Integrand size = 23, antiderivative size = 317 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=-\frac {2 e \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {4 e (2 c d-b e) \sqrt {b x+c x^2}}{3 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
4/3*(-b*e+2*c*d)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b )^(1/2)*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d^2/(-b*e+c*d)^2/(1+ e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/3*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b *e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/ d/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/3*e*(c*x^2+b*x)^(1/2)/d/(-b *e+c*d)/(e*x+d)^(3/2)-4/3*e*(-b*e+2*c*d)*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^ 2/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 16.63 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=-\frac {2 \left (-b e x (b+c x) (b e (3 d+2 e x)-c d (5 d+4 e x))-\sqrt {\frac {b}{c}} c (d+e x) \left (-2 \sqrt {\frac {b}{c}} (-2 c d+b e) (b+c x) (d+e x)+2 i b e (2 c d-b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \left (3 c^2 d^2-5 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b d^2 (c d-b e)^2 \sqrt {x (b+c x)} (d+e x)^{3/2}} \]
(-2*(-(b*e*x*(b + c*x)*(b*e*(3*d + 2*e*x) - c*d*(5*d + 4*e*x))) - Sqrt[b/c ]*c*(d + e*x)*(-2*Sqrt[b/c]*(-2*c*d + b*e)*(b + c*x)*(d + e*x) + (2*I)*b*e *(2*c*d - b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*Arc Sinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*(3*c^2*d^2 - 5*b*c*d*e + 2*b^2*e ^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b /c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b*d^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))
Time = 0.51 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1167, 27, 1237, 27, 1269, 1169, 122, 120, 127, 126}
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.
| \(\displaystyle \int \frac {1}{\sqrt {b x+c x^2} (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle -\frac {2 \int -\frac {3 c d-2 b e-c e x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 c d-2 b e-c e x}{(d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {c (d (3 c d-b e)+2 e (2 c d-b e) x)}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \int \frac {d (3 c d-b e)+2 e (2 c d-b e) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {c \left (2 (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx-d (c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx\right )}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 \sqrt {x} \sqrt {b+c x} (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{\sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}\right )}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{\sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}\right )}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}\right )}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{\sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} (2 c d-b e)}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}-\frac {2 e \sqrt {b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}\) |
(-2*e*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) + ((-4*e*(2*c*d - b*e)*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*Sqrt[d + e*x]) + (c*((4*Sqrt[-b] *(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(S qrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*Sqrt[1 + (e*x)/d]*Sqrt[b *x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sq rt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/(d*(c*d - b*e)))/(3*d*(c*d - b*e) )
3.5.12.3.1 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[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[ (d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (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 = 2.22 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.61
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {4 \left (c e \,x^{2}+b e x \right ) \left (b e -2 c d \right )}{3 d^{2} \left (b e -c d \right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c}{3 d \left (b e -c d \right )}+\frac {-\frac {4 c d}{3}+\frac {2 b e}{3}}{d^{2} \left (b e -c d \right )}-\frac {2 b e \left (b e -2 c d \right )}{3 d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {4 e \left (b e -2 c d \right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{3 d^{2} \left (b e -c d \right )^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(511\) |
| default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d \,e^{2} x -\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2} e x +2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e^{3} x -6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d \,e^{2} x +4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2} e x +\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c \,d^{2} e -\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{3}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} d \,e^{2}-6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c \,d^{2} e +4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{3}+2 b \,c^{2} e^{3} x^{3}-4 c^{3} d \,e^{2} x^{3}+2 b^{2} c \,e^{3} x^{2}-b \,c^{2} d \,e^{2} x^{2}-5 c^{3} d^{2} e \,x^{2}+3 b^{2} d \,e^{2} c x -5 b \,c^{2} d^{2} e x \right )}{3 \left (c x +b \right ) x \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} c \,d^{2}}\) | \(897\) |
(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/3/e/d/(b*e-c* d)*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2+4/3*(c*e*x^2+b*e*x)/d^2 /(b*e-c*d)^2*(b*e-2*c*d)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(1/3*c/d/(b*e-c *d)+2/3*(b*e-2*c*d)/(b*e-c*d)/d^2-2/3*b*e/d^2/(b*e-c*d)^2*(b*e-2*c*d))/c*b *((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^ 3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1 /c*b+d/e))^(1/2))-4/3*e*(b*e-2*c*d)/d^2/(b*e-c*d)^2*b*((1/c*b+x)*c/b)^(1/2 )*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d *x)^(1/2)*((-1/c*b+d/e)*EllipticE(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/ e))^(1/2))-d/e*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2) )))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (5 \, c^{2} d^{4} - 5 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + {\left (5 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 2 \, {\left (5 \, c^{2} d^{3} e - 5 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 6 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{2} + 2 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (5 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + 2 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (c^{3} d^{6} e - 2 \, b c^{2} d^{5} e^{2} + b^{2} c d^{4} e^{3} + {\left (c^{3} d^{4} e^{3} - 2 \, b c^{2} d^{3} e^{4} + b^{2} c d^{2} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{5} e^{2} - 2 \, b c^{2} d^{4} e^{3} + b^{2} c d^{3} e^{4}\right )} x\right )}} \]
2/9*((5*c^2*d^4 - 5*b*c*d^3*e + 2*b^2*d^2*e^2 + (5*c^2*d^2*e^2 - 5*b*c*d*e ^3 + 2*b^2*e^4)*x^2 + 2*(5*c^2*d^3*e - 5*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)*sqr t(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4 /27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3 *(3*c*e*x + c*d + b*e)/(c*e)) - 6*(2*c^2*d^3*e - b*c*d^2*e^2 + (2*c^2*d*e^ 3 - b*c*e^4)*x^2 + 2*(2*c^2*d^2*e^2 - b*c*d*e^3)*x)*sqrt(c*e)*weierstrassZ eta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^ 2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c ^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) - 3*(5*c^2*d^2*e^2 - 3*b*c*d*e^3 + 2*(2*c^2*d*e^3 - b*c*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^3*d^6*e - 2*b*c^2*d^5*e^2 + b^2*c*d^4*e^3 + (c^3*d ^4*e^3 - 2*b*c^2*d^3*e^4 + b^2*c*d^2*e^5)*x^2 + 2*(c^3*d^5*e^2 - 2*b*c^2*d ^4*e^3 + b^2*c*d^3*e^4)*x)
\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]