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {a} - b^{2} - 4 \, a c\right ) + 960 \, {\left (a^{3} b^{2} c^{4} - 2 \, a^{4} c^{5} + {\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b c^{4}\right )} x\right )} \sqrt {a} \log \left (-\frac {2 \, b^{3} c x + b^{2} c^{2} + 4 \, a c^{3} + {\left (b^{4} - 4 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{2} + 4 \, {\left (b c^{2} + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c} \sqrt {a}}{b^{2} x^{2} + 2 \, b c x + c^{2}}\right ) - 4 \, {\left (48 \, a^{4} b^{5} x^{4} + 15 \, a b^{7} c + 28 \, a^{2} b^{5} c^{2} + 400 \, a^{3} b^{3} c^{3} - 960 \, a^{4} b c^{4} + 8 \, {\left (17 \, a^{3} b^{6} - 10 \, a^{4} b^{4} c\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{7} - 32 \, a^{3} b^{5} c + 80 \, a^{4} b^{3} c^{2}\right )} x^{2} + {\left (15 \, a b^{8} + 146 \, a^{2} b^{6} c + 200 \, a^{3} b^{4} c^{2} - 480 \, a^{4} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{768 \, {\left (a^{2} b^{7} x + a^{2} b^{6} c\right )}}, -\frac {960 \, {\left (a^{3} b^{2} c^{4} - 2 \, a^{4} c^{5} + {\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b c^{4}\right )} x\right )} \sqrt {-a} \arctan \left (-\frac {\sqrt {a x^{2} + b x + c} {\left (b c + {\left (b^{2} - 2 \, a c\right )} x\right )} \sqrt {-a}}{2 \, {\left (a^{2} c x^{2} + a b c x + a c^{2}\right )}}\right ) - 15 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 64 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5} + {\left (b^{9} - 8 \, a b^{7} c + 64 \, a^{3} b^{3} c^{3} - 128 \, a^{4} b c^{4}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {-a}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) - 2 \, {\left (48 \, a^{4} b^{5} x^{4} + 15 \, a b^{7} c + 28 \, a^{2} b^{5} c^{2} + 400 \, a^{3} b^{3} c^{3} - 960 \, a^{4} b c^{4} + 8 \, {\left (17 \, a^{3} b^{6} - 10 \, a^{4} b^{4} c\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{7} - 32 \, a^{3} b^{5} c + 80 \, a^{4} b^{3} c^{2}\right )} x^{2} + {\left (15 \, a b^{8} + 146 \, a^{2} b^{6} c + 200 \, a^{3} b^{4} c^{2} - 480 \, a^{4} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{384 \, {\left (a^{2} b^{7} x + a^{2} b^{6} c\right )}}\right ] \]
[-1/768*(15*(b^8*c - 8*a*b^6*c^2 + 64*a^3*b^2*c^4 - 128*a^4*c^5 + (b^9 - 8 *a*b^7*c + 64*a^3*b^3*c^3 - 128*a^4*b*c^4)*x)*sqrt(a)*log(-8*a^2*x^2 - 8*a *b*x - 4*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sqrt(a) - b^2 - 4*a*c) + 960*(a ^3*b^2*c^4 - 2*a^4*c^5 + (a^3*b^3*c^3 - 2*a^4*b*c^4)*x)*sqrt(a)*log(-(2*b^ 3*c*x + b^2*c^2 + 4*a*c^3 + (b^4 - 4*a*b^2*c + 8*a^2*c^2)*x^2 + 4*(b*c^2 + (b^2*c - 2*a*c^2)*x)*sqrt(a*x^2 + b*x + c)*sqrt(a))/(b^2*x^2 + 2*b*c*x + c^2)) - 4*(48*a^4*b^5*x^4 + 15*a*b^7*c + 28*a^2*b^5*c^2 + 400*a^3*b^3*c^3 - 960*a^4*b*c^4 + 8*(17*a^3*b^6 - 10*a^4*b^4*c)*x^3 + 2*(59*a^2*b^7 - 32*a ^3*b^5*c + 80*a^4*b^3*c^2)*x^2 + (15*a*b^8 + 146*a^2*b^6*c + 200*a^3*b^4*c ^2 - 480*a^4*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^2*b^7*x + a^2*b^6*c), - 1/384*(960*(a^3*b^2*c^4 - 2*a^4*c^5 + (a^3*b^3*c^3 - 2*a^4*b*c^4)*x)*sqrt( -a)*arctan(-1/2*sqrt(a*x^2 + b*x + c)*(b*c + (b^2 - 2*a*c)*x)*sqrt(-a)/(a^ 2*c*x^2 + a*b*c*x + a*c^2)) - 15*(b^8*c - 8*a*b^6*c^2 + 64*a^3*b^2*c^4 - 1 28*a^4*c^5 + (b^9 - 8*a*b^7*c + 64*a^3*b^3*c^3 - 128*a^4*b*c^4)*x)*sqrt(-a )*arctan(1/2*sqrt(a*x^2 + b*x + c)*(2*a*x + b)*sqrt(-a)/(a^2*x^2 + a*b*x + a*c)) - 2*(48*a^4*b^5*x^4 + 15*a*b^7*c + 28*a^2*b^5*c^2 + 400*a^3*b^3*c^3 - 960*a^4*b*c^4 + 8*(17*a^3*b^6 - 10*a^4*b^4*c)*x^3 + 2*(59*a^2*b^7 - 32* a^3*b^5*c + 80*a^4*b^3*c^2)*x^2 + (15*a*b^8 + 146*a^2*b^6*c + 200*a^3*b^4* c^2 - 480*a^4*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^2*b^7*x + a^2*b^6*c)]
\[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\int \frac {\left (a x^{2} + b x + c\right )^{\frac {5}{2}}}{\left (b x + c\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Timed out. \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx=\int \frac {{\left (a\,x^2+b\,x+c\right )}^{5/2}}{{\left (c+b\,x\right )}^2} \,d x \]