Integrand size = 26, antiderivative size = 108 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {455}{144} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{12} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {3035}{432} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{27 \sqrt {7}} \]
3035/864*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+2/189*arctan(1/7*(1- 2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-5/12*(3+5*x)^(3/2)*(1-2*x)^(1/2) -455/144*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.01 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=\frac {-210 \sqrt {1-2 x} \left (381+815 x+300 x^2\right )-21245 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+64 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6048 \sqrt {3+5 x}} \]
(-210*Sqrt[1 - 2*x]*(381 + 815*x + 300*x^2) - 21245*Sqrt[30 + 50*x]*ArcTan [Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 64*Sqrt[7]*Sqrt[3 + 5*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6048*Sqrt[3 + 5*x])
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {113, 27, 171, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)} \, dx\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle -\frac {1}{12} \int -\frac {\sqrt {5 x+3} (455 x+306)}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int \frac {\sqrt {5 x+3} (455 x+306)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{24} \left (-\frac {1}{6} \int -\frac {15175 x+10106}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{12} \int \frac {15175 x+10106}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{12} \left (\frac {15175}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {32}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{12} \left (\frac {6070}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {32}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{12} \left (\frac {6070}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {64}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{12} \left (\frac {6070}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {64 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{12} \left (\frac {3035}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {64 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {455}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {5}{12} \sqrt {1-2 x} (5 x+3)^{3/2}\) |
(-5*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12 + ((-455*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] )/6 + ((3035*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (64*ArcTan[Sqr t[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/12)/24
3.25.82.3.1 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*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
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[(((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 = 98, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (21245 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-64 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-25200 x \sqrt {-10 x^{2}-x +3}-53340 \sqrt {-10 x^{2}-x +3}\right )}{12096 \sqrt {-10 x^{2}-x +3}}\) | \(98\) |
| risch | \(\frac {5 \left (127+60 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{144 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {3035 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1728}-\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{189}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(125\) |
1/12096*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(21245*10^(1/2)*arcsin(20/11*x+1/11)-6 4*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-25200*x*(-10* x^2-x+3)^(1/2)-53340*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {5}{144} \, {\left (60 \, x + 127\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3035}{1728} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + \frac {1}{189} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-5/144*(60*x + 127)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3035/1728*sqrt(5)*sqrt( 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10 *x^2 + x - 3)) + 1/189*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {25}{12} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {3035}{1728} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{189} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {635}{144} \, \sqrt {-10 \, x^{2} - x + 3} \]
-25/12*sqrt(-10*x^2 - x + 3)*x + 3035/1728*sqrt(10)*arcsin(20/11*x + 1/11) - 1/189*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 635/1 44*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (76) = 152\).
Time = 0.36 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.60 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=-\frac {1}{1890} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{144} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 91 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {3035}{1728} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
-1/1890*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1/144*(12*sqrt(5)*(5*x + 3) + 91*sqrt(5))*sqrt(5*x + 3) *sqrt(-10*x + 5) + 3035/1728*sqrt(10)*(pi + 2*arctan(-1/4*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))))
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,\left (3\,x+2\right )} \,d x \]