Integrand size = 21, antiderivative size = 216 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^4} \]
1/3*(c*x^2+b*x)^(3/2)/e-1/8*(-b*e+2*c*d)*(-b^2*e^2-8*b*c*d*e+8*c^2*d^2)*ar ctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(3/2)/e^4+d^(3/2)*(-b*e+c*d)^(3/2)*ar ctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2)) /e^4+1/8*(8*c^2*d^2-10*b*c*d*e+b^2*e^2-2*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^( 1/2)/c/e^3
Result contains complex when optimal does not.
Time = 3.74 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.31 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {(x (b+c x))^{3/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (3 b^2 e^2+2 b c e (-15 d+7 e x)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+48 \sqrt {c} \sqrt {d} (c d-b e) \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+48 \sqrt {c} \sqrt {d} (c d-b e) \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+6 \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )\right )}{24 c^{3/2} e^4 x^{3/2} (b+c x)^{3/2}} \]
((x*(b + c*x))^(3/2)*(Sqrt[c]*e*Sqrt[x]*Sqrt[b + c*x]*(3*b^2*e^2 + 2*b*c*e *(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 48*Sqrt[c]*Sqrt[ d]*(c*d - b*e)*(c*d - b*e - I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2* b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(Sqrt[b] - Sqrt[b + c*x]))] + 48*Sqrt[c]*Sqrt[d]*(c*d - b*e)*(c*d - b*e + I*Sqrt[b]*S qrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b*e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c *d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(Sqrt[b] - Sqrt[b + c*x]))] + 6*(16*c^3*d^3 - 24*b *c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*ArcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])]))/(24*c^(3/2)*e^4*x^(3/2)*(b + c*x)^(3/2))
Time = 0.49 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1162, 1231, 27, 1269, 1091, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\int \frac {(b d+(2 c d-b e) x) \sqrt {c x^2+b x}}{d+e x}dx}{2 e}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {-\frac {\int -\frac {b d \left (8 c^2 d^2-10 b c e d+b^2 e^2\right )+(2 c d-b e) \left (8 c^2 d^2-8 b c e d-b^2 e^2\right ) x}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\frac {\int \frac {b d \left (8 c^2 d^2-10 b c e d+b^2 e^2\right )+(2 c d-b e) \left (8 c^2 d^2-8 b c e d-b^2 e^2\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\frac {\frac {(2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {16 c d^2 (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\frac {\frac {2 (2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {16 c d^2 (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right )}{\sqrt {c} e}-\frac {16 c d^2 (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\frac {\frac {32 c d^2 (c d-b e)^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right )}{\sqrt {c} e}-\frac {16 c d^{3/2} (c d-b e)^{3/2} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{4 c e^2}}{2 e}\) |
(b*x + c*x^2)^(3/2)/(3*e) - (-1/4*((8*c^2*d^2 - 10*b*c*d*e + b^2*e^2 - 2*c *e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(c*e^2) + ((2*(2*c*d - b*e)*(8*c^2* d^2 - 8*b*c*d*e - b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c ]*e) - (16*c*d^(3/2)*(c*d - b*e)^(3/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2* Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e)/(8*c*e^2))/(2*e)
3.3.99.3.1 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[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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 = 2.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {2 \left (b e -c d \right )^{2} d^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}-\frac {e \sqrt {x \left (c x +b \right )}\, \left (8 c^{2} e^{2} x^{2}+14 b c \,e^{2} x -12 c^{2} d e x +3 b^{2} e^{2}-30 b c d e +24 c^{2} d^{2}\right )}{24 c}+\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c -24 b \,c^{2} d^{2} e +16 c^{3} d^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}}{e^{4}}\) | \(184\) |
| risch | \(\frac {\left (8 c^{2} e^{2} x^{2}+14 b c \,e^{2} x -12 c^{2} d e x +3 b^{2} e^{2}-30 b c d e +24 c^{2} d^{2}\right ) x \left (c x +b \right )}{24 c \,e^{3} \sqrt {x \left (c x +b \right )}}-\frac {\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c -24 b \,c^{2} d^{2} e +16 c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {16 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) c \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{16 e^{3} c}\) | \(307\) |
| default | \(\frac {\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}}{e}\) | \(549\) |
-1/e^4*(2*(b*e-c*d)^2*d^2/(d*(b*e-c*d))^(1/2)*arctan((x*(c*x+b))^(1/2)/x*d /(d*(b*e-c*d))^(1/2))-1/24*e*(x*(c*x+b))^(1/2)*(8*c^2*e^2*x^2+14*b*c*e^2*x -12*c^2*d*e*x+3*b^2*e^2-30*b*c*d*e+24*c^2*d^2)/c+1/8*(b^3*e^3+6*b^2*c*d*e^ 2-24*b*c^2*d^2*e+16*c^3*d^3)/c^(3/2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2)))
Time = 0.51 (sec) , antiderivative size = 901, normalized size of antiderivative = 4.17 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\left [\frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2} e^{4}}, \frac {96 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2} e^{4}}, \frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - 24 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2} e^{4}}, \frac {48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2} e^{4}}\right ] \]
[1/48*(3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*sqrt(c)*l og(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 48*(c^3*d^2 - b*c^2*d*e)*sqr t(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c *x^2 + b*x))/(e*x + d)) + 2*(8*c^3*e^3*x^2 + 24*c^3*d^2*e - 30*b*c^2*d*e^2 + 3*b^2*c*e^3 - 2*(6*c^3*d*e^2 - 7*b*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2* e^4), 1/48*(96*(c^3*d^2 - b*c^2*d*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c* d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 3*(16*c^3*d^3 - 24*b*c^2 *d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b *x)*sqrt(c)) + 2*(8*c^3*e^3*x^2 + 24*c^3*d^2*e - 30*b*c^2*d*e^2 + 3*b^2*c* e^3 - 2*(6*c^3*d*e^2 - 7*b*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2*e^4), 1/24* (3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*sqrt(-c)*arctan (sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 24*(c^3*d^2 - b*c^2*d*e)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b* x))/(e*x + d)) + (8*c^3*e^3*x^2 + 24*c^3*d^2*e - 30*b*c^2*d*e^2 + 3*b^2*c* e^3 - 2*(6*c^3*d*e^2 - 7*b*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2*e^4), 1/24* (48*(c^3*d^2 - b*c^2*d*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e )*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 3*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6* b^2*c*d*e^2 + b^3*e^3)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (8*c^3*e^3*x^2 + 24*c^3*d^2*e - 30*b*c^2*d*e^2 + 3*b^2*c*e^3 - 2*(6*c^3*d *e^2 - 7*b*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2*e^4)]
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{d+e\,x} \,d x \]