Integrand size = 28, antiderivative size = 63 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}-2 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \] Output:
6/7*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)-2/7*EllipticE(1/11*55^(1/2)* (1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Time = 2.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\frac {2}{7} \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}}+\sqrt {2} E\left (\arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \] Input:
Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]),x]
Output:
(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x] + Sqrt[2]*EllipticE[ArcS in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/7
Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {115, 27, 123}
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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {2}{7} \int \frac {5 \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{7} \int \frac {\sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}-2 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )\) |
Input:
Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]),x]
Output:
(6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) - 2*Sqrt[5/7]*EllipticE[ ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs. \(2(47)=94\).
Time = 0.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.10
| method | result | size |
| default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-30 x^{2}-3 x +9\right )}{7 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(132\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {20 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {10 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{49 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {60}{7} x^{2}-\frac {6}{7} x +\frac {18}{7}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(201\) |
Input:
int(1/(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/7*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(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))-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))-30*x^2-3*x+9)/(30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=-\frac {37 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 90 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) - 270 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{315 \, {\left (3 \, x + 2\right )}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas ")
Output:
-1/315*(37*sqrt(-30)*(3*x + 2)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 90*sqrt(-30)*(3*x + 2)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)) - 270*sqrt(5*x + 3 )*sqrt(3*x + 2)*sqrt(-2*x + 1))/(3*x + 2)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \] Input:
integrate(1/(1-2*x)**(1/2)/(2+3*x)**(3/2)/(3+5*x)**(1/2),x)
Output:
Integral(1/(sqrt(1 - 2*x)*(3*x + 2)**(3/2)*sqrt(5*x + 3)), x)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="maxima ")
Output:
integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(3/2)*sqrt(-2*x + 1)), x)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(3/2)*sqrt(-2*x + 1)), x)
Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \] Input:
int(1/((1 - 2*x)^(1/2)*(3*x + 2)^(3/2)*(5*x + 3)^(1/2)),x)
Output:
int(1/((1 - 2*x)^(1/2)*(3*x + 2)^(3/2)*(5*x + 3)^(1/2)), x)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{90 x^{4}+129 x^{3}+25 x^{2}-32 x -12}d x \right ) \] Input:
int(1/(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2),x)
Output:
- int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(90*x**4 + 129*x**3 + 25*x**2 - 32*x - 12),x)