Integrand size = 21, antiderivative size = 205 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 \sqrt {d} e^4 \sqrt {c d-b e}} \]
-1/2*(c*x^2+b*x)^(3/2)/e/(e*x+d)^2-3*(-b*e+2*c*d)*arctanh(x*c^(1/2)/(c*x^2 +b*x)^(1/2))*c^(1/2)/e^4+3/8*(b^2*e^2-8*b*c*d*e+8*c^2*d^2)*arctanh(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/d^(1/2)/ (-b*e+c*d)^(1/2)+3/4*(2*c*e*x-b*e+4*c*d)*(c*x^2+b*x)^(1/2)/e^3/(e*x+d)
Time = 11.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.16 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {e (-c d+b e) \sqrt {x} \left (-b e (3 d+5 e x)+2 c \left (6 d^2+9 d e x+2 e^2 x^2\right )\right )}{(d+e x)^2}+\frac {12 \sqrt {c} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}-\frac {3 \sqrt {c d-b e} \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {d} \sqrt {b+c x}}\right )}{4 e^4 (-c d+b e) \sqrt {x}} \]
(Sqrt[x*(b + c*x)]*((e*(-(c*d) + b*e)*Sqrt[x]*(-(b*e*(3*d + 5*e*x)) + 2*c* (6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 + (12*Sqrt[c]*(2*c^2*d^2 - 3*b *c*d*e + b^2*e^2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c *x)/b]) - (3*Sqrt[c*d - b*e]*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(Sq rt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[b + c*x]))) /(4*e^4*(-(c*d) + b*e)*Sqrt[x])
Time = 0.43 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1161, 1230, 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.
\(\displaystyle \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx\) |
\(\Big \downarrow \) 1161 |
\(\displaystyle \frac {3 \int \frac {(b+2 c x) \sqrt {c x^2+b x}}{(d+e x)^2}dx}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 1230 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)}-\frac {\int \frac {b (4 c d-b e)+4 c (2 c d-b e) x}{(d+e x) \sqrt {c x^2+b x}}dx}{2 e^2}\right )}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)}-\frac {\frac {4 c (2 c d-b e) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {\left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}\right )}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)}-\frac {\frac {8 c (2 c d-b e) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {\left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}\right )}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)}-\frac {\frac {8 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e)}{e}-\frac {\left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}\right )}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)}-\frac {\frac {2 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {8 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e)}{e}}{2 e^2}\right )}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3 \left (\frac {\sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{e^2 (d+e x)}-\frac {\frac {8 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e)}{e}-\frac {\left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e^2}\right )}{4 e}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}\) |
-1/2*(b*x + c*x^2)^(3/2)/(e*(d + e*x)^2) + (3*(((4*c*d - b*e + 2*c*e*x)*Sq rt[b*x + c*x^2])/(e^2*(d + e*x)) - ((8*Sqrt[c]*(2*c*d - b*e)*ArcTanh[(Sqrt [c]*x)/Sqrt[b*x + c*x^2]])/e - ((8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[ (b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(S qrt[d]*e*Sqrt[c*d - b*e]))/(2*e^2)))/(4*e)
3.4.1.3.1 Defintions of rubi rules used
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 + 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)*(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]
Time = 2.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(-\frac {6 \left (\frac {b x \left (e x +d \right )^{2} \left (b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{8}+\left (\left (e x +d \right )^{2} x \left (c^{\frac {3}{2}} d -\frac {\sqrt {c}\, b e}{2}\right ) b \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+\frac {5 \left (\frac {\left (3 b d e -8 c \,d^{2}\right ) \left (x \left (c x +b \right )\right )^{\frac {3}{2}}}{5}+x \left (-\frac {4 c \left (-2 c x +b \right ) d^{2}}{5}-\frac {21 b c d e x}{5}+b \,e^{2} x \left (-\frac {4 c x}{5}+b \right )\right ) \sqrt {x \left (c x +b \right )}\right ) e}{24}\right ) \sqrt {d \left (b e -c d \right )}\right )}{\sqrt {d \left (b e -c d \right )}\, b \,e^{4} \left (e x +d \right )^{2} x}\) | \(216\) |
risch | \(\frac {x \left (c x +b \right ) c}{e^{3} \sqrt {x \left (c x +b \right )}}+\frac {\frac {3 \sqrt {c}\, \left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e}-\frac {\left (2 b^{2} e^{2}-12 b c d e +12 c^{2} d^{2}\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}}}}-\frac {4 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \left (\frac {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}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {e \left (b e -2 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 )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{3}}+\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \left (\frac {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}}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 e \left (b e -2 c d \right ) \left (\frac {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}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {e \left (b e -2 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 )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \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 )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{4}}}{2 e^{3}}\) | \(971\) |
default | \(\text {Expression too large to display}\) | \(1628\) |
-6/(d*(b*e-c*d))^(1/2)*(1/8*b*x*(e*x+d)^2*(b^2*e^2-8*b*c*d*e+8*c^2*d^2)*ar ctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+((e*x+d)^2*x*(c^(3/2)*d-1/ 2*c^(1/2)*b*e)*b*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+5/24*(1/5*(3*b*d*e-8 *c*d^2)*(x*(c*x+b))^(3/2)+x*(-4/5*c*(-2*c*x+b)*d^2-21/5*b*c*d*e*x+b*e^2*x* (-4/5*c*x+b))*(x*(c*x+b))^(1/2))*e)*(d*(b*e-c*d))^(1/2))/b/e^4/(e*x+d)^2/x
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (177) = 354\).
Time = 0.57 (sec) , antiderivative size = 1749, normalized size of antiderivative = 8.53 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
[-1/8*(12*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c* d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x )*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 3*(8*c^2*d^4 - 8* b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2* (8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*x)*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)) - 2*(12*c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d* e^4)*x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^7)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6)*x), 1/4*(3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2 *e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d *e^3)*x)*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)) - 6*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^ 3*e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + (12 *c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4)* x^2 + (18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x) )/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^7)*x^2 + 2*(c*d^3*e^5 - b*d^ 2*e^6)*x), 1/8*(24*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^...
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (177) = 354\).
Time = 0.33 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.42 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {c x^{2} + b x} c}{e^{3}} + \frac {3 \, {\left (8 \, c^{2} d^{2} - 8 \, b c d e + b^{2} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, \sqrt {-c d^{2} + b d e} e^{4}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2 \, \sqrt {c} e^{4}} + \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2} + 10 \, b^{2} c^{\frac {3}{2}} d^{3} - 5 \, b^{3} \sqrt {c} d^{2} e}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2} e^{4}} \]
sqrt(c*x^2 + b*x)*c/e^3 + 3/4*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*arctan(-(( sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/(sqrt( -c*d^2 + b*d*e)*e^4) + 3/2*(2*c^2*d - b*c*e)*log(abs(2*(sqrt(c)*x - sqrt(c *x^2 + b*x))*sqrt(c) + b))/(sqrt(c)*e^4) + 1/4*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^2*d^2*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 + 5* (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b *x))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2)*d^2*e - (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c)*d*e^2 + 40*(sqrt(c)*x - sq rt(c*x^2 + b*x))*b*c^2*d^3 - 28*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2* e + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*d*e^2 + 10*b^2*c^(3/2)*d^3 - 5*b ^3*sqrt(c)*d^2*e)/(((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - s qrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2*e^4)
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \]