Integrand size = 26, antiderivative size = 91 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)}+\frac {5}{9} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {103 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{63 \sqrt {7}} \]
103/441*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+5/9*arcsin (1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+1/21*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2 +3*x)
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{42+63 x}-\frac {5}{9} \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+\frac {103 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{63 \sqrt {7}} \]
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(42 + 63*x) - (5*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/9 + (103*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]) ])/(63*Sqrt[7])
Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {109, 27, 175, 64, 104, 217, 223}
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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}-\frac {1}{21} \int -\frac {350 x+199}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \int \frac {350 x+199}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{42} \left (\frac {350}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {103}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{42} \left (\frac {140}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {103}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{42} \left (\frac {140}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {206}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{42} \left (\frac {140}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {206 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{42} \left (\frac {70}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {206 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\) |
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)) + ((70*Sqrt[10]*ArcSin[Sqrt[2 /11]*Sqrt[3 + 5*x]])/3 + (206*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]) ])/(3*Sqrt[7]))/42
3.25.73.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Time = 1.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {5 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{18}-\frac {103 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{882}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(127\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (735 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -309 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +490 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-206 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+42 \sqrt {-10 x^{2}-x +3}\right )}{882 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(131\) |
-1/21*(-1+2*x)/(2+3*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3 +5*x))^(1/2)/(1-2*x)^(1/2)+(5/18*10^(1/2)*arcsin(20/11*x+1/11)-103/882*7^( 1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2)))*(( 1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\frac {103 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 245 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{882 \, {\left (3 \, x + 2\right )}} \]
1/882*(103*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3) *sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 245*sqrt(10)*(3*x + 2)*arctan(1/20*sqr t(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*sqrt( 5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{2}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\frac {5}{18} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {103}{882} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{21 \, {\left (3 \, x + 2\right )}} \]
5/18*sqrt(10)*arcsin(20/11*x + 1/11) - 103/882*sqrt(7)*arcsin(37/11*x/abs( 3*x + 2) + 20/11/abs(3*x + 2)) + 1/21*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (67) = 134\).
Time = 0.39 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.86 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=-\frac {103}{8820} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {5}{18} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
-103/8820*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*( (sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 5/18*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqr t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 22/21*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt (5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt( 2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq rt(-10*x + 5) - sqrt(22)))^2 + 280)
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2} \,d x \]