Integrand size = 19, antiderivative size = 89 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {20}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {400 \sqrt {1-2 x}}{3993 (3+5 x)^{3/2}}-\frac {1600 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}} \] Output:
2/33/(1-2*x)^(3/2)/(3+5*x)^(3/2)+20/121/(1-2*x)^(1/2)/(3+5*x)^(3/2)-400/39 93*(1-2*x)^(1/2)/(3+5*x)^(3/2)-1600/43923*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.42 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {2 \left (-361-7140 x+2400 x^2+16000 x^3\right )}{43923 (1-2 x)^{3/2} (3+5 x)^{3/2}} \] Input:
Integrate[1/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
Output:
(-2*(-361 - 7140*x + 2400*x^2 + 16000*x^3))/(43923*(1 - 2*x)^(3/2)*(3 + 5* x)^(3/2))
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {10}{11} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx+\frac {2}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {10}{11} \left (\frac {20}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx+\frac {2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {2}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {10}{11} \left (\frac {20}{11} \left (\frac {4}{33} \int \frac {1}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {2}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {10}{11} \left (\frac {20}{11} \left (-\frac {8 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {2}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
Input:
Int[1/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
Output:
2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (10*(2/(11*Sqrt[1 - 2*x]*(3 + 5*x )^(3/2)) + (20*((-2*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) - (8*Sqrt[1 - 2*x] )/(363*Sqrt[3 + 5*x])))/11))/11
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])
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(-\frac {2 \left (16000 x^{3}+2400 x^{2}-7140 x -361\right )}{43923 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}\) | \(32\) |
orering | \(\frac {2 \left (-1+2 x \right ) \left (16000 x^{3}+2400 x^{2}-7140 x -361\right )}{43923 \left (3+5 x \right )^{\frac {3}{2}} \left (1-2 x \right )^{\frac {5}{2}}}\) | \(37\) |
default | \(\frac {2}{33 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {20}{121 \sqrt {1-2 x}\, \left (3+5 x \right )^{\frac {3}{2}}}-\frac {400 \sqrt {1-2 x}}{3993 \left (3+5 x \right )^{\frac {3}{2}}}-\frac {1600 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}}\) | \(66\) |
Input:
int(1/(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)*(16000*x^3+2400*x^2-7140*x-361)
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {2 \, {\left (16000 \, x^{3} + 2400 \, x^{2} - 7140 \, x - 361\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate(1/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-2/43923*(16000*x^3 + 2400*x^2 - 7140*x - 361)*sqrt(5*x + 3)*sqrt(-2*x + 1 )/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Result contains complex when optimal does not.
Time = 3.71 (sec) , antiderivative size = 391, normalized size of antiderivative = 4.39 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\begin {cases} - \frac {32000 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{3}}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} + \frac {52800 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} - \frac {14520 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} - \frac {2662 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {32000 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{3}}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} + \frac {52800 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} - \frac {14520 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} - \frac {2662 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{5314683 x + 4392300 \left (x + \frac {3}{5}\right )^{3} - 9663060 \left (x + \frac {3}{5}\right )^{2} + \frac {15944049}{5}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)
Output:
Piecewise((-32000*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**3/(5314 683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5) + 52800* sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(5314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5) - 14520*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(5314683*x + 4392300*(x + 3/5)**3 - 9663060* (x + 3/5)**2 + 15944049/5) - 2662*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(5 314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5), 1/Ab s(x + 3/5) > 10/11), (-32000*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3 /5)**3/(5314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049 /5) + 52800*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(5314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5) - 14520*sqrt( 10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(5314683*x + 4392300*(x + 3/5) **3 - 9663060*(x + 3/5)**2 + 15944049/5) - 2662*sqrt(10)*I*sqrt(1 - 11/(10 *(x + 3/5)))/(5314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15 944049/5), True))
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {3200 \, x}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {160}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {40 \, x}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2}{363 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate(1/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
3200/43923*x/sqrt(-10*x^2 - x + 3) + 160/43923/sqrt(-10*x^2 - x + 3) + 40/ 363*x/(-10*x^2 - x + 3)^(3/2) + 2/363/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (65) = 130\).
Time = 0.15 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {5 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{702768 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{5324 \, \sqrt {5 \, x + 3}} - \frac {8 \, {\left (16 \, \sqrt {5} {\left (5 \, x + 3\right )} - 99 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615 \, {\left (2 \, x - 1\right )}^{2}} + \frac {5 \, {\left (\frac {33 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt {10}\right )} {\left (5 \, x + 3\right )}^{\frac {3}{2}}}{43923 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \] Input:
integrate(1/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
-5/702768*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 5/5324*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/2 19615*(16*sqrt(5)*(5*x + 3) - 99*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2 *x - 1)^2 + 5/43923*(33*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5 *x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)) ^3
Time = 2.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {\sqrt {5\,x+3}\,\left (-\frac {640\,x^3}{43923}-\frac {32\,x^2}{14641}+\frac {476\,x}{73205}+\frac {361}{1098075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\sqrt {1-2\,x}} \] Input:
int(1/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)
Output:
((5*x + 3)^(1/2)*((476*x)/73205 - (32*x^2)/14641 - (640*x^3)/43923 + 361/1 098075))/((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 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {\frac {32000}{43923} x^{3}+\frac {1600}{14641} x^{2}-\frac {4760}{14641} x -\frac {722}{43923}}{\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (10 x^{2}+x -3\right )} \] Input:
int(1/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)
Output:
(2*(16000*x**3 + 2400*x**2 - 7140*x - 361))/(43923*sqrt(5*x + 3)*sqrt( - 2 *x + 1)*(10*x**2 + x - 3))