Integrand size = 28, antiderivative size = 156 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{3/2}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 \sqrt {2+3 x}}-\frac {5594 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{1323 \sqrt {35}}-\frac {808 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{1323 \sqrt {35}} \] Output:
2/105*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)-404/2205*(1-2*x)^(1/2)*(3+ 5*x)^(1/2)/(2+3*x)^(3/2)+5594/15435*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1 /2)-5594/46305*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^( 1/2)-808/46305*EllipticF(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 = 6.48 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (8507+29322 x+25173 x^2\right )}{(2+3 x)^{5/2}}+i \sqrt {33} \left (2797 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-3395 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{46305} \] Input:
Integrate[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)),x]
Output:
(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8507 + 29322*x + 25173*x^2))/(2 + 3*x) ^(5/2) + I*Sqrt[33]*(2797*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 33 95*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/46305
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {109, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}-\frac {2}{105} \int -\frac {845 x+496}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \int \frac {845 x+496}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{105} \left (\frac {2}{21} \int \frac {2020 x+2279}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{21} \int \frac {2020 x+2279}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (2797 x+2336)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {2797 x+2336}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {3289}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2797}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {3289}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2797}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {598}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2797}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\) |
Input:
Int[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)),x]
Output:
(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^(5/2)) + ((-404*Sqrt[1 - 2* x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + ((5594*Sqrt[1 - 2*x]*Sqrt[3 + 5*x ])/(7*Sqrt[2 + 3*x]) + (10*((-2797*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*S qrt[1 - 2*x]], 35/33])/5 - (598*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt [1 - 2*x]], 35/33])/5))/7)/21)/105
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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(248\) vs. \(2(116)=232\).
Time = 0.73 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.60
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {4672 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{64827 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5594 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{64827 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2835 \left (\frac {2}{3}+x \right )^{3}}-\frac {404 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19845 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {11188}{3087} x^{2}-\frac {5594}{15435} x +\frac {5594}{5145}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(249\) |
| default | \(-\frac {2 \left (88803 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-25173 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+118404 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-33564 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+39468 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11188 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-755190 x^{4}-955179 x^{3}-116619 x^{2}+238377 x +76563\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{46305 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(308\) |
Input:
int((3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2),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 )*(4672/64827*(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)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+5594/64827*(2 8+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*EllipticE(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)))+2/2835*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-4 04/19845*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+5594/46305*(-30*x^2-3*x+9) /((2/3+x)*(-30*x^2-3*x+9))^(1/2))
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\frac {270 \, {\left (25173 \, x^{2} + 29322 \, x + 8507\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 145909 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 251730 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{2083725 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2),x, algorithm="fricas")
Output:
1/2083725*(270*(25173*x^2 + 29322*x + 8507)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq rt(-2*x + 1) - 145909*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrassPI nverse(1159/675, 38998/91125, x + 23/90) + 251730*sqrt(-30)*(27*x^3 + 54*x ^2 + 36*x + 8)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse( 1159/675, 38998/91125, x + 23/90)))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((3+5*x)**(3/2)/(1-2*x)**(1/2)/(2+3*x)**(7/2),x)
Output:
Integral((5*x + 3)**(3/2)/(sqrt(1 - 2*x)*(3*x + 2)**(7/2)), x)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(3/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)), x)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(3/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)), x)
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}} \,d x \] Input:
int((5*x + 3)^(3/2)/((1 - 2*x)^(1/2)*(3*x + 2)^(7/2)),x)
Output:
int((5*x + 3)^(3/2)/((1 - 2*x)^(1/2)*(3*x + 2)^(7/2)), x)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\frac {-30 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-45900 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{3}-91800 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{2}-61200 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x -13600 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right )+15633 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{3}+31266 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{2}+20844 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x +4632 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right )}{378 x^{3}+756 x^{2}+504 x +112} \] Input:
int((3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2),x)
Output:
( - 30*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 45900*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x** 4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x**3 - 91800*int((sqrt(3*x + 2)*s qrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 52 8*x**3 - 392*x**2 - 272*x - 48),x)*x**2 - 61200*int((sqrt(3*x + 2)*sqrt(5* x + 3)*sqrt( - 2*x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x - 13600*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sq rt( - 2*x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x* *2 - 272*x - 48),x) + 15633*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48) ,x)*x**3 + 31266*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x **6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x**2 + 20844*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241* x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x + 4632*int((sqrt (3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x** 4 + 528*x**3 - 392*x**2 - 272*x - 48),x))/(14*(27*x**3 + 54*x**2 + 36*x + 8))