Integrand size = 26, antiderivative size = 122 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {185 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {19415 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {222185 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
-222185/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/7* (1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+185/196*(1-2*x)^(1/2)*(3+5*x)^(1/2)/ (2+3*x)^2+19415/2744*(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 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {\frac {63 \sqrt {1-2 x} \sqrt {3+5 x} \left (9248+26750 x+19415 x^2\right )}{(2+3 x)^3}-222185 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208} \]
((63*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9248 + 26750*x + 19415*x^2))/(2 + 3*x)^3 - 222185*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/19208
Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {114, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {15 (7-8 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \int \frac {7-8 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{14} \int \frac {801-740 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \int \frac {801-740 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {44437}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3883 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {44437}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3883 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {44437}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {3883 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {3883 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {44437 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\) |
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (5*((37*Sqrt[1 - 2*x]*Sqrt [3 + 5*x])/(14*(2 + 3*x)^2) + ((3883*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (44437*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])) /28))/14
3.25.97.3.1 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_)^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 = 1.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02
| method | result | size |
| risch | \(-\frac {9 \left (-1+2 x \right ) \sqrt {3+5 x}\, \left (19415 x^{2}+26750 x +9248\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2744 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {222185 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{38416 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(124\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5998995 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+11997990 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+7998660 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2446290 x^{2} \sqrt {-10 x^{2}-x +3}+1777480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3370500 x \sqrt {-10 x^{2}-x +3}+1165248 \sqrt {-10 x^{2}-x +3}\right )}{38416 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(202\) |
-9/2744*(-1+2*x)*(3+5*x)^(1/2)*(19415*x^2+26750*x+9248)/(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)+222185/38416*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 = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=-\frac {222185 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 126 \, {\left (19415 \, x^{2} + 26750 \, x + 9248\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
-1/38416*(222185*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)) - 126*(19415*x^ 2 + 26750*x + 9248)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \sqrt {5 x + 3}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {222185}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {185 \, \sqrt {-10 \, x^{2} - x + 3}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {19415 \, \sqrt {-10 \, x^{2} - x + 3}}{2744 \, {\left (3 \, x + 2\right )}} \]
222185/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1 /7*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 185/196*sqrt(-10*x ^2 - x + 3)/(9*x^2 + 12*x + 4) + 19415/2744*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 = 0.41 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.54 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\frac {44437}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {495 \, \sqrt {10} {\left (937 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 333760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {35170240 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {140680960 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
44437/76832*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)))) + 495/1372*sqrt(10)*(937*((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)))^5 + 333760*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 35170240*(sqrt(2)*s qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 140680960*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 = 19.35 (sec) , antiderivative size = 1273, normalized size of antiderivative = 10.43 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx=\text {Too large to display} \]
((133036146*((1 - 2*x)^(1/2) - 1)^5)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^ 5) - (52971444*((1 - 2*x)^(1/2) - 1)^3)/(5359375*(3^(1/2) - (5*x + 3)^(1/2 ))^3) - (885996*((1 - 2*x)^(1/2) - 1))/(5359375*(3^(1/2) - (5*x + 3)^(1/2) )) - (66518073*((1 - 2*x)^(1/2) - 1)^7)/(1071875*(3^(1/2) - (5*x + 3)^(1/2 ))^7) + (13242861*((1 - 2*x)^(1/2) - 1)^9)/(85750*(3^(1/2) - (5*x + 3)^(1/ 2))^9) + (221499*((1 - 2*x)^(1/2) - 1)^11)/(13720*(3^(1/2) - (5*x + 3)^(1/ 2))^11) + (6657318*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5359375*(3^(1/2) - (5 *x + 3)^(1/2))^2) + (6000759*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(765625*(3^( 1/2) - (5*x + 3)^(1/2))^4) - (181223838*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/( 5359375*(3^(1/2) - (5*x + 3)^(1/2))^6) + (6000759*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(122500*(3^(1/2) - (5*x + 3)^(1/2))^8) + (3328659*3^(1/2)*((1 - 2 *x)^(1/2) - 1)^10)/(68600*(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)/(15 625*(3^(1/2) - (5*x + 3)^(1/2))^5) - (17352*3^(1/2)*((1 - 2*x)^(1/2) - ...