Integrand size = 23, antiderivative size = 451 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/3*(b*(-b*e+c*d)+c*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/b^2/d/(-b*e+c*d)/(c*x^2 +b*x)^(3/2)+2/3*(b*(-b*e+c*d)*(-2*b^2*e^2-5*b*c*d*e+8*c^2*d^2)+2*c*(-b*e+2 *c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*x)*(e*x+d)^(1/2)/b^4/d^2/(-b*e+c*d)^2 /(c*x^2+b*x)^(1/2)-4/3*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*Ellipti cE(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)^(7/2)/d^2/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x) ^(1/2)+2/3*(-b^2*e^2-16*b*c*d*e+16*c^2*d^2)*EllipticF(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)^(7/2)/d/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 10.15 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b (d+e x) \left (b c^3 d^2 (c d-b e) x^2+2 c^3 d^2 (4 c d-5 b e) x^2 (b+c x)-b d (c d-b e)^2 (b+c x)^2+2 (c d-b e)^2 (4 c d+b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} c x (b+c x) \left (2 \sqrt {\frac {b}{c}} \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (b+c x) (d+e x)+2 i b e \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^3 d^3-13 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^2 (c d-b e)^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
(2*(b*(d + e*x)*(b*c^3*d^2*(c*d - b*e)*x^2 + 2*c^3*d^2*(4*c*d - 5*b*e)*x^2 *(b + c*x) - b*d*(c*d - b*e)^2*(b + c*x)^2 + 2*(c*d - b*e)^2*(4*c*d + b*e) *x*(b + c*x)^2) - Sqrt[b/c]*c*x*(b + c*x)*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c ^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3 *d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*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^5*d^2*(c*d - b*e)^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])
Time = 0.69 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1165, 27, 1235, 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.
| \(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{5/2} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {8 c^2 d^2-5 b c e d-2 b^2 e^2+3 c e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {8 c^2 d^2-5 b c e d-2 b^2 e^2+3 c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {-\frac {2 \int -\frac {c e \left (b d \left (8 c^2 d^2-11 b c e d+b^2 e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {c e \int \frac {b d \left (8 c^2 d^2-11 b c e d+b^2 e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {\frac {c e \left (\frac {2 (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle -\frac {\frac {c e \left (\frac {2 \sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {\frac {c e \left (\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {\frac {c e \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {\frac {c e \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {\frac {c e \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{b^2 d \sqrt {b x+c x^2} (c d-b e)}}{3 b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}\) |
(-2*Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e )*(b*x + c*x^2)^(3/2)) - ((-2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*c^2*d^2 - 5* b*c*d*e - 2*b^2*e^2) + 2*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2) *x))/(b^2*d*(c*d - b*e)*Sqrt[b*x + c*x^2]) + (c*e*((4*Sqrt[-b]*(2*c*d - b* e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e* x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[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)*(16*c^2*d ^2 - 16*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ell ipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/(b^2*d*(c*d - b*e)))/(3*b^2*d*(c*d - b*e))
3.5.26.3.1 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[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*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 *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -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 = 2.50 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.45
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 d \,b^{3} x^{2}}+\frac {4 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (b e +4 c d \right )}{3 b^{4} d^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )^{2}}-\frac {4 \left (c e \,x^{2}+c d x \right ) c^{2} \left (5 b e -4 c d \right )}{3 b^{4} \left (b e -c d \right )^{2} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {c e}{3 b^{3} d}-\frac {c^{2} e}{3 \left (b e -c d \right ) b^{3}}+\frac {2 c^{2} \left (5 b e -4 c d \right )}{3 \left (b e -c d \right ) b^{4}}+\frac {2 c^{3} d \left (5 b e -4 c d \right )}{3 b^{4} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {2 c e \left (b e +4 c d \right )}{3 b^{4} d^{2}}+\frac {2 e \,c^{3} \left (5 b e -4 c d \right )}{3 \left (b e -c d \right )^{2} b^{4}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(653\) |
| default | \(\text {Expression too large to display}\) | \(1763\) |
(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3/d/b^3*(c*e *x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/x^2+4/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4/d^ 2*(b*e+4*c*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2/3/b^3/(b*e-c*d)*c*(c*e *x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(1/c*b+x)^2-4/3*(c*e*x^2+c*d*x)/b^4/(b*e -c*d)^2*c^2*(5*b*e-4*c*d)/((1/c*b+x)*(c*e*x^2+c*d*x))^(1/2)+2*(-1/3/b^3/d* c*e-1/3*c^2*e/(b*e-c*d)/b^3+2/3/(b*e-c*d)*c^2*(5*b*e-4*c*d)/b^4+2/3*c^3*d/ b^4/(b*e-c*d)^2*(5*b*e-4*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))+2*(-2/3*c*e*(b*e+4*c*d )/b^4/d^2+2/3*e*c^3*(5*b*e-4*c*d)/(b*e-c*d)^2/b^4)/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 = 0.12 (sec) , antiderivative size = 953, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{6} d^{4} - 32 \, b c^{5} d^{3} e + 13 \, b^{2} c^{4} d^{2} e^{2} + 3 \, b^{3} c^{3} d e^{3} + 2 \, b^{4} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (16 \, b c^{5} d^{4} - 32 \, b^{2} c^{4} d^{3} e + 13 \, b^{3} c^{3} d^{2} e^{2} + 3 \, b^{4} c^{2} d e^{3} + 2 \, b^{5} c e^{4}\right )} x^{3} + {\left (16 \, b^{2} c^{4} d^{4} - 32 \, b^{3} c^{3} d^{3} e + 13 \, b^{4} c^{2} d^{2} e^{2} + 3 \, b^{5} c d e^{3} + 2 \, b^{6} e^{4}\right )} x^{2}\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 (8 \, c^{6} d^{3} e - 12 \, b c^{5} d^{2} e^{2} + 2 \, b^{2} c^{4} d e^{3} + b^{3} c^{3} e^{4}\right )} x^{4} + 2 \, {\left (8 \, b c^{5} d^{3} e - 12 \, b^{2} c^{4} d^{2} e^{2} + 2 \, b^{3} c^{3} d e^{3} + b^{4} c^{2} e^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d^{3} e - 12 \, b^{3} c^{3} d^{2} e^{2} + 2 \, b^{4} c^{2} d e^{3} + b^{5} c e^{4}\right )} x^{2}\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^{3} c^{3} d^{3} e - 2 \, b^{4} c^{2} d^{2} e^{2} + b^{5} c d e^{3} - 2 \, {\left (8 \, c^{6} d^{3} e - 12 \, b c^{5} d^{2} e^{2} + 2 \, b^{2} c^{4} d e^{3} + b^{3} c^{3} e^{4}\right )} x^{3} - {\left (24 \, b c^{5} d^{3} e - 37 \, b^{2} c^{4} d^{2} e^{2} + 7 \, b^{3} c^{3} d e^{3} + 4 \, b^{4} c^{2} e^{4}\right )} x^{2} - 2 \, {\left (3 \, b^{2} c^{4} d^{3} e - 5 \, b^{3} c^{3} d^{2} e^{2} + b^{4} c^{2} d e^{3} + b^{5} c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left ({\left (b^{4} c^{5} d^{4} e - 2 \, b^{5} c^{4} d^{3} e^{2} + b^{6} c^{3} d^{2} e^{3}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} d^{4} e - 2 \, b^{6} c^{3} d^{3} e^{2} + b^{7} c^{2} d^{2} e^{3}\right )} x^{3} + {\left (b^{6} c^{3} d^{4} e - 2 \, b^{7} c^{2} d^{3} e^{2} + b^{8} c d^{2} e^{3}\right )} x^{2}\right )}} \]
2/9*(((16*c^6*d^4 - 32*b*c^5*d^3*e + 13*b^2*c^4*d^2*e^2 + 3*b^3*c^3*d*e^3 + 2*b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 32*b^2*c^4*d^3*e + 13*b^3*c^3*d^2 *e^2 + 3*b^4*c^2*d*e^3 + 2*b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 32*b^3*c^3*d ^3*e + 13*b^4*c^2*d^2*e^2 + 3*b^5*c*d*e^3 + 2*b^6*e^4)*x^2)*sqrt(c*e)*weie rstrassPInverse(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*((8*c^6*d^3*e - 12*b*c^5*d^2*e^2 + 2*b^2*c^4*d*e^3 + b^3*c^3*e^4)*x^4 + 2*(8*b*c^5*d^3*e - 12*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e ^3 + b^4*c^2*e^4)*x^3 + (8*b^2*c^4*d^3*e - 12*b^3*c^3*d^2*e^2 + 2*b^4*c^2* d*e^3 + b^5*c*e^4)*x^2)*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^3*c^3*d^3*e - 2*b^4*c^2 *d^2*e^2 + b^5*c*d*e^3 - 2*(8*c^6*d^3*e - 12*b*c^5*d^2*e^2 + 2*b^2*c^4*d*e ^3 + b^3*c^3*e^4)*x^3 - (24*b*c^5*d^3*e - 37*b^2*c^4*d^2*e^2 + 7*b^3*c^3*d *e^3 + 4*b^4*c^2*e^4)*x^2 - 2*(3*b^2*c^4*d^3*e - 5*b^3*c^3*d^2*e^2 + b^4*c ^2*d*e^3 + b^5*c*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/((b^4*c^5*d^4*e - 2*b^5*c^4*d^3*e^2 + b^6*c^3*d^2*e^3)*x^4 + 2*(b^5*c^4*d^4*e - 2*b^6*c^3* d^3*e^2 + b^7*c^2*d^2*e^3)*x^3 + (b^6*c^3*d^4*e - 2*b^7*c^2*d^3*e^2 + b...
\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \]
\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,\sqrt {d+e\,x}} \,d x \]