Integrand size = 28, antiderivative size = 129 \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{55 \sqrt {3+5 x}}-\frac {27}{275} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {438}{275} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {9}{275} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
-2/55*(1-2*x)^(1/2)*(2+3*x)^(3/2)/(3+5*x)^(1/2)-27/275*(1-2*x)^(1/2)*(2+3* x)^(1/2)*(3+5*x)^(1/2)-438/1375*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35 *1155^(1/2))*35^(1/2)+9/1375*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*11 55^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 5.35 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x} (101+165 x)}{\sqrt {3+5 x}}+438 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-455 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1375} \] Input:
Integrate[(2 + 3*x)^(5/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]
Output:
((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(101 + 165*x))/Sqrt[3 + 5*x] + (438*I)*Sq rt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (455*I)*Sqrt[33]*Elli pticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/1375
Time = 0.24 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 27, 171, 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.
\(\displaystyle \int \frac {(3 x+2)^{5/2}}{\sqrt {1-2 x} (5 x+3)^{3/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle -\frac {2}{55} \int -\frac {3 \sqrt {3 x+2} (27 x+25)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{55} \int \frac {\sqrt {3 x+2} (27 x+25)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {3}{55} \left (-\frac {1}{15} \int -\frac {3 (876 x+563)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{55} \left (\frac {1}{10} \int \frac {876 x+563}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {3}{55} \left (\frac {1}{10} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {9}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {3}{55} \left (\frac {1}{10} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {9}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle \frac {3}{55} \left (\frac {1}{10} \left (-\frac {34}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {9}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\) |
Input:
Int[(2 + 3*x)^(5/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)),x]
Output:
(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(55*Sqrt[3 + 5*x]) + (3*((-9*Sqrt[1 - 2 *x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((-292*Sqrt[33]*EllipticE[ArcSin[Sqrt [3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (34*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7 ]*Sqrt[1 - 2*x]], 35/33])/5)/10))/55
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[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 = 0.52 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (561 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-876 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9900 x^{3}+7710 x^{2}-2290 x -2020\right )}{2750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(138\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {3 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}+\frac {563 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3850 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {438 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1925 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{1375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(220\) |
Input:
int((2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(561*2^(1/2)*(2+3*x)^(1/ 2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2) )-876*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28 +42*x)^(1/2),1/2*70^(1/2))+9900*x^3+7710*x^2-2290*x-2020)/(30*x^3+23*x^2-7 *x-6)
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {150 \, {\left (165 \, x + 101\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 5087 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 13140 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{41250 \, {\left (5 \, x + 3\right )}} \] Input:
integrate((2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
-1/41250*(150*(165*x + 101)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 5 087*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/91125, x + 23/ 90) - 13140*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/91125, wei erstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(5*x + 3)
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((2+3*x)**(5/2)/(1-2*x)**(1/2)/(3+5*x)**(3/2),x)
Output:
Integral((3*x + 2)**(5/2)/(sqrt(1 - 2*x)*(5*x + 3)**(3/2)), x)
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(5/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)), x)
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(5/2)/((5*x + 3)^(3/2)*sqrt(-2*x + 1)), x)
Timed out. \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int((3*x + 2)^(5/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(3/2)),x)
Output:
int((3*x + 2)^(5/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {-72 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -242 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+510 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x +306 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )+905 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x +543 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )}{600 x +360} \] Input:
int((2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(3/2),x)
Output:
( - 72*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 242*sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1) + 510*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt ( - 2*x + 1)*x**2)/(150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x)*x + 306* int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(150*x**4 + 205*x* *3 + 34*x**2 - 51*x - 18),x) + 905*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x)*x + 543*int((sq rt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x))/(120*(5*x + 3))