Integrand size = 26, antiderivative size = 94 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {343}{44 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {128633 \sqrt {1-2 x}}{36300 (3+5 x)^{3/2}}-\frac {129853 \sqrt {1-2 x}}{199650 \sqrt {3+5 x}}-\frac {27 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25 \sqrt {10}} \] Output:
343/44/(1-2*x)^(1/2)/(3+5*x)^(3/2)-128633/36300*(1-2*x)^(1/2)/(3+5*x)^(3/2 )-129853/199650*(1-2*x)^(1/2)/(3+5*x)^(1/2)-27/250*arcsin(1/11*22^(1/2)*(3 +5*x)^(1/2))*10^(1/2)
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {229661+772408 x+649265 x^2}{99825 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {27 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{25 \sqrt {10}} \] Input:
Integrate[(2 + 3*x)^3/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]
Output:
(229661 + 772408*x + 649265*x^2)/(99825*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ( 27*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(25*Sqrt[10])
Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {109, 27, 162, 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.
\(\displaystyle \int \frac {(3 x+2)^3}{(1-2 x)^{3/2} (5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {1}{11} \int \frac {(3 x+2) (99 x+38)}{2 \sqrt {1-2 x} (5 x+3)^{5/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2) (99 x+38)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx\) |
\(\Big \downarrow \) 162 |
\(\displaystyle \frac {1}{22} \left (-\frac {297}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} (38770 x+24439)}{9075 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {1}{22} \left (-\frac {594}{125} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {2 \sqrt {1-2 x} (38770 x+24439)}{9075 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{22} \left (-\frac {297}{25} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x} (38770 x+24439)}{9075 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
Input:
Int[(2 + 3*x)^3/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]
Output:
(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-2*Sqrt[1 - 2*x]*(2 4439 + 38770*x))/(9075*(3 + 5*x)^(3/2)) - (297*Sqrt[2/5]*ArcSin[Sqrt[2/11] *Sqrt[3 + 5*x]])/25)/22
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(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^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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 = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {5390550 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+3773385 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-1293732 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +12985300 x^{2} \sqrt {-10 x^{2}-x +3}-970299 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+15448160 x \sqrt {-10 x^{2}-x +3}+4593220 \sqrt {-10 x^{2}-x +3}}{1996500 \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(127\) |
Input:
int((2+3*x)^3/(1-2*x)^(3/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/1996500*(5390550*10^(1/2)*arcsin(20/11*x+1/11)*x^3+3773385*10^(1/2)*arcs in(20/11*x+1/11)*x^2-1293732*10^(1/2)*arcsin(20/11*x+1/11)*x+12985300*x^2* (-10*x^2-x+3)^(1/2)-970299*10^(1/2)*arcsin(20/11*x+1/11)+15448160*x*(-10*x ^2-x+3)^(1/2)+4593220*(-10*x^2-x+3)^(1/2))/(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/ 2)/(3+5*x)^(3/2)
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {107811 \, \sqrt {10} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (649265 \, x^{2} + 772408 \, x + 229661\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1996500 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \] Input:
integrate((2+3*x)^3/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
1/1996500*(107811*sqrt(10)*(50*x^3 + 35*x^2 - 12*x - 9)*arctan(1/20*sqrt(1 0)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(649265* x^2 + 772408*x + 229661)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)
\[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((2+3*x)**3/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)
Output:
Integral((3*x + 2)**3/((1 - 2*x)**(3/2)*(5*x + 3)**(5/2)), x)
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {27}{500} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {129853 \, x}{99825 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {382849}{499125 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{4125 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:
integrate((2+3*x)^3/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
-27/500*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 129853/99825*x/sqrt(-10*x ^2 - x + 3) + 382849/499125/sqrt(-10*x^2 - x + 3) - 2/4125/(5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt(-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (67) = 134\).
Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.77 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{7986000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {27}{250} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {41 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{133100 \, \sqrt {5 \, x + 3}} - \frac {343 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6655 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {615 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{499125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \] Input:
integrate((2+3*x)^3/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
-1/7986000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 27/250*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 41/133100*sqrt(10 )*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 343/6655*sqrt(5)*sq rt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/499125*sqrt(10)*(5*x + 3)^(3/2)* (615*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(- 10*x + 5) - sqrt(22))^3
Timed out. \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int((3*x + 2)^3/((1 - 2*x)^(3/2)*(5*x + 3)^(5/2)),x)
Output:
int((3*x + 2)^3/((1 - 2*x)^(3/2)*(5*x + 3)^(5/2)), x)
Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {539055 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x +323433 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )+6492650 x^{2}+7724080 x +2296610}{998250 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (5 x +3\right )} \] Input:
int((2+3*x)^3/(1-2*x)^(3/2)/(3+5*x)^(5/2),x)
Output:
(539055*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqr t(5))/sqrt(11))*x + 323433*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(10)*asin((s qrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 6492650*x**2 + 7724080*x + 2296610)/( 998250*sqrt(5*x + 3)*sqrt( - 2*x + 1)*(5*x + 3))