Integrand size = 33, antiderivative size = 162 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\frac {95}{18} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {1397 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{27 \sqrt {5-2 x}}-\frac {4543 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{36 \sqrt {-5+2 x}} \]
-4543/216*EllipticF(1/11*33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*66^(1/2)*(5-2 *x)^(1/2)/(-5+2*x)^(1/2)+1/4*(1+4*x)^(3/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)+13 97/27*EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*11^(1/2)*(-5+2* x)^(1/2)/(5-2*x)^(1/2)+95/18*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)
Time = 2.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \left (-995+218 x+72 x^2\right )+5588 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )-4543 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{216 \sqrt {-5+2 x}} \]
(6*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-995 + 218*x + 72*x^2) + 5588*Sqrt[66]*Sqr t[5 - 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] - 4543*Sqrt[66 ]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(216*Sqr t[-5 + 2*x])
Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {171, 27, 171, 27, 176, 124, 123, 131, 27, 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.
| \(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {4 x+1} (5 x+7)}{\sqrt {2 x-5}} \, dx\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{20} \int \frac {5 (213-380 x) \sqrt {4 x+1}}{2 \sqrt {2-3 x} \sqrt {2 x-5}}dx+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \frac {(213-380 x) \sqrt {4 x+1}}{\sqrt {2-3 x} \sqrt {2 x-5}}dx+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{8} \left (\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}-\frac {1}{9} \int -\frac {11 (537-2032 x)}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \int \frac {537-2032 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \left (-4543 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-1016 \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx\right )+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \left (-\frac {1016 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{\sqrt {5-2 x}}-4543 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \left (-4543 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {508 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \left (-\frac {413 \sqrt {22} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {508 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \left (-\frac {4543 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {508 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{8} \left (\frac {11}{9} \left (-\frac {413 \sqrt {\frac {22}{3}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {508 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {380}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}\) |
(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*(1 + 4*x)^(3/2))/4 + ((380*Sqrt[2 - 3*x]*Sqr t[-5 + 2*x]*Sqrt[1 + 4*x])/9 + (11*((-508*Sqrt[22/3]*Sqrt[-5 + 2*x]*Ellipt icE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[5 - 2*x] - (413*Sqrt[22/3 ]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[-5 + 2*x]))/9)/8
3.1.46.3.1 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[(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]
Time = 1.63 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(\frac {\sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (2453 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-5588 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+5184 x^{4}+13536 x^{3}-79044 x^{2}+27234 x +11940\right )}{5184 x^{3}-15120 x^{2}+4536 x +2160}\) | \(139\) |
| elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}+\frac {199 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{36}+\frac {179 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{264 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {254 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{99 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(205\) |
| risch | \(-\frac {\left (199+36 x \right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{36 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (\frac {179 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{792 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {254 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{297 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(247\) |
1/216*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(2453*(1+4*x)^(1/2)*(2-3* x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2))-55 88*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*EllipticE(1/11*(11+4 4*x)^(1/2),3^(1/2))+5184*x^4+13536*x^3-79044*x^2+27234*x+11940)/(24*x^3-70 *x^2+21*x+10)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\frac {1}{36} \, {\left (36 \, x + 199\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} + \frac {142417}{3888} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) - \frac {1397}{27} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
1/36*(36*x + 199)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2) + 142417/3888 *sqrt(-6)*weierstrassPInverse(847/108, 6655/2916, x - 35/36) - 1397/27*sqr t(-6)*weierstrassZeta(847/108, 6655/2916, weierstrassPInverse(847/108, 665 5/2916, x - 35/36))
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1} \cdot \left (5 x + 7\right )}{\sqrt {2 x - 5}}\, dx \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\left (5\,x+7\right )}{\sqrt {2\,x-5}} \,d x \]