Integrand size = 28, antiderivative size = 156 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=-\frac {12758 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{315 (2+3 x)^{3/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {31588 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{567 \sqrt {35}}-\frac {13834 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{567 \sqrt {35}} \] Output:
-12758/6615*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)-118/315*(1-2*x)^(1/2 )*(3+5*x)^(3/2)/(2+3*x)^(3/2)-2/15*(1-2*x)^(1/2)*(3+5*x)^(5/2)/(2+3*x)^(5/ 2)+31588/19845*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^( 1/2)-13834/19845*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 = 7.79 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (36919+113319 x+87021 x^2\right )}{(2+3 x)^{5/2}}-i \sqrt {33} \left (15794 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-9415 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{19845} \] Input:
Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(7/2),x]
Output:
(2*((-3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(36919 + 113319*x + 87021*x^2))/(2 + 3 *x)^(5/2) - I*Sqrt[33]*(15794*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 9415*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/19845
Time = 0.27 (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 = {108, 27, 167, 27, 167, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int \frac {(19-60 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^{5/2}}dx-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \int \frac {(19-60 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{5/2}}dx-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{15} \left (\frac {2}{63} \int \frac {3 (333-2690 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{3/2}}dx-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \int \frac {(333-2690 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{3/2}}dx-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \left (\frac {2}{21} \int -\frac {5 (31588 x+4919)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {12758 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \left (-\frac {5}{21} \int \frac {31588 x+4919}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {12758 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \left (-\frac {5}{21} \left (\frac {31588}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {70169}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {12758 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \left (-\frac {5}{21} \left (-\frac {70169}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {31588}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {12758 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{21} \left (-\frac {5}{21} \left (\frac {12758}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {31588}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {12758 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\) |
Input:
Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(7/2),x]
Output:
(-2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(15*(2 + 3*x)^(5/2)) + ((-118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(21*(2 + 3*x)^(3/2)) + ((-12758*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*Sqrt[2 + 3*x]) - (5*((-31588*Sqrt[11/3]*EllipticE[ArcSin[Sqrt [3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (12758*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[ 3/7]*Sqrt[1 - 2*x]], 35/33])/5))/21)/21)/15
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(248\) vs. \(2(116)=232\).
Time = 0.37 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.60
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3645 \left (\frac {2}{3}+x \right )^{3}}+\frac {86 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2835 \left (\frac {2}{3}+x \right )^{2}}-\frac {6446 \left (-30 x^{2}-3 x +9\right )}{6615 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {4919 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {31588 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(249\) |
| default | \(-\frac {\left (1894563 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+284292 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2526084 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+379056 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+842028 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+126352 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+5221260 x^{4}+7321266 x^{3}+1328676 x^{2}-1818228 x -664542\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{19845 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(308\) |
Input:
int((1-2*x)^(1/2)*(3+5*x)^(5/2)/(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 )*(-2/3645*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+86/2835*(-30*x^3-23*x^2+ 7*x+6)^(1/2)/(2/3+x)^2-6446/6615*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9)) ^(1/2)-4919/27783*(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))-31588/277 83*(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*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))))
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=-\frac {270 \, {\left (87021 \, x^{2} + 113319 \, x + 36919\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 141907 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 1421460 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{893025 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="fricas")
Output:
-1/893025*(270*(87021*x^2 + 113319*x + 36919)*sqrt(5*x + 3)*sqrt(3*x + 2)* sqrt(-2*x + 1) + 141907*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrass PInverse(1159/675, 38998/91125, x + 23/90) + 1421460*sqrt(-30)*(27*x^3 + 5 4*x^2 + 36*x + 8)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInver se(1159/675, 38998/91125, x + 23/90)))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((1-2*x)**(1/2)*(3+5*x)**(5/2)/(2+3*x)**(7/2),x)
Output:
Integral(sqrt(1 - 2*x)*(5*x + 3)**(5/2)/(3*x + 2)**(7/2), x)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2), x)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(7/2), x)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \] Input:
int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(7/2),x)
Output:
int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(7/2), x)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx=\frac {2800 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+7560 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -7482 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-17157150 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{3}-34314300 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{2}-22876200 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x -5083600 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right )+5849577 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{3}+11699154 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{2}+7799436 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x +1733208 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right )}{4536 x^{3}+9072 x^{2}+6048 x +1344} \] Input:
int((1-2*x)^(1/2)*(3+5*x)^(5/2)/(2+3*x)^(7/2),x)
Output:
(2800*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 7560*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 7482*sqrt(3*x + 2)*sqrt(5*x + 3)*sqr t( - 2*x + 1) - 17157150*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 - 4 8),x)*x**3 - 34314300*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**2 - 22876200*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 - 5083600*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) + 58495 77*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 + 11699154*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 + 7799436*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x**4 + 5 28*x**3 - 392*x**2 - 272*x - 48),x)*x + 1733208*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 - 39 2*x**2 - 272*x - 48),x))/(168*(27*x**3 + 54*x**2 + 36*x + 8))