Integrand size = 28, antiderivative size = 249 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{9587193 \sqrt {35}}-\frac {139423864 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{9587193 \sqrt {35}} \] Output:
4/231/(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+1616/17787/(1-2*x)^(1/2)/( 2+3*x)^(5/2)/(3+5*x)^(1/2)-2206/207515*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x) ^(1/2)+499564/1452605*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+72709316/1 0168235*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-4839325048/67110351*(1-2 *x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+4839325048/335551755*EllipticE(1/11* 55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-139423864/335551755*Ellip ticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 8.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {91855922241+53503915182 x-673871013766 x^2-559512908172 x^3+1263428429256 x^4+1306617762960 x^5}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (604915631 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-623092960 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{335551755} \] Input:
Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]
Output:
(2*(-((91855922241 + 53503915182*x - 673871013766*x^2 - 559512908172*x^3 + 1263428429256*x^4 + 1306617762960*x^5)/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*S qrt[3 + 5*x])) - (4*I)*Sqrt[33]*(604915631*EllipticE[I*ArcSinh[Sqrt[9 + 15 *x]], -2/33] - 623092960*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/33 5551755
Time = 0.34 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {115, 27, 169, 27, 169, 169, 27, 169, 27, 169, 27, 176, 123, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^{7/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}-\frac {2}{231} \int -\frac {270 x+269}{2 (1-2 x)^{3/2} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \int \frac {270 x+269}{(1-2 x)^{3/2} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}-\frac {2}{77} \int -\frac {84840 x+55457}{2 \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \int \frac {84840 x+55457}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \int \frac {82725 x+429823}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {2}{21} \int \frac {3 (10683869-11240190 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \int \frac {10683869-11240190 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (88263934-54531987 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \int \frac {88263934-54531987 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {2}{11} \int \frac {3 (1209831262 x+765888881)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1209831262 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \int \frac {1209831262 x+765888881}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1209831262 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {199950619}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1209831262}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1209831262 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {199950619}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1209831262}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1209831262 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (-\frac {36354658}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1209831262}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1209831262 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {109063974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {749346 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )-\frac {6618 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {1616}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\) |
Input:
Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]
Output:
4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (1616/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + ((-6618*Sqrt[1 - 2*x])/(35*(2 + 3*x )^(5/2)*Sqrt[3 + 5*x]) + (2*((749346*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*Sqr t[3 + 5*x]) + ((109063974*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (10*((-1209831262*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (3*(( -1209831262*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/ 5 - (36354658*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] )/5))/11))/7)/7))/35)/77)/231
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[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[((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]
Time = 1.13 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.32
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {16 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{871563 \left (x -\frac {1}{2}\right )^{2}}-\frac {17216 \left (-30 x^{2}-38 x -12\right )}{67110351 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {3063555524 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{469772457 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4839325048 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{469772457 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35 \left (\frac {2}{3}+x \right )^{2}}-\frac {817326 \left (-30 x^{2}-3 x +9\right )}{84035 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {6250 \left (-30 x^{2}-5 x +10\right )}{1331 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(329\) |
| default | \(\frac {\frac {1308767688 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}}{10168235}-\frac {29035950288 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}}{111850585}+\frac {218127948 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}}{2033647}-\frac {4839325048 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}}{22370117}-\frac {290837264 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}}{10168235}+\frac {19357300192 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}}{335551755}-\frac {290837264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{10168235}+\frac {19357300192 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{335551755}-\frac {174215701728 x^{5}}{22370117}-\frac {120326517072 x^{4}}{15978655}+\frac {373008605448 x^{3}}{111850585}+\frac {1347742027532 x^{2}}{335551755}-\frac {35669276788 x}{111850585}-\frac {61237281494}{111850585}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}\) | \(391\) |
Input:
int(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(16/871563*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x-1/2)^2-17216/67110351*(-30*x^ 2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)-3063555524/469772457*(28+42*x )^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*Ellip ticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-4839325048/469772457*(28+42*x)^(1/2 )*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-1/15*Elli pticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF(1/7*(28+42*x)^(1/2), 1/2*70^(1/2)))-6/1715*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-6/35*(-30*x^3 -23*x^2+7*x+6)^(1/2)/(2/3+x)^2-817326/84035*(-30*x^2-3*x+9)/((2/3+x)*(-30* x^2-3*x+9))^(1/2)-6250/1331*(-30*x^2-5*x+10)/((x+3/5)*(-30*x^2-5*x+10))^(1 /2))
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (1306617762960 \, x^{5} + 1263428429256 \, x^{4} - 559512908172 \, x^{3} - 673871013766 \, x^{2} + 53503915182 \, x + 91855922241\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 41103880264 \, \sqrt {-30} {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 108884813580 \, \sqrt {-30} {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{15099828975 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} \] Input:
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="fricas ")
Output:
-2/15099828975*(45*(1306617762960*x^5 + 1263428429256*x^4 - 559512908172*x ^3 - 673871013766*x^2 + 53503915182*x + 91855922241)*sqrt(5*x + 3)*sqrt(3* x + 2)*sqrt(-2*x + 1) - 41103880264*sqrt(-30)*(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)*weierstrassPInverse(1159/675, 38998/91125 , x + 23/90) + 108884813580*sqrt(-30)*(540*x^6 + 864*x^5 + 99*x^4 - 425*x^ 3 - 154*x^2 + 52*x + 24)*weierstrassZeta(1159/675, 38998/91125, weierstras sPInverse(1159/675, 38998/91125, x + 23/90)))/(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="maxima ")
Output:
integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)),x)
Output:
int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\left (-2 x +1\right )^{\frac {5}{2}} \left (3 x +2\right )^{\frac {7}{2}} \left (5 x +3\right )^{\frac {3}{2}}}d x \] Input:
int(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x)
Output:
int(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x)