Integrand size = 26, antiderivative size = 144 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=-\frac {32735 \sqrt {3+5 x}}{15092 \sqrt {1-2 x}}+\frac {\sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}+\frac {27 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^2}+\frac {2865 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)}-\frac {102345 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
-102345/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-3273 5/15092*(3+5*x)^(1/2)/(1-2*x)^(1/2)+1/7*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1 /2)+27/28*(3+5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)+2865/392*(3+5*x)^(1/2)/(2+ 3*x)/(1-2*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {-7 \sqrt {3+5 x} \left (-421184-377658 x+1549935 x^2+1767690 x^3\right )-1125795 \sqrt {7-14 x} (2+3 x)^3 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{211288 \sqrt {1-2 x} (2+3 x)^3} \]
(-7*Sqrt[3 + 5*x]*(-421184 - 377658*x + 1549935*x^2 + 1767690*x^3) - 11257 95*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]) ])/(211288*Sqrt[1 - 2*x]*(2 + 3*x)^3)
Time = 0.24 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {114, 27, 168, 27, 168, 27, 169, 27, 104, 217}
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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {3 (23-60 x)}{2 (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {23-60 x}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \int \frac {35 (47-216 x)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \int \frac {47-216 x}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{7} \int -\frac {11460 x+817}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {573 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {573 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {1}{14} \int \frac {11460 x+817}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {2}{77} \int \frac {225159}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {26188 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {573 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {20469}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {26188 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {573 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {40938}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {26188 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {573 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (-\frac {40938 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {26188 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {573 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {27 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {\sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
Sqrt[3 + 5*x]/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + ((27*Sqrt[3 + 5*x])/(2*Sqrt[ 1 - 2*x]*(2 + 3*x)^2) + (5*((573*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x) ) + ((-26188*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (40938*ArcTan[Sqrt[1 - 2* x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/14))/4)/14
3.26.59.3.1 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[(((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 )/((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] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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] && ILtQ[m, -1]
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[((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])
Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(111)=222\).
Time = 1.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(\frac {\left (60792930 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+91189395 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+20264310 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+24747660 x^{3} \sqrt {-10 x^{2}-x +3}-22515900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +21699090 x^{2} \sqrt {-10 x^{2}-x +3}-9006360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-5287212 x \sqrt {-10 x^{2}-x +3}-5896576 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{422576 \left (2+3 x \right )^{3} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(257\) |
1/422576*(60792930*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x^4+91189395*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2) )*x^3+20264310*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x^2+24747660*x^3*(-10*x^2-x+3)^(1/2)-22515900*7^(1/2)*arctan(1/14*(37*x+20 )*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+21699090*x^2*(-10*x^2-x+3)^(1/2)-9006360* 7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-5287212*x*(-10* x^2-x+3)^(1/2)-5896576*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2 +3*x)^3/(-1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=-\frac {1125795 \, \sqrt {7} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1767690 \, x^{3} + 1549935 \, x^{2} - 377658 \, x - 421184\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{422576 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
-1/422576*(1125795*sqrt(7)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/ 14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14 *(1767690*x^3 + 1549935*x^2 - 377658*x - 421184)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{4} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{4} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (111) = 222\).
Time = 0.47 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.33 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {20469}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{132055 \, {\left (2 \, x - 1\right )}} + \frac {297 \, \sqrt {10} {\left (4937 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1785280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {188708800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {754835200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
20469/76832*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)))) - 32/132055*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2* x - 1) + 297/9604*sqrt(10)*(4937*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 178 5280*((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)))^3 + 188708800*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 754835200*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)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^4\,\sqrt {5\,x+3}} \,d x \]