Integrand size = 26, antiderivative size = 122 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}-\frac {4477 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}} \]
1/7*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^3-4477/2744*arctan(1/7*(1-2*x)^(1/ 2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+37/28*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x )^2-407/392*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (1648+4902 x+3547 x^2\right )}{(2+3 x)^3}-4477 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1648 + 4902*x + 3547*x^2))/(2 + 3*x)^3 - 4477*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2744
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {107, 105, 105, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {37}{14} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}dx+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {37}{14} \left (\frac {11}{4} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {37}{14} \left (\frac {11}{4} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {37}{14} \left (\frac {11}{4} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {37}{14} \left (\frac {11}{4} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}\) |
((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(7*(2 + 3*x)^3) + (37*((Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2*(2 + 3*x)^2) + (11*(-1/7*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/( 2 + 3*x) - (11*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])) )/4))/14
3.23.70.3.1 Defintions of rubi rules used
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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])
Time = 1.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (3547 x^{2}+4902 x +1648\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{392 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {4477 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5488 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(124\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (120879 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+241758 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+161172 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +49658 x^{2} \sqrt {-10 x^{2}-x +3}+35816 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+68628 x \sqrt {-10 x^{2}-x +3}+23072 \sqrt {-10 x^{2}-x +3}\right )}{5488 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(202\) |
-1/392*(-1+2*x)*(3+5*x)^(1/2)*(3547*x^2+4902*x+1648)/(2+3*x)^3/(-(-1+2*x)* (3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+4477/5488*7^(1/2)*arc tan(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 = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=-\frac {4477 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (3547 \, x^{2} + 4902 \, x + 1648\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5488 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
-1/5488*(4477*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37 *x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(3547*x^2 + 4 902*x + 1648)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{4}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\frac {4477}{5488} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {185}{294} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1369 \, \sqrt {-10 \, x^{2} - x + 3}}{1176 \, {\left (3 \, x + 2\right )}} \]
4477/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 185/ 294*sqrt(-10*x^2 - x + 3) + 1/7*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 111/196*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1369/117 6*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (95) = 190\).
Time = 0.39 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.54 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\frac {4477}{54880} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (37 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 24640 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2900800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {11603200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{196 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
4477/54880*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)))) - 121/196*sqrt(10)*(37*((sqrt(2)*sqrt(-10*x + 5) - sqr t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22) ))^5 - 24640*((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 - 2900800*(sqrt(2)*sqrt(- 10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11603200*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
Time = 17.48 (sec) , antiderivative size = 1273, normalized size of antiderivative = 10.43 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx=\text {Too large to display} \]
((2630142*((1 - 2*x)^(1/2) - 1)^5)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^5) - (1060932*((1 - 2*x)^(1/2) - 1)^3)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^3) - (16732*((1 - 2*x)^(1/2) - 1))/(765625*(3^(1/2) - (5*x + 3)^(1/2))) - (1 315071*((1 - 2*x)^(1/2) - 1)^7)/(153125*(3^(1/2) - (5*x + 3)^(1/2))^7) + ( 265233*((1 - 2*x)^(1/2) - 1)^9)/(12250*(3^(1/2) - (5*x + 3)^(1/2))^9) + (4 183*((1 - 2*x)^(1/2) - 1)^11)/(1960*(3^(1/2) - (5*x + 3)^(1/2))^11) + (131 174*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^2 ) + (121551*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(109375*(3^(1/2) - (5*x + 3)^ (1/2))^4) - (3688612*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(765625*(3^(1/2) - ( 5*x + 3)^(1/2))^6) + (121551*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(17500*(3^(1 /2) - (5*x + 3)^(1/2))^8) + (65587*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(9800 *(3^(1/2) - (5*x + 3)^(1/2))^10))/((5856*((1 - 2*x)^(1/2) - 1)^2)/(15625*( 3^(1/2) - (5*x + 3)^(1/2))^2) - (4224*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^( 1/2) - (5*x + 3)^(1/2))^4) - (14776*((1 - 2*x)^(1/2) - 1)^6)/(15625*(3^(1/ 2) - (5*x + 3)^(1/2))^6) - (1056*((1 - 2*x)^(1/2) - 1)^8)/(625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (366*((1 - 2*x)^(1/2) - 1)^10)/(25*(3^(1/2) - (5*x + 3)^(1/2))^10) + ((1 - 2*x)^(1/2) - 1)^12/(3^(1/2) - (5*x + 3)^(1/2))^12 - (7776*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(15625*(3^(1/2) - (5*x + 3)^(1/2)) ^3) + (34704*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(15625*(3^(1/2) - (5*x + 3)^ (1/2))^5) - (17352*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(3125*(3^(1/2) - (5...