Integrand size = 24, antiderivative size = 89 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {7}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {37}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {740 \sqrt {1-2 x}}{3993 (3+5 x)^{3/2}}-\frac {2960 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}} \] Output:
7/33/(1-2*x)^(3/2)/(3+5*x)^(3/2)+37/121/(1-2*x)^(1/2)/(3+5*x)^(3/2)-740/39 93*(1-2*x)^(1/2)/(3+5*x)^(3/2)-2960/43923*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.42 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {5728+26418 x-8880 x^2-59200 x^3}{43923 (1-2 x)^{3/2} (3+5 x)^{3/2}} \] Input:
Integrate[(2 + 3*x)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
Output:
(5728 + 26418*x - 8880*x^2 - 59200*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^( 3/2))
Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {87, 55, 55, 48}
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.
\(\displaystyle \int \frac {3 x+2}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {37}{55} \int \frac {1}{(1-2 x)^{5/2} (5 x+3)^{3/2}}dx-\frac {2}{165 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {37}{55} \left (\frac {20}{33} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)^{3/2}}dx+\frac {2}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\right )-\frac {2}{165 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {37}{55} \left (\frac {20}{33} \left (\frac {10}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {2}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\right )-\frac {2}{165 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {37}{55} \left (\frac {20}{33} \left (\frac {2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {20 \sqrt {1-2 x}}{121 \sqrt {5 x+3}}\right )+\frac {2}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\right )-\frac {2}{165 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
Input:
Int[(2 + 3*x)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
Output:
-2/(165*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (37*(2/(33*(1 - 2*x)^(3/2)*Sqrt [3 + 5*x]) + (20*(2/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x])/ (121*Sqrt[3 + 5*x])))/33))/55
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(-\frac {2 \left (29600 x^{3}+4440 x^{2}-13209 x -2864\right )}{43923 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}\) | \(32\) |
default | \(-\frac {2 \left (29600 x^{3}+4440 x^{2}-13209 x -2864\right )}{43923 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}\) | \(32\) |
orering | \(\frac {2 \left (-1+2 x \right ) \left (29600 x^{3}+4440 x^{2}-13209 x -2864\right )}{43923 \left (3+5 x \right )^{\frac {3}{2}} \left (1-2 x \right )^{\frac {5}{2}}}\) | \(37\) |
Input:
int((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/43923/(1-2*x)^(3/2)/(3+5*x)^(3/2)*(29600*x^3+4440*x^2-13209*x-2864)
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {2 \, {\left (29600 \, x^{3} + 4440 \, x^{2} - 13209 \, x - 2864\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-2/43923*(29600*x^3 + 4440*x^2 - 13209*x - 2864)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
\[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {3 x + 2}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((2+3*x)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)
Output:
Integral((3*x + 2)/((1 - 2*x)**(5/2)*(5*x + 3)**(5/2)), x)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {5920 \, x}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {296}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {74 \, x}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {40}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
5920/43923*x/sqrt(-10*x^2 - x + 3) + 296/43923/sqrt(-10*x^2 - x + 3) + 74/ 363*x/(-10*x^2 - x + 3)^(3/2) + 40/363/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (65) = 130\).
Time = 0.18 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.81 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{702768 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {7 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{5324 \, \sqrt {5 \, x + 3}} - \frac {8 \, {\left (181 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1188 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1098075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {231 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{43923 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \] Input:
integrate((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
-1/702768*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 7/5324*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/1 098075*(181*sqrt(5)*(5*x + 3) - 1188*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5 )/(2*x - 1)^2 + 1/43923*sqrt(10)*(5*x + 3)^(3/2)*(231*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {\sqrt {5\,x+3}\,\left (-\frac {1184\,x^3}{43923}-\frac {296\,x^2}{73205}+\frac {4403\,x}{366025}+\frac {2864}{1098075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\sqrt {1-2\,x}} \] Input:
int((3*x + 2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)
Output:
((5*x + 3)^(1/2)*((4403*x)/366025 - (296*x^2)/73205 - (1184*x^3)/43923 + 2 864/1098075))/((6*x*(1 - 2*x)^(1/2))/25 + (9*(1 - 2*x)^(1/2))/50 - (7*x^2* (1 - 2*x)^(1/2))/10 - x^3*(1 - 2*x)^(1/2))
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.48 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {\frac {59200}{43923} x^{3}+\frac {2960}{14641} x^{2}-\frac {8806}{14641} x -\frac {5728}{43923}}{\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (10 x^{2}+x -3\right )} \] Input:
int((2+3*x)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)
Output:
(2*(29600*x**3 + 4440*x**2 - 13209*x - 2864))/(43923*sqrt(5*x + 3)*sqrt( - 2*x + 1)*(10*x**2 + x - 3))