Integrand size = 23, antiderivative size = 370 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \sqrt {c} \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2*(b*(-b*e+c*d)+c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b *x)^(1/2)+4*(b^2*e^2-b*c*d*e+c^2*d^2)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2) ,(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(3/2) /d^2/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2*(-b*e+2*c*d)*Ellipti cF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(1+c*x/b)^( 1/2)*(1+e*x/d)^(1/2)/(-b)^(3/2)/d/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/ 2)-4*e*(b^2*e^2-b*c*d*e+c^2*d^2)*(c*x^2+b*x)^(1/2)/b^2/d^2/(-b*e+c*d)^2/(e *x+d)^(1/2)
Result contains complex when optimal does not.
Time = 11.76 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 b d \left (b^2 e^2+b c e^2 x+c^2 d (d+e x)\right )+4 i \sqrt {\frac {b}{c}} c e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-2 i \sqrt {\frac {b}{c}} c e \left (c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b^2 d^2 (c d-b e)^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(2*b*d*(b^2*e^2 + b*c*e^2*x + c^2*d*(d + e*x)) + (4*I)*Sqrt[b/c]*c*e*(c^2* d^2 - b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip ticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(c^2 *d^2 - 3*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)])/(b^2*d^2*(c*d - b*e) ^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
Time = 0.60 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1165, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {e (b (c d-2 b e)+c (2 c d-b e) x)}{2 (d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {b (c d-2 b e)+c (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {2 \int \frac {c \left (b d (c d+b e)+2 \left (c^2 d^2-b c e d+b^2 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \int \frac {b d (c d+b e)+2 \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {2 \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {2 \sqrt {x} \sqrt {b+c x} \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}\right )}{b^2 d (c d-b e)}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}\) |
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*Sqrt[d + e*x]* Sqrt[b*x + c*x^2]) - (e*((4*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[b*x + c*x^2 ])/(d*(c*d - b*e)*Sqrt[d + e*x]) - (c*((4*Sqrt[-b]*(c^2*d^2 - b*c*d*e + b^ 2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*S qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c *x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(2*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)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/(d*(c*d - b*e))))/(b^2* d*(c*d - b*e))
3.5.19.3.1 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[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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 = 2.14 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.75
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right )}{d^{2} b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 c e x \left (\frac {\left (b^{2} e^{2}+c^{2} d^{2}\right ) x}{d^{2} b^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (b e +c d \right ) \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2} d^{2} c e}\right )}{\sqrt {\left (\frac {\left (b e +c d \right ) x}{c e}+x^{2}+\frac {b d}{c e}\right ) c e x}}+\frac {2 \left (-\frac {b e +c d}{b^{2} d^{2}}+\frac {\left (b e +c d \right ) \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2} d^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c e}{b^{2} d^{2}}+\frac {\left (b^{2} e^{2}+c^{2} d^{2}\right ) c e}{d^{2} b^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(649\) |
| default | \(-\frac {2 \left (\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c d \,e^{2}-3 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{2} d^{2} e +2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{3} d^{3}+2 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} e^{3}-4 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} c d \,e^{2}+4 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c^{2} d^{2} e -2 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{3} d^{3}+2 b^{2} c^{2} e^{3} x^{2}-2 b \,c^{3} d \,e^{2} x^{2}+2 c^{4} d^{2} e \,x^{2}+2 x \,b^{3} c \,e^{3}-b^{2} c^{2} d \,e^{2} x -b \,c^{3} d^{2} e x +2 c^{4} d^{3} x +b^{3} c d \,e^{2}-2 b^{2} d^{2} e \,c^{2}+c^{3} b \,d^{3}\right ) \sqrt {x \left (c x +b \right )}}{x \left (c x +b \right ) \left (b e -c d \right )^{2} c \,b^{2} d^{2} \sqrt {e x +d}}\) | \(698\) |
(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2*(c*e*x^2+b*e *x+c*d*x+b*d)/d^2/b^2/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2*c*e*x*((b^2*e^ 2+c^2*d^2)/d^2/b^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*x+(b*e+c*d)*(b^2*e^2-b*c*d* e+c^2*d^2)/(b^2*e^2-2*b*c*d*e+c^2*d^2)/b^2/d^2/c/e)/(((b*e+c*d)/c/e*x+x^2+ b*d/c/e)*c*e*x)^(1/2)+2*(-(b*e+c*d)/b^2/d^2+(b*e+c*d)*(b^2*e^2-b*c*d*e+c^2 *d^2)/(b^2*e^2-2*b*c*d*e+c^2*d^2)/b^2/d^2)/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))+2*(c*e/b^2 /d^2+(b^2*e^2+c^2*d^2)*c*e/d^2/b^2/(b^2*e^2-2*b*c*d*e+c^2*d^2))/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)*((-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 = 0.10 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} - 3 \, b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x^{3} + {\left (2 \, c^{4} d^{4} - b c^{3} d^{3} e - 6 \, b^{2} c^{2} d^{2} e^{2} - b^{3} c d e^{3} + 2 \, b^{4} e^{4}\right )} x^{2} + {\left (2 \, b c^{3} d^{4} - 3 \, b^{2} c^{2} d^{3} e - 3 \, b^{3} c d^{2} e^{2} + 2 \, b^{4} 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 ({\left (c^{4} d^{2} e^{2} - b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} + {\left (c^{4} d^{3} e + b^{3} c e^{4}\right )} x^{2} + {\left (b c^{3} d^{3} e - b^{2} c^{2} d^{2} e^{2} + b^{3} 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 (b c^{3} d^{3} e - 2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} + 2 \, {\left (c^{4} d^{2} e^{2} - b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{2} + {\left (2 \, c^{4} d^{3} e - b c^{3} d^{2} e^{2} - b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left ({\left (b^{2} c^{4} d^{4} e^{2} - 2 \, b^{3} c^{3} d^{3} e^{3} + b^{4} c^{2} d^{2} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{5} e - b^{3} c^{3} d^{4} e^{2} - b^{4} c^{2} d^{3} e^{3} + b^{5} c d^{2} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{5} e - 2 \, b^{4} c^{2} d^{4} e^{2} + b^{5} c d^{3} e^{3}\right )} x\right )}} \]
-2/3*(((2*c^4*d^3*e - 3*b*c^3*d^2*e^2 - 3*b^2*c^2*d*e^3 + 2*b^3*c*e^4)*x^3 + (2*c^4*d^4 - b*c^3*d^3*e - 6*b^2*c^2*d^2*e^2 - b^3*c*d*e^3 + 2*b^4*e^4) *x^2 + (2*b*c^3*d^4 - 3*b^2*c^2*d^3*e - 3*b^3*c*d^2*e^2 + 2*b^4*d*e^3)*x)* sqrt(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*((c^4*d^2*e^2 - b*c^3*d*e^3 + b^2*c^2 *e^4)*x^3 + (c^4*d^3*e + b^3*c*e^4)*x^2 + (b*c^3*d^3*e - b^2*c^2*d^2*e^2 + b^3*c*d*e^3)*x)*sqrt(c*e)*weierstrassZeta(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*(b*c^3*d^3*e - 2*b^2*c^2*d^2*e^2 + b^3*c*d*e^3 + 2*(c^4*d^2*e^2 - b*c^3*d*e^3 + b^2*c^2*e^4)*x^2 + (2*c^4*d ^3*e - b*c^3*d^2*e^2 - b^2*c^2*d*e^3 + 2*b^3*c*e^4)*x)*sqrt(c*x^2 + b*x)*s qrt(e*x + d))/((b^2*c^4*d^4*e^2 - 2*b^3*c^3*d^3*e^3 + b^4*c^2*d^2*e^4)*x^3 + (b^2*c^4*d^5*e - b^3*c^3*d^4*e^2 - b^4*c^2*d^3*e^3 + b^5*c*d^2*e^4)*x^2 + (b^3*c^3*d^5*e - 2*b^4*c^2*d^4*e^2 + b^5*c*d^3*e^3)*x)
\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]