Integrand size = 24, antiderivative size = 74 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {1331}{294 (1-2 x)^{3/2}}-\frac {4719}{686 \sqrt {1-2 x}}+\frac {\sqrt {1-2 x}}{1029 (2+3 x)}-\frac {200 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \] Output:
1331/294/(1-2*x)^(3/2)-4719/686/(1-2*x)^(1/2)+(1-2*x)^(1/2)/(2058+3087*x)- 200/21609*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {-65219+127050 (1-2 x)-42475 (1-2 x)^2}{2058 (-7+3 (1-2 x)) (1-2 x)^{3/2}}-\frac {200 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \] Input:
Integrate[(3 + 5*x)^3/((1 - 2*x)^(5/2)*(2 + 3*x)^2),x]
Output:
(-65219 + 127050*(1 - 2*x) - 42475*(1 - 2*x)^2)/(2058*(-7 + 3*(1 - 2*x))*( 1 - 2*x)^(3/2)) - (200*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 27, 161, 73, 219}
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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{5/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {1}{21} \int \frac {10 (5 x+3) (18 x+13)}{(1-2 x)^{3/2} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10}{21} \int \frac {(5 x+3) (18 x+13)}{(1-2 x)^{3/2} (3 x+2)^2}dx\) |
\(\Big \downarrow \) 161 |
\(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10}{21} \left (\frac {1450 x+969}{49 \sqrt {1-2 x} (3 x+2)}-\frac {10}{49} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10}{21} \left (\frac {10}{49} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {1450 x+969}{49 \sqrt {1-2 x} (3 x+2)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)}-\frac {10}{21} \left (\frac {20 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}+\frac {1450 x+969}{49 \sqrt {1-2 x} (3 x+2)}\right )\) |
Input:
Int[(3 + 5*x)^3/((1 - 2*x)^(5/2)*(2 + 3*x)^2),x]
Output:
(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (10*((969 + 1450*x)/(49* Sqrt[1 - 2*x]*(2 + 3*x)) + (20*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[ 21])))/21
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) *((g_.) + (h_.)*(x_)), x_] :> Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 ) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( n + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c , d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {42475 x^{2}+21050 x -4839}{1029 \left (2+3 x \right ) \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {200 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}\) | \(53\) |
derivativedivides | \(\frac {1331}{294 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {4719}{686 \sqrt {1-2 x}}-\frac {2 \sqrt {1-2 x}}{3087 \left (-\frac {4}{3}-2 x \right )}-\frac {200 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}\) | \(54\) |
default | \(\frac {1331}{294 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {4719}{686 \sqrt {1-2 x}}-\frac {2 \sqrt {1-2 x}}{3087 \left (-\frac {4}{3}-2 x \right )}-\frac {200 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}\) | \(54\) |
pseudoelliptic | \(\frac {200 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (6 x^{2}+x -2\right ) \sqrt {21}+891975 x^{2}+442050 x -101619}{\left (1-2 x \right )^{\frac {3}{2}} \left (43218+64827 x \right )}\) | \(60\) |
trager | \(\frac {\left (42475 x^{2}+21050 x -4839\right ) \sqrt {1-2 x}}{1029 \left (-1+2 x \right )^{2} \left (2+3 x \right )}+\frac {100 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{21609}\) | \(79\) |
Input:
int((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/1029*(42475*x^2+21050*x-4839)/(2+3*x)/(1-2*x)^(1/2)/(-1+2*x)-200/21609* 21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {100 \, \sqrt {21} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (42475 \, x^{2} + 21050 \, x - 4839\right )} \sqrt {-2 \, x + 1}}{21609 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \] Input:
integrate((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^2,x, algorithm="fricas")
Output:
1/21609*(100*sqrt(21)*(12*x^3 - 4*x^2 - 5*x + 2)*log((3*x + sqrt(21)*sqrt( -2*x + 1) - 5)/(3*x + 2)) + 21*(42475*x^2 + 21050*x - 4839)*sqrt(-2*x + 1) )/(12*x^3 - 4*x^2 - 5*x + 2)
Time = 62.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.50 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {101 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{21609} + \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{147} - \frac {4719}{686 \sqrt {1 - 2 x}} + \frac {1331}{294 \left (1 - 2 x\right )^{\frac {3}{2}}} \] Input:
integrate((3+5*x)**3/(1-2*x)**(5/2)/(2+3*x)**2,x)
Output:
101*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/21609 + 4*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21 )/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/147 - 4719/(686*sqrt(1 - 2*x)) + 133 1/(294*(1 - 2*x)**(3/2))
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {100}{21609} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {42475 \, {\left (2 \, x - 1\right )}^{2} + 254100 \, x - 61831}{2058 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \] Input:
integrate((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^2,x, algorithm="maxima")
Output:
100/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(- 2*x + 1))) - 1/2058*(42475*(2*x - 1)^2 + 254100*x - 61831)/(3*(-2*x + 1)^( 5/2) - 7*(-2*x + 1)^(3/2))
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {100}{21609} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, {\left (117 \, x - 20\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {\sqrt {-2 \, x + 1}}{1029 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^2,x, algorithm="giac")
Output:
100/21609*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 121/1029*(117*x - 20)/((2*x - 1)*sqrt(-2*x + 1)) + 1 /1029*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {\frac {6050\,x}{147}+\frac {42475\,{\left (2\,x-1\right )}^2}{6174}-\frac {8833}{882}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}-\frac {200\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609} \] Input:
int((5*x + 3)^3/((1 - 2*x)^(5/2)*(3*x + 2)^2),x)
Output:
((6050*x)/147 + (42475*(2*x - 1)^2)/6174 - 8833/882)/((7*(1 - 2*x)^(3/2))/ 3 - (1 - 2*x)^(5/2)) - (200*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/ 21609
Time = 0.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.38 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {600 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+100 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -200 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-600 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-100 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +200 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )-891975 x^{2}-442050 x +101619}{21609 \sqrt {-2 x +1}\, \left (6 x^{2}+x -2\right )} \] Input:
int((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^2,x)
Output:
(600*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 + 1 00*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 200*sq rt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 600*sqrt( - 2 *x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 - 100*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 200*sqrt( - 2*x + 1) *sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) - 891975*x**2 - 442050*x + 10 1619)/(21609*sqrt( - 2*x + 1)*(6*x**2 + x - 2))