\(\int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx\) [703]

Optimal result

Integrand size = 24, antiderivative size = 88 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\sqrt {1-2 x}}{567 (2+3 x)^3}-\frac {155 \sqrt {1-2 x}}{3969 (2+3 x)^2}+\frac {3520 \sqrt {1-2 x}}{9261 (2+3 x)}-\frac {78710 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \] Output:

1/567*(1-2*x)^(1/2)/(2+3*x)^3-155/3969*(1-2*x)^(1/2)/(2+3*x)^2+3520*(1-2*x 
)^(1/2)/(18522+27783*x)-78710/194481*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x) 
^(1/2))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (13373+41155 x+31680 x^2\right )}{(2+3 x)^3}-78710 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{194481} \] Input:

Integrate[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]
 

Output:

((21*Sqrt[1 - 2*x]*(13373 + 41155*x + 31680*x^2))/(2 + 3*x)^3 - 78710*Sqrt 
[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/194481
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 27, 162, 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.

\(\displaystyle \int \frac {(5 x+3)^3}{\sqrt {1-2 x} (3 x+2)^4} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}-\frac {1}{63} \int -\frac {10 (5 x+3) (52 x+29)}{\sqrt {1-2 x} (3 x+2)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {10}{63} \int \frac {(5 x+3) (52 x+29)}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {10}{63} \left (\frac {7871}{294} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x} (1867 x+1205)}{294 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {10}{63} \left (\frac {\sqrt {1-2 x} (1867 x+1205)}{294 (3 x+2)^2}-\frac {7871}{294} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {10}{63} \left (\frac {\sqrt {1-2 x} (1867 x+1205)}{294 (3 x+2)^2}-\frac {7871 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\)

Input:

Int[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]
 

Output:

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (10*((Sqrt[1 - 2*x]*(1205 + 
 1867*x))/(294*(2 + 3*x)^2) - (7871*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147 
*Sqrt[21])))/63
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 109

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])
 

rule 162

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]))
 

rule 219

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])
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.58

method	result	size
risch	\(-\frac {63360 x^{3}+50630 x^{2}-14409 x -13373}{9261 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {78710 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{194481}\)	\(51\)
pseudoelliptic	\(\frac {-78710 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (31680 x^{2}+41155 x +13373\right )}{194481 \left (2+3 x \right )^{3}}\)	\(55\)
derivativedivides	\(\frac {-\frac {7040 \left (1-2 x \right )^{\frac {5}{2}}}{1029}+\frac {41620 \left (1-2 x \right )^{\frac {3}{2}}}{1323}-\frac {6836 \sqrt {1-2 x}}{189}}{\left (-4-6 x \right )^{3}}-\frac {78710 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{194481}\)	\(57\)
default	\(\frac {-\frac {7040 \left (1-2 x \right )^{\frac {5}{2}}}{1029}+\frac {41620 \left (1-2 x \right )^{\frac {3}{2}}}{1323}-\frac {6836 \sqrt {1-2 x}}{189}}{\left (-4-6 x \right )^{3}}-\frac {78710 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{194481}\)	\(57\)
trager	\(\frac {\left (31680 x^{2}+41155 x +13373\right ) \sqrt {1-2 x}}{9261 \left (2+3 x \right )^{3}}+\frac {39355 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{194481}\)	\(72\)

Input:

int((3+5*x)^3/(1-2*x)^(1/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/9261*(63360*x^3+50630*x^2-14409*x-13373)/(2+3*x)^3/(1-2*x)^(1/2)-78710/ 
194481*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {39355 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (31680 \, x^{2} + 41155 \, x + 13373\right )} \sqrt {-2 \, x + 1}}{194481 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

integrate((3+5*x)^3/(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="fricas")
 

Output:

1/194481*(39355*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)* 
sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(31680*x^2 + 41155*x + 13373)*sqrt(-2* 
x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\text {Timed out} \] Input:

integrate((3+5*x)**3/(1-2*x)**(1/2)/(2+3*x)**4,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {39355}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (15840 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 72835 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83741 \, \sqrt {-2 \, x + 1}\right )}}{9261 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:

integrate((3+5*x)^3/(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="maxima")
 

Output:

39355/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr 
t(-2*x + 1))) + 4/9261*(15840*(-2*x + 1)^(5/2) - 72835*(-2*x + 1)^(3/2) + 
83741*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {39355}{194481} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {15840 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 72835 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83741 \, \sqrt {-2 \, x + 1}}{18522 \, {\left (3 \, x + 2\right )}^{3}} \] Input:

integrate((3+5*x)^3/(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="giac")
 

Output:

39355/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 
) + 3*sqrt(-2*x + 1))) + 1/18522*(15840*(2*x - 1)^2*sqrt(-2*x + 1) - 72835 
*(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(3*x + 2)^3
 

Mupad [B] (verification not implemented)

Time = 1.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {\frac {6836\,\sqrt {1-2\,x}}{5103}-\frac {41620\,{\left (1-2\,x\right )}^{3/2}}{35721}+\frac {7040\,{\left (1-2\,x\right )}^{5/2}}{27783}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {78710\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \] Input:

int((5*x + 3)^3/((1 - 2*x)^(1/2)*(3*x + 2)^4),x)
 

Output:

((6836*(1 - 2*x)^(1/2))/5103 - (41620*(1 - 2*x)^(3/2))/35721 + (7040*(1 - 
2*x)^(5/2))/27783)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27) - (787 
10*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/194481
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.24 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx=\frac {665280 \sqrt {-2 x +1}\, x^{2}+864255 \sqrt {-2 x +1}\, x +280833 \sqrt {-2 x +1}+1062585 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}+2125170 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+1416780 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +314840 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-1062585 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}-2125170 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-1416780 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -314840 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{5250987 x^{3}[phone] x^{2}+7001316 x +1555848} \] Input:

int((3+5*x)^3/(1-2*x)^(1/2)/(2+3*x)^4,x)
 

Output:

(665280*sqrt( - 2*x + 1)*x**2 + 864255*sqrt( - 2*x + 1)*x + 280833*sqrt( - 
 2*x + 1) + 1062585*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**3 + 212 
5170*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 + 1416780*sqrt(21)*l 
og(3*sqrt( - 2*x + 1) - sqrt(21))*x + 314840*sqrt(21)*log(3*sqrt( - 2*x + 
1) - sqrt(21)) - 1062585*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 
- 2125170*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 - 1416780*sqrt( 
21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 314840*sqrt(21)*log(3*sqrt( - 2 
*x + 1) + sqrt(21)))/(194481*(27*x**3 + 54*x**2 + 36*x + 8))