Integrand size = 33, antiderivative size = 367 \[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {8 (d e-3 c f) \sqrt {e+f x} \sqrt {c^2-d^2 x^2}}{15 f^2}-\frac {2 (c+d x) \sqrt {e+f x} \sqrt {c^2-d^2 x^2}}{5 f}+\frac {4 \sqrt {c} \left (4 d^2 e^2-15 c d e f+27 c^2 f^2\right ) \sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right )|-\frac {2 c f}{d e-c f}\right )}{15 f^3 \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {c^2-d^2 x^2}}-\frac {4 \sqrt {c} (d e-c f) \left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right ) \sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{15 d f^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \] Output:
8/15*(-3*c*f+d*e)*(f*x+e)^(1/2)*(-d^2*x^2+c^2)^(1/2)/f^2-2/5*(d*x+c)*(f*x+ e)^(1/2)*(-d^2*x^2+c^2)^(1/2)/f+4/15*c^(1/2)*(27*c^2*f^2-15*c*d*e*f+4*d^2* e^2)*(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))/f^3/(d*(f*x+e)/(-c*f+d*e))^( 1/2)/(-d^2*x^2+c^2)^(1/2)-4/15*c^(1/2)*(-c*f+d*e)*(15*c^2*f^2-11*c*d*e*f+4 *d^2*e^2)*(d*x+c)^(1/2)*(1-d*x/c)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*Ellip ticF(1/2*(d*x+c)^(1/2)*2^(1/2)/c^(1/2),(-2*c*f/(-c*f+d*e))^(1/2))/d/f^3/(f *x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2)
Result contains complex when optimal does not.
Time = 24.20 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-\frac {2 (e+f x) (-4 d e+15 c f+3 d f x)}{f^2}-\frac {4 \left (f^2 \sqrt {-\frac {d e+c f}{d}} \left (4 d^2 e^2-15 c d e f+27 c^2 f^2\right ) \left (c^2-d^2 x^2\right )+i d \left (4 d^3 e^3-11 c d^2 e^2 f+12 c^2 d e f^2+27 c^3 f^3\right ) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-c f}{d e+c f}\right )-2 i c d f \left (2 d^2 e^2-7 c d e f+21 c^2 f^2\right ) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-c f}{d e+c f}\right )\right )}{d f^4 \sqrt {-\frac {d e+c f}{d}} \left (c^2-d^2 x^2\right )}\right )}{15 \sqrt {e+f x}} \] Input:
Integrate[(c + d*x)^3/(Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
Output:
(Sqrt[c^2 - d^2*x^2]*((-2*(e + f*x)*(-4*d*e + 15*c*f + 3*d*f*x))/f^2 - (4* (f^2*Sqrt[-((d*e + c*f)/d)]*(4*d^2*e^2 - 15*c*d*e*f + 27*c^2*f^2)*(c^2 - d ^2*x^2) + I*d*(4*d^3*e^3 - 11*c*d^2*e^2*f + 12*c^2*d*e*f^2 + 27*c^3*f^3)*S qrt[(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)] - (2*I)*c*d*f*(2*d^2*e^2 - 7*c*d*e*f + 21*c^2*f^2)*Sqr t[(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)]))/(d*f^4*Sqrt[-((d*e + c*f)/d)]*(c^2 - d^2*x^2))))/(15*S qrt[e + f*x])
Time = 0.63 (sec) , antiderivative size = 403, 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, 113, 25, 27, 171, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{\sqrt {c^2-d^2 x^2} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 718 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \int \frac {(c+d x)^{5/2}}{\sqrt {c-d x} \sqrt {e+f x}}dx}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (-\frac {2 \int -\frac {d \sqrt {c+d x} (c (d e+3 c f)-2 d (d e-3 c f) x)}{\sqrt {c-d x} \sqrt {e+f x}}dx}{5 d f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \int \frac {d \sqrt {c+d x} (c (d e+3 c f)-2 d (d e-3 c f) x)}{\sqrt {c-d x} \sqrt {e+f x}}dx}{5 d f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \int \frac {\sqrt {c+d x} (c (d e+3 c f)-2 d (d e-3 c f) x)}{\sqrt {c-d x} \sqrt {e+f x}}dx}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}-\frac {2 \int -\frac {d \left (f (d e+15 c f) c^2+d \left (4 d^2 e^2-15 c d f e+27 c^2 f^2\right ) x\right )}{2 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 d f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {\int \frac {f (d e+15 c f) c^2+d \left (4 d^2 e^2-15 c d f e+27 c^2 f^2\right ) x}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \int \frac {\sqrt {c-d x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\frac {\sqrt {2} \sqrt {c-d x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {2} \sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\frac {\sqrt {c-d x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {\frac {\sqrt {2} c \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {\sqrt {2}}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {c-d x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {\frac {2 c \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {c-d x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {\frac {2 c \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {c-d x} \sqrt {e+f x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {2 \left (\frac {\frac {4 c^{3/2} \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{d \sqrt {c-d x} \sqrt {e+f x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}}{3 f}+\frac {4 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x} (d e-3 c f)}{3 f}\right )}{5 f}-\frac {2 \sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}{5 f}\right )}{\sqrt {c^2-d^2 x^2}}\) |
Input:
Int[(c + d*x)^3/(Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
Output:
(Sqrt[c - d*x]*Sqrt[c + d*x]*((-2*Sqrt[c - d*x]*(c + d*x)^(3/2)*Sqrt[e + f *x])/(5*f) + (2*((4*(d*e - 3*c*f)*Sqrt[c - d*x]*Sqrt[c + d*x]*Sqrt[e + f*x ])/(3*f) + ((-2*Sqrt[2]*Sqrt[-(d*e) + c*f]*(4*d^2*e^2 - 15*c*d*e*f + 27*c^ 2*f^2)*Sqrt[c - d*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticE[ArcSin[(Sqr t[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], (1 - (d*e)/(c*f))/2])/(d*Sqrt[f]* Sqrt[(c - d*x)/c]*Sqrt[e + f*x]) + (4*c^(3/2)*(2*d^2*e^2 - 7*c*d*e*f + 21* c^2*f^2)*Sqrt[(c - d*x)/c]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[ArcSi n[Sqrt[c + d*x]/(Sqrt[2]*Sqrt[c])], (-2*c*f)/(d*e - c*f)])/(d*Sqrt[c - d*x ]*Sqrt[e + f*x]))/(3*f)))/(5*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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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]
Time = 3.85 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.72
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (-\frac {2 d x \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}{5 f}-\frac {2 \left (3 c \,d^{2}-\frac {4 d^{3} e}{5 f}\right ) \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}{3 d^{2} f}+\frac {2 \left (c^{3}+\frac {2 e \,c^{2} d}{5 f}+\frac {c^{2} \left (3 c \,d^{2}-\frac {4 d^{3} e}{5 f}\right )}{3 d^{2}}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {2 \left (\frac {18 c^{2} d}{5}-\frac {2 e \left (3 c \,d^{2}-\frac {4 d^{3} e}{5 f}\right )}{3 f}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}\) | \(631\) |
| risch | \(-\frac {2 \left (3 d f x +15 c f -4 d e \right ) \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{15 f^{2}}+\frac {2 \left (\frac {2 d \left (27 c^{2} f^{2}-15 c d e f +4 d^{2} e^{2}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {2 e \,c^{2} d f \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {30 f^{2} c^{3} \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\right ) \sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}}{15 f^{2} \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}\) | \(725\) |
| default | \(\text {Expression too large to display}\) | \(1177\) |
Input:
int((d*x+c)^3/(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)*(-2/5*d/ f*x*(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)-2/3*(3*c*d^2-4/5*d^3*e/f)/d ^2/f*(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)+2*(c^3+2/5*e*c^2*d/f+1/3*c ^2/d^2*(3*c*d^2-4/5*d^3*e/f))*(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))+2*(18/5*c^2*d-2/3*e/f*(3*c*d^2-4/5*d^3*e/f))*(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 = 0.08 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {2 \, {\left (2 \, {\left (4 \, d^{3} e^{3} - 15 \, c d^{2} e^{2} f + 24 \, c^{2} d e f^{2} - 45 \, c^{3} f^{3}\right )} \sqrt {-d^{2} f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right ) + 6 \, {\left (4 \, d^{3} e^{2} f - 15 \, c d^{2} e f^{2} + 27 \, c^{2} d f^{3}\right )} \sqrt {-d^{2} f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) - 3 \, {\left (3 \, d^{3} f^{3} x - 4 \, d^{3} e f^{2} + 15 \, c d^{2} f^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {f x + e}\right )}}{45 \, d^{2} f^{4}} \] Input:
integrate((d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="frica s")
Output:
2/45*(2*(4*d^3*e^3 - 15*c*d^2*e^2*f + 24*c^2*d*e*f^2 - 45*c^3*f^3)*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) + 6*(4*d^3*e^2*f - 15*c*d ^2*e*f^2 + 27*c^2*d*f^3)*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), weierstrassPInve rse(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*(3*d^3*f^3*x - 4*d^3*e*f^2 + 15*c*d^2*f^3) *sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e))/(d^2*f^4)
\[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {\left (c + d x\right )^{3}}{\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \sqrt {e + f x}}\, dx \] Input:
integrate((d*x+c)**3/(f*x+e)**(1/2)/(-d**2*x**2+c**2)**(1/2),x)
Output:
Integral((c + d*x)**3/(sqrt(-(-c + d*x)*(c + d*x))*sqrt(e + f*x)), x)
\[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{\sqrt {-d^{2} x^{2} + c^{2}} \sqrt {f x + e}} \,d x } \] Input:
integrate((d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="maxim a")
Output:
integrate((d*x + c)^3/(sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e)), x)
\[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{\sqrt {-d^{2} x^{2} + c^{2}} \sqrt {f x + e}} \,d x } \] Input:
integrate((d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="giac" )
Output:
integrate((d*x + c)^3/(sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e)), x)
Timed out. \[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{\sqrt {e+f\,x}\,\sqrt {c^2-d^2\,x^2}} \,d x \] Input:
int((c + d*x)^3/((e + f*x)^(1/2)*(c^2 - d^2*x^2)^(1/2)),x)
Output:
int((c + d*x)^3/((e + f*x)^(1/2)*(c^2 - d^2*x^2)^(1/2)), x)
\[ \int \frac {(c+d x)^3}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {-18 \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} f -2 \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, d^{2} e x -27 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, x^{2}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \right ) c^{2} d^{2} f^{2}+15 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, x^{2}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \right ) c \,d^{3} e f -4 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, x^{2}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \right ) d^{4} e^{2}+9 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \right ) c^{4} f^{2}+5 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \right ) c^{3} d e f +2 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \right ) c^{2} d^{2} e^{2}}{5 d e f} \] Input:
int((d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x)
Output:
( - 18*sqrt(e + f*x)*sqrt(c**2 - d**2*x**2)*c**2*f - 2*sqrt(e + f*x)*sqrt( c**2 - d**2*x**2)*d**2*e*x - 27*int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2)* x**2)/(c**2*e + c**2*f*x - d**2*e*x**2 - d**2*f*x**3),x)*c**2*d**2*f**2 + 15*int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2)*x**2)/(c**2*e + c**2*f*x - d* *2*e*x**2 - d**2*f*x**3),x)*c*d**3*e*f - 4*int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2)*x**2)/(c**2*e + c**2*f*x - d**2*e*x**2 - d**2*f*x**3),x)*d**4*e **2 + 9*int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2))/(c**2*e + c**2*f*x - d* *2*e*x**2 - d**2*f*x**3),x)*c**4*f**2 + 5*int((sqrt(e + f*x)*sqrt(c**2 - d **2*x**2))/(c**2*e + c**2*f*x - d**2*e*x**2 - d**2*f*x**3),x)*c**3*d*e*f + 2*int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2))/(c**2*e + c**2*f*x - d**2*e* x**2 - d**2*f*x**3),x)*c**2*d**2*e**2)/(5*d*e*f)