Integrand size = 22, antiderivative size = 334 \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\frac {\sqrt {c+b x+a x^2} \left (-45 b^8+390 a b^6 c+24 a^2 b^4 c^2+160 a^3 b^2 c^3+1920 a^4 c^4+30 a b^7 x+1308 a^2 b^5 c x-80 a^3 b^3 c^2 x-960 a^4 b c^3 x+744 a^2 b^6 x^2+48 a^3 b^4 c x^2+640 a^4 b^2 c^2 x^2+1008 a^3 b^5 x^3-480 a^4 b^3 c x^3+384 a^4 b^4 x^4\right )}{1920 a^2 b^5}-\frac {a^{5/2} c^5 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}\right )}{b^6}+\frac {\left (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{256 a^{5/2} b^6}+\frac {a^{5/2} c^5 \log \left (2 \sqrt {a} c+\sqrt {a} b x-b \sqrt {c+b x+a x^2}\right )}{b^6} \]
1/1920*(a*x^2+b*x+c)^(1/2)*(384*a^4*b^4*x^4-480*a^4*b^3*c*x^3+1008*a^3*b^5 *x^3+640*a^4*b^2*c^2*x^2+48*a^3*b^4*c*x^2+744*a^2*b^6*x^2-960*a^4*b*c^3*x- 80*a^3*b^3*c^2*x+1308*a^2*b^5*c*x+30*a*b^7*x+1920*a^4*c^4+160*a^3*b^2*c^3+ 24*a^2*b^4*c^2+390*a*b^6*c-45*b^8)/a^2/b^5-a^(5/2)*c^5*ln(-x*a^(1/2)+(a*x^ 2+b*x+c)^(1/2))/b^6+1/256*(256*a^5*c^5-80*a^2*b^6*c^2+30*a*b^8*c-3*b^10)*l n(b+2*a*x-2*a^(1/2)*(a*x^2+b*x+c)^(1/2))/a^(5/2)/b^6+a^(5/2)*c^5*ln(2*a^(1 /2)*c+a^(1/2)*b*x-b*(a*x^2+b*x+c)^(1/2))/b^6
Time = 1.49 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\frac {2 \sqrt {a} b \sqrt {c+x (b+a x)} \left (-45 b^8+1920 a^4 c^4+30 a b^7 x-960 a^4 b c^3 x+160 a^3 b^2 c^2 \left (c+4 a x^2\right )-80 a^3 b^3 c x \left (c+6 a x^2\right )+12 a^2 b^5 x \left (109 c+84 a x^2\right )+6 a b^6 \left (65 c+124 a x^2\right )+24 a^2 b^4 \left (c^2+2 a c x^2+16 a^2 x^4\right )\right )-3840 a^5 c^5 \log \left (-\sqrt {a} x+\sqrt {c+x (b+a x)}\right )-15 \left (3 b^{10}-30 a b^8 c+80 a^2 b^6 c^2-256 a^5 c^5\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )+3840 a^5 c^5 \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+x (b+a x)}\right )}{3840 a^{5/2} b^6} \]
(2*Sqrt[a]*b*Sqrt[c + x*(b + a*x)]*(-45*b^8 + 1920*a^4*c^4 + 30*a*b^7*x - 960*a^4*b*c^3*x + 160*a^3*b^2*c^2*(c + 4*a*x^2) - 80*a^3*b^3*c*x*(c + 6*a* x^2) + 12*a^2*b^5*x*(109*c + 84*a*x^2) + 6*a*b^6*(65*c + 124*a*x^2) + 24*a ^2*b^4*(c^2 + 2*a*c*x^2 + 16*a^2*x^4)) - 3840*a^5*c^5*Log[-(Sqrt[a]*x) + S qrt[c + x*(b + a*x)]] - 15*(3*b^10 - 30*a*b^8*c + 80*a^2*b^6*c^2 - 256*a^5 *c^5)*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + x*(b + a*x)]] + 3840*a^5*c^5*Log[ Sqrt[a]*(2*c + b*x) - b*Sqrt[c + x*(b + a*x)]])/(3840*a^(5/2)*b^6)
Time = 0.63 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1162, 25, 1231, 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.
| \(\displaystyle \int \frac {\left (a x^2+b x+c\right )^{5/2}}{b x+c} \, dx\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}-\frac {\int -\frac {\left (b c+\left (b^2-2 a c\right ) x\right ) \left (a x^2+b x+c\right )^{3/2}}{c+b x}dx}{2 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b c+\left (b^2-2 a c\right ) x\right ) \left (a x^2+b x+c\right )^{3/2}}{c+b x}dx}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\int \frac {\left (b c \left (3 b^4-18 a c b^2+8 a^2 c^2\right )+\left (b^2-2 a c\right ) \left (3 b^4-12 a c b^2-16 a^2 c^2\right ) x\right ) \sqrt {a x^2+b x+c}}{2 (c+b x)}dx}{8 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\int \frac {\left (b c \left (3 b^4-18 a c b^2+8 a^2 c^2\right )+\left (b^2-2 a c\right ) \left (3 b^4-12 a c b^2-16 a^2 c^2\right ) x\right ) \sqrt {a x^2+b x+c}}{c+b x}dx}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\int \frac {c \left (3 b^4-30 a c b^2+80 a^2 c^2\right ) b^5+\left (3 b^{10}-30 a c b^8+80 a^2 c^2 b^6-256 a^5 c^5\right ) x}{2 (c+b x) \sqrt {a x^2+b x+c}}dx}{4 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\int \frac {c \left (3 b^4-30 a c b^2+80 a^2 c^2\right ) b^5+\left (3 b^{10}-30 a c b^8+80 a^2 c^2 b^6-256 a^5 c^5\right ) x}{(c+b x) \sqrt {a x^2+b x+c}}dx}{8 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {256 a^5 c^6 \int \frac {1}{(c+b x) \sqrt {a x^2+b x+c}}dx}{b}+\frac {\left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \int \frac {1}{\sqrt {a x^2+b x+c}}dx}{b}}{8 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {256 a^5 c^6 \int \frac {1}{(c+b x) \sqrt {a x^2+b x+c}}dx}{b}+\frac {2 \left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \int \frac {1}{4 a-\frac {(b+2 a x)^2}{a x^2+b x+c}}d\frac {b+2 a x}{\sqrt {a x^2+b x+c}}}{b}}{8 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {256 a^5 c^6 \int \frac {1}{(c+b x) \sqrt {a x^2+b x+c}}dx}{b}+\frac {\left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} b}}{8 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {\left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} b}-\frac {512 a^5 c^6 \int \frac {1}{4 a c^2-\frac {\left (b c+\left (b^2-2 a c\right ) x\right )^2}{a x^2+b x+c}}d\frac {b c+\left (b^2-2 a c\right ) x}{\sqrt {a x^2+b x+c}}}{b}}{8 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{24 a b^2}-\frac {\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{4 a b^2}-\frac {\frac {\left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a} b}-\frac {256 a^{9/2} c^5 \text {arctanh}\left (\frac {x \left (b^2-2 a c\right )+b c}{2 \sqrt {a} c \sqrt {a x^2+b x+c}}\right )}{b}}{8 a b^2}}{16 a b^2}}{2 b}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b}\) |
(c + b*x + a*x^2)^(5/2)/(5*b) + (((3*b^4 - 6*a*b^2*c + 16*a^2*c^2 + 6*a*b* (b^2 - 2*a*c)*x)*(c + b*x + a*x^2)^(3/2))/(24*a*b^2) - (((3*b^8 - 18*a*b^6 *c + 8*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 2*a*b*(b^2 - 2*a*c)*(3 *b^4 - 12*a*b^2*c - 16*a^2*c^2)*x)*Sqrt[c + b*x + a*x^2])/(4*a*b^2) - (((3 *b^10 - 30*a*b^8*c + 80*a^2*b^6*c^2 - 256*a^5*c^5)*ArcTanh[(b + 2*a*x)/(2* Sqrt[a]*Sqrt[c + b*x + a*x^2])])/(Sqrt[a]*b) - (256*a^(9/2)*c^5*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))/ (16*a*b^2))/(2*b)
3.30.21.3.1 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 + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 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 = 0.40 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {\sqrt {a \,x^{2}+b x +c}\, \left (384 a^{4} b^{4} x^{4}-480 a^{4} b^{3} c \,x^{3}+1008 a^{3} b^{5} x^{3}+640 a^{4} b^{2} c^{2} x^{2}+48 a^{3} b^{4} c \,x^{2}+744 a^{2} b^{6} x^{2}-960 a^{4} b \,c^{3} x -80 a^{3} b^{3} c^{2} x +1308 a^{2} b^{5} c x +30 a \,b^{7} x +1920 a^{4} c^{4}+160 a^{3} b^{2} c^{3}+24 a^{2} b^{4} c^{2}+390 a \,b^{6} c -45 b^{8}\right )}{1920 a^{2} b^{5}}-\frac {\frac {256 a^{5} c^{6} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {\left (x +\frac {c}{b}\right )^{2} a -\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{2} \sqrt {\frac {a \,c^{2}}{b^{2}}}}+\frac {\left (256 a^{5} c^{5}-80 a^{2} b^{6} c^{2}+30 a \,b^{8} c -3 b^{10}\right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{b \sqrt {a}}}{256 a^{2} b^{5}}\) | \(368\) |
| default | \(\frac {\frac {\left (\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}}\right )^{\frac {5}{2}}}{5}-\frac {\left (2 a c -b^{2}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \left (\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}}\right )^{\frac {3}{2}}}{8 a}+\frac {3 \left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \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}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\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}}}\right )}{8 a^{\frac {3}{2}}}\right )}{16 a}\right )}{2 b}+\frac {a \,c^{2} \left (\frac {\left (\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}}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 a c -b^{2}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \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}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\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}}}\right )}{8 a^{\frac {3}{2}}}\right )}{2 b}+\frac {a \,c^{2} \left (\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}}}-\frac {\left (2 a c -b^{2}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\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}}}\right )}{2 b \sqrt {a}}-\frac {a \,c^{2} \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}}}}\right )}{b^{2}}\right )}{b^{2}}}{b}\) | \(881\) |
1/1920*(a*x^2+b*x+c)^(1/2)*(384*a^4*b^4*x^4-480*a^4*b^3*c*x^3+1008*a^3*b^5 *x^3+640*a^4*b^2*c^2*x^2+48*a^3*b^4*c*x^2+744*a^2*b^6*x^2-960*a^4*b*c^3*x- 80*a^3*b^3*c^2*x+1308*a^2*b^5*c*x+30*a*b^7*x+1920*a^4*c^4+160*a^3*b^2*c^3+ 24*a^2*b^4*c^2+390*a*b^6*c-45*b^8)/a^2/b^5-1/256/a^2/b^5*(256*a^5*c^6/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))+(256*a^5*c ^5-80*a^2*b^6*c^2+30*a*b^8*c-3*b^10)/b*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c )^(1/2))/a^(1/2))
Time = 47.04 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.12 \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\left [\frac {3840 \, a^{\frac {11}{2}} c^{5} \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 ) - 15 \, {\left (3 \, b^{10} - 30 \, a b^{8} c + 80 \, a^{2} b^{6} c^{2} - 256 \, a^{5} c^{5}\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 ) + 4 \, {\left (384 \, a^{5} b^{5} x^{4} - 45 \, a b^{9} + 390 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 160 \, a^{4} b^{3} c^{3} + 1920 \, a^{5} b c^{4} + 48 \, {\left (21 \, a^{4} b^{6} - 10 \, a^{5} b^{4} c\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{7} + 6 \, a^{4} b^{5} c + 80 \, a^{5} b^{3} c^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{2} b^{8} + 654 \, a^{3} b^{6} c - 40 \, a^{4} b^{4} c^{2} - 480 \, a^{5} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{7680 \, a^{3} b^{6}}, -\frac {3840 \, \sqrt {-a} a^{5} c^{5} \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 (3 \, b^{10} - 30 \, a b^{8} c + 80 \, a^{2} b^{6} c^{2} - 256 \, a^{5} c^{5}\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 (384 \, a^{5} b^{5} x^{4} - 45 \, a b^{9} + 390 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 160 \, a^{4} b^{3} c^{3} + 1920 \, a^{5} b c^{4} + 48 \, {\left (21 \, a^{4} b^{6} - 10 \, a^{5} b^{4} c\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{7} + 6 \, a^{4} b^{5} c + 80 \, a^{5} b^{3} c^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{2} b^{8} + 654 \, a^{3} b^{6} c - 40 \, a^{4} b^{4} c^{2} - 480 \, a^{5} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{3840 \, a^{3} b^{6}}\right ] \]
[1/7680*(3840*a^(11/2)*c^5*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)) - 15*(3*b^10 - 30*a*b^8*c + 80* a^2*b^6*c^2 - 256*a^5*c^5)*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) + 4*(384*a^5*b^5*x^4 - 45*a *b^9 + 390*a^2*b^7*c + 24*a^3*b^5*c^2 + 160*a^4*b^3*c^3 + 1920*a^5*b*c^4 + 48*(21*a^4*b^6 - 10*a^5*b^4*c)*x^3 + 8*(93*a^3*b^7 + 6*a^4*b^5*c + 80*a^5 *b^3*c^2)*x^2 + 2*(15*a^2*b^8 + 654*a^3*b^6*c - 40*a^4*b^4*c^2 - 480*a^5*b ^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^3*b^6), -1/3840*(3840*sqrt(-a)*a^5*c^ 5*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*(3*b^10 - 30*a*b^8*c + 80*a^2*b^6*c^2 - 256 *a^5*c^5)*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*(384*a^5*b^5*x^4 - 45*a*b^9 + 390*a^2*b^7*c + 24*a^3*b^5*c^2 + 160*a^4*b^3*c^3 + 1920*a^5*b*c^4 + 48*(21*a^4*b^6 - 10*a^ 5*b^4*c)*x^3 + 8*(93*a^3*b^7 + 6*a^4*b^5*c + 80*a^5*b^3*c^2)*x^2 + 2*(15*a ^2*b^8 + 654*a^3*b^6*c - 40*a^4*b^4*c^2 - 480*a^5*b^2*c^3)*x)*sqrt(a*x^2 + b*x + c))/(a^3*b^6)]
\[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\int \frac {\left (a x^{2} + b x + c\right )^{\frac {5}{2}}}{b x + c}\, dx \]
Exception generated. \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx=\int \frac {{\left (a\,x^2+b\,x+c\right )}^{5/2}}{c+b\,x} \,d x \]