Integrand size = 26, antiderivative size = 86 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {33 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}-\frac {(3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {33 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \] Output:
33/49*(3+5*x)^(1/2)/(1-2*x)^(1/2)-1/7*(3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)+ 33/343*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 1.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {1}{343} \left (-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (45+64 x)}{-2+x+6 x^2}-33 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-33 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \] Input:
Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^2),x]
Output:
((-7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(45 + 64*x))/(-2 + x + 6*x^2) - 33*Sqrt[7 ]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 1 0*x])] - 33*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11 ] + Sqrt[5 - 10*x]))])/343
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.14, 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.
\(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{3/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {2 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {3}{7} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {2 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {3}{7} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {2 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {3}{7} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {3}{7} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )\) |
Input:
Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^2),x]
Output:
(2*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)) - (3*(-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])))/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. \(153\) vs. \(2(65)=130\).
Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.79
method | result | size |
default | \(\frac {\left (198 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+33 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -66 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+896 x \sqrt {-10 x^{2}-x +3}+630 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{686 \left (2+3 x \right ) \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}}\) | \(154\) |
Input:
int((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^2,x,method=_RETURNVERBOSE)
Output:
1/686*(198*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+ 33*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-66*7^(1/2) *arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+896*x*(-10*x^2-x+3)^(1 /2)+630*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2)/(-10*x^2- x+3)^(1/2)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {33 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (64 \, x + 45\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{686 \, {\left (6 \, x^{2} + x - 2\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^2,x, algorithm="fricas")
Output:
1/686*(33*sqrt(7)*(6*x^2 + x - 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(64*x + 45)*sqrt(5*x + 3)*sqrt (-2*x + 1))/(6*x^2 + x - 2)
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2}}\, dx \] Input:
integrate((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**2,x)
Output:
Integral((5*x + 3)**(3/2)/((1 - 2*x)**(3/2)*(3*x + 2)**2), x)
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {33}{686} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {320 \, x}{147 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {611}{441 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{63 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^2,x, algorithm="maxima")
Output:
-33/686*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 320/14 7*x/sqrt(-10*x^2 - x + 3) + 611/441/sqrt(-10*x^2 - x + 3) - 1/63/(3*sqrt(- 10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (65) = 130\).
Time = 0.22 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.55 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {33}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{245 \, {\left (2 \, x - 1\right )}} + \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^2,x, algorithm="giac")
Output:
-33/6860*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)))) - 22/245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 22/49*sqrt(10)*((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)))/(((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)
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2} \,d x \] Input:
int((5*x + 3)^(3/2)/((1 - 2*x)^(3/2)*(3*x + 2)^2),x)
Output:
int((5*x + 3)^(3/2)/((1 - 2*x)^(3/2)*(3*x + 2)^2), x)
Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.27 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {-99 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -66 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+99 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +66 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+448 \sqrt {5 x +3}\, x +315 \sqrt {5 x +3}}{343 \sqrt {-2 x +1}\, \left (3 x +2\right )} \] Input:
int((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^2,x)
Output:
( - 99*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 - 66*sqrt( - 2*x + 1)*sqrt(7) *atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2 ))/sqrt(2)) + 99*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(as in((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 66*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s qrt(11))/2))/sqrt(2)) + 448*sqrt(5*x + 3)*x + 315*sqrt(5*x + 3))/(343*sqrt ( - 2*x + 1)*(3*x + 2))