Integrand size = 35, antiderivative size = 243 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\frac {46134551 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{38880}+\frac {26291}{540} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1679}{756} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {1}{9} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3+\frac {2629157597 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{163296 \sqrt {5-2 x}}-\frac {2161804579 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{54432 \sqrt {-5+2 x}} \]
-2161804579/326592*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)+2629157597/163296*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)+46134 551/38880*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)+26291/540*(7+5*x)*(2- 3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)+1679/756*(7+5*x)^2*(2-3*x)^(1/2)*( -5+2*x)^(1/2)*(1+4*x)^(1/2)+1/9*(7+5*x)^3*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+ 4*x)^(1/2)
Time = 5.51 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\frac {6 \sqrt {2-3 x} \sqrt {1+4 x} \left (-455686385+51484034 x+21329208 x^2+8614800 x^3+1512000 x^4\right )+2629157597 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )-2161804579 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{326592 \sqrt {-5+2 x}} \]
(6*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-455686385 + 51484034*x + 21329208*x^2 + 8 614800*x^3 + 1512000*x^4) + 2629157597*Sqrt[66]*Sqrt[5 - 2*x]*EllipticE[Ar cSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] - 2161804579*Sqrt[66]*Sqrt[5 - 2*x]*E llipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(326592*Sqrt[-5 + 2*x])
Time = 0.65 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {180, 25, 2103, 27, 2103, 27, 2118, 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.
| \(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {4 x+1} (5 x+7)^3}{\sqrt {2 x-5}} \, dx\) |
| \(\Big \downarrow \) 180 |
| \(\displaystyle \frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3-\frac {1}{18} \int -\frac {(5 x+7)^2 \left (-3358 x^2+565 x+699\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{18} \int \frac {(5 x+7)^2 \left (-3358 x^2+565 x+699\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 2103 |
| \(\displaystyle \frac {1}{18} \left (\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {1}{168} \int -\frac {2 (5 x+7) \left (-4416888 x^2-138145 x+993625\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \int \frac {(5 x+7) \left (-4416888 x^2-138145 x+993625\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 2103 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)-\frac {1}{120} \int -\frac {24 \left (-322941857 x^2-102379055 x+80234014\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \int \frac {-322941857 x^2-102379055 x+80234014}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 2118 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {1}{108} \int \frac {165 (228338691-956057308 x)}{2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \int \frac {228338691-956057308 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \left (-2161804579 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-478028654 \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx\right )+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \left (-\frac {478028654 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{\sqrt {5-2 x}}-2161804579 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \left (-2161804579 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {239014327 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \left (-\frac {196527689 \sqrt {22} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {239014327 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \left (-\frac {2161804579 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {239014327 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{84} \left (\frac {1}{5} \left (\frac {55}{72} \left (-\frac {196527689 \sqrt {\frac {22}{3}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {239014327 \sqrt {\frac {22}{3}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {322941857}{36} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {368074}{5} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1679}{42} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\right )+\frac {1}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^3\) |
(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3)/9 + ((1679*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^2)/42 + ((368074*Sqrt[2 - 3 *x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/5 + ((322941857*Sqrt[2 - 3*x]* Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/36 + (55*((-239014327*Sqrt[22/3]*Sqrt[-5 + 2 *x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[5 - 2*x] - (196 527689*Sqrt[22/3]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]] , 1/3])/Sqrt[-5 + 2*x]))/72)/5)/84)/18
3.1.44.3.1 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[((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[(((a_.) + (b_.)*(x_))^(m_)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)* (x_)])/Sqrt[(c_.) + (d_.)*(x_)], x_] :> Simp[2*(a + b*x)^m*Sqrt[c + d*x]*Sq rt[e + f*x]*(Sqrt[g + h*x]/(d*(2*m + 3))), x] - Simp[1/(d*(2*m + 3)) Int[ ((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*b*c* e*g*m + a*(c*(f*g + e*h) - 2*d*e*g*(m + 1)) - (b*(2*d*e*g - c*(f*g + e*h)*( 2*m + 1)) - a*(2*c*f*h - d*(2*m + 1)*(f*g + e*h)))*x - (2*a*d*f*h*m + b*(d* (f*g + e*h) - 2*c*f*h*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && GtQ[m, 0]
Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[ (c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S ymbol] :> Simp[2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/( d*f*h*(2*m + 3))), x] + Simp[1/(d*f*h*(2*m + 3)) Int[((a + b*x)^(m - 1)/( Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a *(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C* (2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b *B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x ^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2 *m] && GtQ[m, 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Time = 1.63 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(\frac {\sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (108864000 x^{6}+574905600 x^{5}+1227098543 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-2629157597 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+1259114976 x^{4}+2963596608 x^{3}-34609891236 x^{2}+13052783142 x +5468236620\right )}{7838208 x^{3}-22861440 x^{2}+6858432 x +3265920}\) | \(149\) |
| elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {51901 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{108}+\frac {13019611 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{7776}+\frac {10873271 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{57024 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {239014327 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{299376 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {86075 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{756}+\frac {125 x^{3} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{9}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(250\) |
| risch | \(-\frac {\left (756000 x^{3}+6197400 x^{2}+26158104 x +91137277\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{54432 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (\frac {10873271 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{171072 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {239014327 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{898128 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(257\) |
1/326592*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(108864000*x^6+5749056 00*x^5+1227098543*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*Ellip ticF(1/11*(11+44*x)^(1/2),3^(1/2))-2629157597*(1+4*x)^(1/2)*(2-3*x)^(1/2)* 22^(1/2)*(5-2*x)^(1/2)*EllipticE(1/11*(11+44*x)^(1/2),3^(1/2))+1259114976* x^4+2963596608*x^3-34609891236*x^2+13052783142*x+5468236620)/(24*x^3-70*x^ 2+21*x+10)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\frac {1}{54432} \, {\left (756000 \, x^{3} + 6197400 \, x^{2} + 26158104 \, x + 91137277\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} + \frac {4958213249}{419904} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) - \frac {2629157597}{163296} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]
1/54432*(756000*x^3 + 6197400*x^2 + 26158104*x + 91137277)*sqrt(4*x + 1)*s qrt(2*x - 5)*sqrt(-3*x + 2) + 4958213249/419904*sqrt(-6)*weierstrassPInver se(847/108, 6655/2916, x - 35/36) - 2629157597/163296*sqrt(-6)*weierstrass Zeta(847/108, 6655/2916, weierstrassPInverse(847/108, 6655/2916, x - 35/36 ))
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1} \left (5 x + 7\right )^{3}}{\sqrt {2 x - 5}}\, dx \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\int { \frac {{\left (5 \, x + 7\right )}^{3} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^3}{\sqrt {-5+2 x}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,{\left (5\,x+7\right )}^3}{\sqrt {2\,x-5}} \,d x \]