Integrand size = 26, antiderivative size = 79 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {2 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \] Output:
-2/49*(3+5*x)^(1/2)/(1-2*x)^(1/2)+2/21*(3+5*x)^(3/2)/(1-2*x)^(3/2)-2/343*7 ^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {2 (3+5 x)^{3/2} \left (-7+\frac {3 (1-2 x)}{3+5 x}\right )}{147 (1-2 x)^{3/2}}-\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \] Input:
Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(5/2)*(2 + 3*x)),x]
Output:
(-2*(3 + 5*x)^(3/2)*(-7 + (3*(1 - 2*x))/(3 + 5*x)))/(147*(1 - 2*x)^(3/2)) - (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {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.
\(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{5/2} (3 x+2)} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{7} \int \frac {\sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)}dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{7} \left (-\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
Input:
Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(5/2)*(2 + 3*x)),x]
Output:
(2*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)) + ((-2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) - (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/7
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_)^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. \(146\) vs. \(2(58)=116\).
Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {\left (12 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-12 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +3 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+574 x \sqrt {-10 x^{2}-x +3}+252 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{1029 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\) | \(147\) |
Input:
int((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x),x,method=_RETURNVERBOSE)
Output:
1/1029*(12*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2- 12*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3*7^(1/2)* arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+574*x*(-10*x^2-x+3)^(1/ 2)+252*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(-10*x^2-x+3)^(1/2 )
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.09 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {3 \, \sqrt {7} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (41 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1029 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="fricas")
Output:
-1/1029*(3*sqrt(7)*(4*x^2 - 4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt( 5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(41*x + 18)*sqrt(5*x + 3)*s qrt(-2*x + 1))/(4*x^2 - 4*x + 1)
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \cdot \left (3 x + 2\right )}\, dx \] Input:
integrate((3+5*x)**(3/2)/(1-2*x)**(5/2)/(2+3*x),x)
Output:
Integral((5*x + 3)**(3/2)/((1 - 2*x)**(5/2)*(3*x + 2)), x)
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {1}{343} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {205 \, x}{147 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x^{2}}{6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {37}{588 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1385 \, x}{84 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {67}{28 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="maxima")
Output:
1/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 205/147* x/sqrt(-10*x^2 - x + 3) + 125/6*x^2/(-10*x^2 - x + 3)^(3/2) - 37/588/sqrt( -10*x^2 - x + 3) + 1385/84*x/(-10*x^2 - x + 3)^(3/2) + 67/28/(-10*x^2 - x + 3)^(3/2)
Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.43 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {1}{3430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {2 \, {\left (41 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3675 \, {\left (2 \, x - 1\right )}^{2}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="giac")
Output:
1/3430*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22)))) + 2/3675*(41*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3) *sqrt(-10*x + 5)/(2*x - 1)^2
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )} \,d x \] Input:
int((5*x + 3)^(3/2)/((1 - 2*x)^(5/2)*(3*x + 2)),x)
Output:
int((5*x + 3)^(3/2)/((1 - 2*x)^(5/2)*(3*x + 2)), x)
Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.47 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {\frac {4 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x}{343}-\frac {2 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{343}-\frac {4 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x}{343}+\frac {2 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{343}-\frac {82 \sqrt {5 x +3}\, x}{147}-\frac {12 \sqrt {5 x +3}}{49}}{\sqrt {-2 x +1}\, \left (2 x -1\right )} \] Input:
int((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x),x)
Output:
(2*(6*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 3*sqrt( - 2*x + 1)*sqrt(7)*a tan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2)) /sqrt(2)) - 6*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin( (sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 3*sqrt( - 2*x + 1)*s qrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 11))/2))/sqrt(2)) - 287*sqrt(5*x + 3)*x - 126*sqrt(5*x + 3)))/(1029*sqrt( - 2*x + 1)*(2*x - 1))