Integrand size = 28, antiderivative size = 129 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62}{165} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {4}{165} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
-2/15*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)-62/165*(1-2*x)^(1/2)*(2+3* x)^(1/2)/(3+5*x)^(1/2)+62/825*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1 155^(1/2))*35^(1/2)+4/825*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 = 2.97 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {2}{825} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x} (104+155 x)}{(3+5 x)^{3/2}}-31 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:
Integrate[(Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(5/2),x]
Output:
(2*((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(104 + 155*x))/(3 + 5*x)^(3/2) - (31*I )*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (35*I)*Sqrt[33]*E llipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/825
Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {12 x+1}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{15} \int \frac {12 x+1}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{15} \left (\frac {2}{11} \int -\frac {3 (31 x+23)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{11} \int \frac {31 x+23}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{11} \left (\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {31}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{11} \left (\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {31}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{11} \left (-\frac {4}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {31}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
Input:
Int[(Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3 + 5*x)^(5/2),x]
Output:
(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) + ((-62*Sqrt[1 - 2*x ]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (6*((-31*Sqrt[11/3]*EllipticE[ArcSin [Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt [3/7]*Sqrt[1 - 2*x]], 35/33])/5))/11)/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 + 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. \(214\) vs. \(2(93)=186\).
Time = 0.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(\frac {2 \left (330 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-155 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+198 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-93 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4650 x^{3}-3895 x^{2}+1030 x +1040\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{825 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(215\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{375 \left (x +\frac {3}{5}\right )^{2}}-\frac {62 \left (-30 x^{2}-5 x +10\right )}{825 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {46 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1155 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {62 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1155 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(225\) |
Input:
int((1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/825*(330*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^( 1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-155*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2 ),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+198*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))-93*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Elliptic E(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-4650*x^3-3895*x^2+1030*x+1040)*(1-2*x) ^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (155 \, x + 104\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1357 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2790 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{37125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/37125*(450*(155*x + 104)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1 357*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassPInverse(1159/675, 38998/9112 5, x + 23/90) + 2790*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassZeta(1159/67 5, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2 5*x^2 + 30*x + 9)
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {3 x + 2}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1-2*x)**(1/2)*(2+3*x)**(1/2)/(3+5*x)**(5/2),x)
Output:
Integral(sqrt(1 - 2*x)*sqrt(3*x + 2)/(5*x + 3)**(5/2), x)
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(3*x + 2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2), x)
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(3*x + 2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int(((1 - 2*x)^(1/2)*(3*x + 2)^(1/2))/(5*x + 3)^(5/2),x)
Output:
int(((1 - 2*x)^(1/2)*(3*x + 2)^(1/2))/(5*x + 3)^(5/2), x)
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+4650 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}+5580 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x +1674 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )-2125 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}-2550 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x -765 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )}{650 x^{2}+780 x +234} \] Input:
int((1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x)
Output:
(2*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 4650*int((sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475*x**4 + 785*x**3 - 15 3*x**2 - 243*x - 54),x)*x**2 + 5580*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54 ),x)*x + 1674*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750 *x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x) - 2125*int((sqrt( 3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x**2 - 2550*int((sqrt(3*x + 2)*sqrt(5*x + 3)*s qrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54) ,x)*x - 765*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x))/(26*(25*x**2 + 30*x + 9 ))