Integrand size = 24, antiderivative size = 116 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {231 \sqrt {1-2 x} \sqrt {3+5 x}}{1000}+\frac {7}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {1}{25} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {2541 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1000 \sqrt {10}} \] Output:
-2/25*(1-2*x)^(5/2)/(3+5*x)^(1/2)+231/1000*(1-2*x)^(1/2)*(3+5*x)^(1/2)+7/1 00*(1-2*x)^(3/2)*(3+5*x)^(1/2)+1/25*(1-2*x)^(5/2)*(3+5*x)^(1/2)+2541/10000 *arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (943+1125 x-1340 x^2+800 x^3\right )-2541 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{10000 \sqrt {3+5 x}} \] Input:
Integrate[((1 - 2*x)^(5/2)*(2 + 3*x))/(3 + 5*x)^(3/2),x]
Output:
(10*Sqrt[1 - 2*x]*(943 + 1125*x - 1340*x^2 + 800*x^3) - 2541*Sqrt[30 + 50* x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(10000*Sqrt[3 + 5*x])
Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 60, 60, 60, 64, 223}
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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)}{(5 x+3)^{3/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {21}{55} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{7/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {21}{55} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {2 (1-2 x)^{7/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {21}{55} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {2 (1-2 x)^{7/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {21}{55} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {2 (1-2 x)^{7/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {21}{55} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {2 (1-2 x)^{7/2}}{55 \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {21}{55} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {2 (1-2 x)^{7/2}}{55 \sqrt {5 x+3}}\) |
Input:
Int[((1 - 2*x)^(5/2)*(2 + 3*x))/(3 + 5*x)^(3/2),x]
Output:
(-2*(1 - 2*x)^(7/2))/(55*Sqrt[3 + 5*x]) + (21*(((1 - 2*x)^(5/2)*Sqrt[3 + 5 *x])/15 + (11*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sq rt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20)) /6))/55
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
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]))))
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\left (16000 x^{3} \sqrt {-10 x^{2}-x +3}+12705 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -26800 x^{2} \sqrt {-10 x^{2}-x +3}+7623 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+22500 x \sqrt {-10 x^{2}-x +3}+18860 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{20000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(116\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/20000*(16000*x^3*(-10*x^2-x+3)^(1/2)+12705*10^(1/2)*arcsin(20/11*x+1/11) *x-26800*x^2*(-10*x^2-x+3)^(1/2)+7623*10^(1/2)*arcsin(20/11*x+1/11)+22500* x*(-10*x^2-x+3)^(1/2)+18860*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+ 3)^(1/2)/(3+5*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=-\frac {2541 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (800 \, x^{3} - 1340 \, x^{2} + 1125 \, x + 943\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20000 \, {\left (5 \, x + 3\right )}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
-1/20000*(2541*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(800*x^3 - 1340*x^2 + 1125*x + 943)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \cdot \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)/(3+5*x)**(3/2),x)
Output:
Integral((1 - 2*x)**(5/2)*(3*x + 2)/(5*x + 3)**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=-\frac {8 \, x^{4}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {87 \, x^{3}}{25 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {359 \, x^{2}}{100 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2541}{20000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {761 \, x}{1000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {943}{1000 \, \sqrt {-10 \, x^{2} - x + 3}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
-8/5*x^4/sqrt(-10*x^2 - x + 3) + 87/25*x^3/sqrt(-10*x^2 - x + 3) - 359/100 *x^2/sqrt(-10*x^2 - x + 3) - 2541/20000*sqrt(10)*arcsin(-20/11*x - 1/11) - 761/1000*x/sqrt(-10*x^2 - x + 3) + 943/1000/sqrt(-10*x^2 - x + 3)
Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\frac {1}{25000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 139 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3597 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {2541}{10000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{3125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
1/25000*(4*(8*sqrt(5)*(5*x + 3) - 139*sqrt(5))*(5*x + 3) + 3597*sqrt(5))*s qrt(5*x + 3)*sqrt(-10*x + 5) + 2541/10000*sqrt(10)*arcsin(1/11*sqrt(22)*sq rt(5*x + 3)) - 121/6250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt (5*x + 3) + 242/3125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(22))
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )}{{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(3*x + 2))/(5*x + 3)^(3/2),x)
Output:
int(((1 - 2*x)^(5/2)*(3*x + 2))/(5*x + 3)^(3/2), x)
Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\frac {-2541 \sqrt {5 x +3}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )+8000 \sqrt {-2 x +1}\, x^{3}-13400 \sqrt {-2 x +1}\, x^{2}+11250 \sqrt {-2 x +1}\, x +9430 \sqrt {-2 x +1}}{10000 \sqrt {5 x +3}} \] Input:
int((1-2*x)^(5/2)*(2+3*x)/(3+5*x)^(3/2),x)
Output:
( - 2541*sqrt(5*x + 3)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 8000*sqrt( - 2*x + 1)*x**3 - 13400*sqrt( - 2*x + 1)*x**2 + 11250*sqrt( - 2*x + 1)*x + 9430*sqrt( - 2*x + 1))/(10000*sqrt(5*x + 3))