Integrand size = 33, antiderivative size = 372 \[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {(c-d x)^2 \sqrt {e+f x}}{3 c (d e-c f) \left (c^2-d^2 x^2\right )^{3/2}}-\frac {(d e-3 c f) (c-d x) \sqrt {e+f x}}{3 c^2 (d e-c f)^2 \sqrt {c^2-d^2 x^2}}-\frac {(d e-3 c f) \sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right )|-\frac {2 c f}{d e-c f}\right )}{3 c^{3/2} (d e-c f)^2 \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {c^2-d^2 x^2}}+\frac {(d e-2 c f) \sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{3 c^{3/2} d (d e-c f) \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \] Output:
-1/3*(-d*x+c)^2*(f*x+e)^(1/2)/c/(-c*f+d*e)/(-d^2*x^2+c^2)^(3/2)-1/3*(-3*c* f+d*e)*(-d*x+c)*(f*x+e)^(1/2)/c^2/(-c*f+d*e)^2/(-d^2*x^2+c^2)^(1/2)-1/3*(- 3*c*f+d*e)*(d*x+c)^(1/2)*(1-d*x/c)^(1/2)*(f*x+e)^(1/2)*EllipticE(1/2*(d*x+ c)^(1/2)*2^(1/2)/c^(1/2),(-2*c*f/(-c*f+d*e))^(1/2))/c^(3/2)/(-c*f+d*e)^2/( d*(f*x+e)/(-c*f+d*e))^(1/2)/(-d^2*x^2+c^2)^(1/2)+1/3*(-2*c*f+d*e)*(d*x+c)^ (1/2)*(1-d*x/c)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*EllipticF(1/2*(d*x+c)^( 1/2)*2^(1/2)/c^(1/2),(-2*c*f/(-c*f+d*e))^(1/2))/c^(3/2)/d/(-c*f+d*e)/(f*x+ e)^(1/2)/(-d^2*x^2+c^2)^(1/2)
Result contains complex when optimal does not.
Time = 12.24 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (\frac {f (d e-3 c f)}{d}+\frac {(e+f x) \left (4 c^2 f-d^2 e x+c d (-2 e+3 f x)\right )}{(c+d x)^2}+\frac {i \left (d^2 e^2-2 c d e f-3 c^2 f^2\right ) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-c f}{d e+c f}\right )}{f \sqrt {-\frac {d e+c f}{d}} \left (c^2-d^2 x^2\right )}-\frac {i c (d e-5 c f) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-c f}{d e+c f}\right )}{\sqrt {-\frac {d e+c f}{d}} \left (c^2-d^2 x^2\right )}\right )}{3 c^2 (d e-c f)^2 \sqrt {e+f x}} \] Input:
Integrate[1/((c + d*x)^2*Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
Output:
(Sqrt[c^2 - d^2*x^2]*((f*(d*e - 3*c*f))/d + ((e + f*x)*(4*c^2*f - d^2*e*x + c*d*(-2*e + 3*f*x)))/(c + d*x)^2 + (I*(d^2*e^2 - 2*c*d*e*f - 3*c^2*f^2)* Sqrt[(f*(-c + d*x))/(d*(e + f*x))]*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-((d*e + c*f)/d)]/Sqrt[e + f*x]], (d*e - c*f)/(d*e + c*f)])/(f*Sqrt[-((d*e + c*f)/d)]*(c^2 - d^2*x^2)) - (I*c*(d *e - 5*c*f)*Sqrt[(f*(-c + d*x))/(d*(e + f*x))]*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-((d*e + c*f)/d)]/Sqrt[e + f*x]], (d*e - c*f)/(d*e + c*f)])/(Sqrt[-((d*e + c*f)/d)]*(c^2 - d^2*x^2)) ))/(3*c^2*(d*e - c*f)^2*Sqrt[e + f*x])
Time = 0.56 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {718, 115, 27, 169, 25, 27, 176, 124, 27, 123, 131, 27, 131, 129}
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.
| \(\displaystyle \int \frac {1}{(c+d x)^2 \sqrt {c^2-d^2 x^2} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 718 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \int \frac {1}{\sqrt {c-d x} (c+d x)^{5/2} \sqrt {e+f x}}dx}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (-\frac {\int -\frac {d (2 d e-5 c f+d f x)}{2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}dx}{3 c d (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\int \frac {2 d e-5 c f+d f x}{\sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}dx}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {-\frac {\int -\frac {d f \left (2 c^2 f-d (d e-3 c f) x\right )}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{c d (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {\int \frac {d f \left (2 c^2 f-d (d e-3 c f) x\right )}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{c d (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \int \frac {2 c^2 f-d (d e-3 c f) x}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left ((d e-3 c f) \int \frac {\sqrt {c-d x}}{\sqrt {c+d x} \sqrt {e+f x}}dx-c (d e-5 c f) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {\sqrt {2} \sqrt {c-d x} (d e-3 c f) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {2} \sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-c (d e-5 c f) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {\sqrt {c-d x} (d e-3 c f) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-c (d e-5 c f) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {2 \sqrt {2} \sqrt {c-d x} (d e-3 c f) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-c (d e-5 c f) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {2 \sqrt {2} \sqrt {c-d x} (d e-3 c f) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-\frac {c \sqrt {\frac {c-d x}{c}} (d e-5 c f) \int \frac {\sqrt {2}}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {2} \sqrt {c-d x}}\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {2 \sqrt {2} \sqrt {c-d x} (d e-3 c f) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-\frac {c \sqrt {\frac {c-d x}{c}} (d e-5 c f) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {c-d x}}\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {2 \sqrt {2} \sqrt {c-d x} (d e-3 c f) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-\frac {c \sqrt {\frac {c-d x}{c}} (d e-5 c f) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {c-d x} \sqrt {e+f x}}\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {f \left (\frac {2 \sqrt {2} \sqrt {c-d x} (d e-3 c f) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}-\frac {2 c^{3/2} \sqrt {\frac {c-d x}{c}} (d e-5 c f) \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{d \sqrt {c-d x} \sqrt {e+f x}}\right )}{c (d e-c f)}-\frac {2 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{c \sqrt {c+d x} (d e-c f)}}{6 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{3 c (c+d x)^{3/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\) |
Input:
Int[1/((c + d*x)^2*Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
Output:
(Sqrt[c - d*x]*Sqrt[c + d*x]*(-1/3*(Sqrt[c - d*x]*Sqrt[e + f*x])/(c*(d*e - c*f)*(c + d*x)^(3/2)) + ((-2*(d*e - 3*c*f)*Sqrt[c - d*x]*Sqrt[e + f*x])/( c*(d*e - c*f)*Sqrt[c + d*x]) + (f*((2*Sqrt[2]*(d*e - 3*c*f)*Sqrt[-(d*e) + c*f]*Sqrt[c - d*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticE[ArcSin[(Sqrt[ f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], (1 - (d*e)/(c*f))/2])/(d*Sqrt[f]*Sq rt[(c - d*x)/c]*Sqrt[e + f*x]) - (2*c^(3/2)*(d*e - 5*c*f)*Sqrt[(c - d*x)/c ]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[ArcSin[Sqrt[c + d*x]/(Sqrt[2]* Sqrt[c])], (-2*c*f)/(d*e - c*f)])/(d*Sqrt[c - d*x]*Sqrt[e + f*x])))/(c*(d* e - c*f)))/(6*c*(d*e - c*f))))/Sqrt[c^2 - d^2*x^2]
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) ^2)^(p_), x_Symbol] :> Simp[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]* (a/d + (c*x)/e)^FracPart[p]) Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/ e)*x)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(324)=648\).
Time = 6.25 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.96
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\frac {\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}{3 c \left (c f -d e \right ) d^{2} \left (x +\frac {c}{d}\right )^{2}}+\frac {\left (-d^{2} f \,x^{2}+c d f x -d^{2} e x +d e c \right ) \left (3 c f -d e \right )}{3 c^{2} \left (c f -d e \right )^{2} d \sqrt {\left (x +\frac {c}{d}\right ) \left (-d^{2} f \,x^{2}+c d f x -d^{2} e x +d e c \right )}}+\frac {2 \left (-\frac {f}{6 c \left (c f -d e \right )}+\frac {\left (2 c f -d e \right ) \left (3 c f -d e \right )}{6 c^{2} \left (c f -d e \right )^{2}}-\frac {\left (d f c -d^{2} e \right ) \left (3 c f -d e \right )}{6 c^{2} \left (c f -d e \right )^{2} d}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {\left (3 c f -d e \right ) d f \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{d}\right )}{3 c^{2} \left (c f -d e \right )^{2} \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}\) | \(729\) |
| default | \(\text {Expression too large to display}\) | \(1743\) |
Input:
int(1/(d*x+c)^2/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x,method=_RETURNVERBOSE )
Output:
((f*x+e)*(-d^2*x^2+c^2))^(1/2)/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2)*(1/3/c/( c*f-d*e)/d^2*(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)/(x+c/d)^2+1/3*(-d^ 2*f*x^2+c*d*f*x-d^2*e*x+c*d*e)/c^2/(c*f-d*e)^2*(3*c*f-d*e)/d/((x+c/d)*(-d^ 2*f*x^2+c*d*f*x-d^2*e*x+c*d*e))^(1/2)+2*(-1/6*f/c/(c*f-d*e)+1/6*(2*c*f-d*e )*(3*c*f-d*e)/c^2/(c*f-d*e)^2-1/6*(c*d*f-d^2*e)/c^2/(c*f-d*e)^2*(3*c*f-d*e )/d)*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x-c/d)/(-e/f-c/d))^(1/2)*((x+c/ d)/(-e/f+c/d))^(1/2)/(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)*EllipticF( ((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f-c/d))^(1/2))+1/3*(3*c*f-d*e)*d *f/c^2/(c*f-d*e)^2*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x-c/d)/(-e/f-c/d) )^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1 /2)*((-e/f-c/d)*EllipticE(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f-c/d) )^(1/2))+c/d*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f-c/d))^( 1/2))))
Time = 0.09 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {{\left (c^{2} d^{2} e^{2} - 3 \, c^{3} d e f + 6 \, c^{4} f^{2} + {\left (d^{4} e^{2} - 3 \, c d^{3} e f + 6 \, c^{2} d^{2} f^{2}\right )} x^{2} + 2 \, {\left (c d^{3} e^{2} - 3 \, c^{2} d^{2} e f + 6 \, c^{3} d f^{2}\right )} x\right )} \sqrt {-d^{2} f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right ) + 3 \, {\left (c^{2} d^{2} e f - 3 \, c^{3} d f^{2} + {\left (d^{4} e f - 3 \, c d^{3} f^{2}\right )} x^{2} + 2 \, {\left (c d^{3} e f - 3 \, c^{2} d^{2} f^{2}\right )} x\right )} \sqrt {-d^{2} f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) + 3 \, {\left (2 \, c d^{3} e f - 4 \, c^{2} d^{2} f^{2} + {\left (d^{4} e f - 3 \, c d^{3} f^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {f x + e}}{9 \, {\left (c^{4} d^{4} e^{2} f - 2 \, c^{5} d^{3} e f^{2} + c^{6} d^{2} f^{3} + {\left (c^{2} d^{6} e^{2} f - 2 \, c^{3} d^{5} e f^{2} + c^{4} d^{4} f^{3}\right )} x^{2} + 2 \, {\left (c^{3} d^{5} e^{2} f - 2 \, c^{4} d^{4} e f^{2} + c^{5} d^{3} f^{3}\right )} x\right )}} \] Input:
integrate(1/(d*x+c)^2/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="fri cas")
Output:
-1/9*((c^2*d^2*e^2 - 3*c^3*d*e*f + 6*c^4*f^2 + (d^4*e^2 - 3*c*d^3*e*f + 6* c^2*d^2*f^2)*x^2 + 2*(c*d^3*e^2 - 3*c^2*d^2*e*f + 6*c^3*d*f^2)*x)*sqrt(-d^ 2*f)*weierstrassPInverse(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f^2), -8/27*(d^2*e ^3 - 9*c^2*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f) + 3*(c^2*d^2*e*f - 3*c^3*d *f^2 + (d^4*e*f - 3*c*d^3*f^2)*x^2 + 2*(c*d^3*e*f - 3*c^2*d^2*f^2)*x)*sqrt (-d^2*f)*weierstrassZeta(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f^2), -8/27*(d^2*e ^3 - 9*c^2*e*f^2)/(d^2*f^3), weierstrassPInverse(4/3*(d^2*e^2 + 3*c^2*f^2) /(d^2*f^2), -8/27*(d^2*e^3 - 9*c^2*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f)) + 3*(2*c*d^3*e*f - 4*c^2*d^2*f^2 + (d^4*e*f - 3*c*d^3*f^2)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e))/(c^4*d^4*e^2*f - 2*c^5*d^3*e*f^2 + c^6*d^2*f^3 + (c ^2*d^6*e^2*f - 2*c^3*d^5*e*f^2 + c^4*d^4*f^3)*x^2 + 2*(c^3*d^5*e^2*f - 2*c ^4*d^4*e*f^2 + c^5*d^3*f^3)*x)
\[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \left (c + d x\right )^{2} \sqrt {e + f x}}\, dx \] Input:
integrate(1/(d*x+c)**2/(f*x+e)**(1/2)/(-d**2*x**2+c**2)**(1/2),x)
Output:
Integral(1/(sqrt(-(-c + d*x)*(c + d*x))*(c + d*x)**2*sqrt(e + f*x)), x)
\[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} + c^{2}} {\left (d x + c\right )}^{2} \sqrt {f x + e}} \,d x } \] Input:
integrate(1/(d*x+c)^2/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="max ima")
Output:
integrate(1/(sqrt(-d^2*x^2 + c^2)*(d*x + c)^2*sqrt(f*x + e)), x)
\[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} + c^{2}} {\left (d x + c\right )}^{2} \sqrt {f x + e}} \,d x } \] Input:
integrate(1/(d*x+c)^2/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="gia c")
Output:
integrate(1/(sqrt(-d^2*x^2 + c^2)*(d*x + c)^2*sqrt(f*x + e)), x)
Timed out. \[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {c^2-d^2\,x^2}\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(1/((e + f*x)^(1/2)*(c^2 - d^2*x^2)^(1/2)*(c + d*x)^2),x)
Output:
int(1/((e + f*x)^(1/2)*(c^2 - d^2*x^2)^(1/2)*(c + d*x)^2), x)
\[ \int \frac {1}{(c+d x)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{-d^{4} f \,x^{5}-2 c \,d^{3} f \,x^{4}-d^{4} e \,x^{4}-2 c \,d^{3} e \,x^{3}+2 c^{3} d f \,x^{2}+c^{4} f x +2 c^{3} d e x +c^{4} e}d x \] Input:
int(1/(d*x+c)^2/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x)
Output:
int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2))/(c**4*e + c**4*f*x + 2*c**3*d*e *x + 2*c**3*d*f*x**2 - 2*c*d**3*e*x**3 - 2*c*d**3*f*x**4 - d**4*e*x**4 - d **4*f*x**5),x)