Integrand size = 26, antiderivative size = 93 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {29 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}-\frac {451 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}} \] Output:
-1/14*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+29*(1-2*x)^(1/2)*(3+5*x)^(1/2) /(392+588*x)-451/1372*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/ 2))
Time = 1.53 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (44+87 x)}{(2+3 x)^2}+451 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+451 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )}{1372} \] Input:
Integrate[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]
Output:
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(44 + 87*x))/(2 + 3*x)^2 + 451*Sqrt[7]*Arc Tan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x]) ] + 451*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))])/1372
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {107, 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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {41}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {41}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {41}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {41}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\) |
Input:
Int[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]
Output:
(3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(14*(2 + 3*x)^2) + (41*(-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])))/28
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 = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (44+87 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{196 \left (2+3 x \right )^{2} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {451 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2744 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(119\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4059 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+5412 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1804 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1218 x \sqrt {-10 x^{2}-x +3}+616 \sqrt {-10 x^{2}-x +3}\right )}{2744 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(154\) |
Input:
int((3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/196*(-1+2*x)*(3+5*x)^(1/2)*(44+87*x)/(2+3*x)^2/(-(-1+2*x)*(3+5*x))^(1/2 )*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+451/2744*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 = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {451 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (87 \, x + 44\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2744 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="fricas")
Output:
-1/2744*(451*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sq rt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(87*x + 44)*sqrt(5*x + 3 )*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3}}\, dx \] Input:
integrate((3+5*x)**(1/2)/(1-2*x)**(1/2)/(2+3*x)**3,x)
Output:
Integral(sqrt(5*x + 3)/(sqrt(1 - 2*x)*(3*x + 2)**3), x)
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {451}{2744} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {29 \, \sqrt {-10 \, x^{2} - x + 3}}{196 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="maxima")
Output:
451/2744*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/14* sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 29/196*sqrt(-10*x^2 - x + 3)/(3 *x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (72) = 144\).
Time = 0.21 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {451}{27440} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (41 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {7000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {28000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \] Input:
integrate((3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="giac")
Output:
451/27440*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)))) - 11/98*sqrt(10)*(41*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^ 3 - 7000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 28000*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)^2
Time = 8.87 (sec) , antiderivative size = 1037, normalized size of antiderivative = 11.15 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\text {Too large to display} \] Input:
int((5*x + 3)^(1/2)/((1 - 2*x)^(1/2)*(3*x + 2)^3),x)
Output:
((199*((1 - 2*x)^(1/2) - 1)^5)/(245*(3^(1/2) - (5*x + 3)^(1/2))^5) - (398* ((1 - 2*x)^(1/2) - 1)^3)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^3) - (314*((1 - 2*x)^(1/2) - 1))/(30625*(3^(1/2) - (5*x + 3)^(1/2))) + (157*((1 - 2*x)^(1 /2) - 1)^7)/(980*(3^(1/2) - (5*x + 3)^(1/2))^7) + (2197*3^(1/2)*((1 - 2*x) ^(1/2) - 1)^2)/(30625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (4276*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(30625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (2197*3^(1/2)* ((1 - 2*x)^(1/2) - 1)^6)/(4900*(3^(1/2) - (5*x + 3)^(1/2))^6))/((544*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (1764*((1 - 2*x) ^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (136*((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^( 1/2) - (5*x + 3)^(1/2))^8 - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(625*(3^( 1/2) - (5*x + 3)^(1/2))^3) + (48*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(125*(3^ (1/2) - (5*x + 3)^(1/2))^5) + (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(5*(3^( 1/2) - (5*x + 3)^(1/2))^7) - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(625*(3^(1 /2) - (5*x + 3)^(1/2))) + 16/625) - (451*7^(1/2)*atan(((451*7^(1/2)*((2706 *3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^( 1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^ (1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^ (1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(12 25*(3^(1/2) - (5*x + 3)^(1/2))^2)))/2744 + (451*7^(1/2)*((2706*3^(1/2))...
Time = 0.26 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {4059 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+5412 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +1804 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-4059 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-5412 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -1804 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+609 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +308 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{12348 x^{2}+16464 x +5488} \] Input:
int((3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3,x)
Output:
(4059*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) )/sqrt(11))/2))/sqrt(2))*x**2 + 5412*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan (asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 1804*sqrt(7)*a tan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2)) /sqrt(2)) - 4059*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 5412*sqrt(7)*atan((sqrt(33) + s qrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 180 4*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sq rt(11))/2))/sqrt(2)) + 609*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 308*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(1372*(9*x**2 + 12*x + 4))