3.22.26 \(\int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\) [2126]

3.22.26.1 Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=-\frac {71 \sqrt {1-2 x} (2+3 x)^3}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1344 \sqrt {1-2 x} (2+3 x)^2}{33275 (3+5 x)}+\frac {441 \sqrt {1-2 x} (3344+1125 x)}{332750}-\frac {4557 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{166375 \sqrt {55}} \]

output

-4557/9150625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/11*(2+3*x)^4 
/(3+5*x)^2/(1-2*x)^(1/2)-71/1210*(2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^2-1344/33 
275*(2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)+441/332750*(3344+1125*x)*(1-2*x)^(1/2)
 

3.22.26.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {-\frac {55 \left (-16342856-41668993 x-764310 x^2+42046290 x^3+5390550 x^4\right )}{\sqrt {1-2 x} (3+5 x)^2}-9114 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18301250} \]

input

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

output

((-55*(-16342856 - 41668993*x - 764310*x^2 + 42046290*x^3 + 5390550*x^4))/ 
(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 9114*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2* 
x]])/18301250
 

3.22.26.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {109, 166, 27, 166, 27, 164, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {1}{11} \int \frac {(3 x+2)^3 (207 x+110)}{\sqrt {1-2 x} (5 x+3)^3}dx\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{11} \left (-\frac {1}{110} \int \frac {21 (3 x+2)^2 (681 x+383)}{\sqrt {1-2 x} (5 x+3)^2}dx-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (-\frac {21}{110} \int \frac {(3 x+2)^2 (681 x+383)}{\sqrt {1-2 x} (5 x+3)^2}dx-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{11} \left (-\frac {21}{110} \left (\frac {1}{55} \int \frac {7 (3 x+2) (3375 x+1994)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {128 \sqrt {1-2 x} (3 x+2)^2}{55 (5 x+3)}\right )-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (-\frac {21}{110} \left (\frac {7}{55} \int \frac {(3 x+2) (3375 x+1994)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {128 \sqrt {1-2 x} (3 x+2)^2}{55 (5 x+3)}\right )-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{11} \left (-\frac {21}{110} \left (\frac {7}{55} \left (-\frac {31}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {3}{5} \sqrt {1-2 x} (1125 x+3344)\right )+\frac {128 \sqrt {1-2 x} (3 x+2)^2}{55 (5 x+3)}\right )-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{11} \left (-\frac {21}{110} \left (\frac {7}{55} \left (\frac {31}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {3}{5} \sqrt {1-2 x} (1125 x+3344)\right )+\frac {128 \sqrt {1-2 x} (3 x+2)^2}{55 (5 x+3)}\right )-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{11} \left (-\frac {21}{110} \left (\frac {7}{55} \left (\frac {62 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {3}{5} \sqrt {1-2 x} (1125 x+3344)\right )+\frac {128 \sqrt {1-2 x} (3 x+2)^2}{55 (5 x+3)}\right )-\frac {71 \sqrt {1-2 x} (3 x+2)^3}{110 (5 x+3)^2}\right )+\frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^2}\)

input

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

output

(7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) + ((-71*Sqrt[1 - 2*x]*(2 + 
3*x)^3)/(110*(3 + 5*x)^2) - (21*((128*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*(3 + 
5*x)) + (7*((-3*Sqrt[1 - 2*x]*(3344 + 1125*x))/5 + (62*ArcTanh[Sqrt[5/11]* 
Sqrt[1 - 2*x]])/(5*Sqrt[55])))/55))/110)/11
 

3.22.26.3.1 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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(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*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

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])
 

3.22.26.4 Maple [A] (verified)

Time = 3.47 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.44

method	result	size
risch	\(-\frac {5390550 x^{4}[phone] x^{3}-764310 x^{2}-41668993 x -16342856}{332750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {4557 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9150625}\)	\(56\)
pseudoelliptic	\(-\frac {81 \left (\frac {1519 \sqrt {55}\, \left (x +\frac {3}{5}\right )^{2} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1976535}+x^{4}+\frac {39 x^{3}}{5}-\frac {25477 x^{2}}{179685}-\frac {41668993 x}{5390550}-\frac {8171428}{2695275}\right )}{5 \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}\)	\(65\)
derivativedivides	\(-\frac {81 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {1539 \sqrt {1-2 x}}{625}+\frac {\frac {337 \left (1-2 x \right )^{\frac {3}{2}}}{166375}-\frac {339 \sqrt {1-2 x}}{75625}}{\left (-6-10 x \right )^{2}}-\frac {4557 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9150625}+\frac {16807}{5324 \sqrt {1-2 x}}\)	\(75\)
default	\(-\frac {81 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {1539 \sqrt {1-2 x}}{625}+\frac {\frac {337 \left (1-2 x \right )^{\frac {3}{2}}}{166375}-\frac {339 \sqrt {1-2 x}}{75625}}{\left (-6-10 x \right )^{2}}-\frac {4557 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9150625}+\frac {16807}{5324 \sqrt {1-2 x}}\)	\(75\)
trager	\(\frac {\left (5390550 x^{4}[phone] x^{3}-764310 x^{2}-41668993 x -16342856\right ) \sqrt {1-2 x}}{332750 \left (3+5 x \right )^{2} \left (-1+2 x \right )}+\frac {4557 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{18301250}\)	\(89\)

input

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

output

-1/332750*(5390550*x^4+42046290*x^3-764310*x^2-41668993*x-16342856)/(3+5*x 
)^2/(1-2*x)^(1/2)-4557/9150625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/ 
2)
 

3.22.26.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {4557 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (5390550 \, x^{4} + 42046290 \, x^{3} - 764310 \, x^{2} - 41668993 \, x - 16342856\right )} \sqrt {-2 \, x + 1}}{18301250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

input

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

output

1/18301250*(4557*sqrt(55)*(50*x^3 + 35*x^2 - 12*x - 9)*log((5*x + sqrt(55) 
*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(5390550*x^4 + 42046290*x^3 - 764310* 
x^2 - 41668993*x - 16342856)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)
 

3.22.26.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.22.26.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=-\frac {81}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4557}{18301250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1539}{625} \, \sqrt {-2 \, x + 1} + \frac {262616115 \, {\left (2 \, x - 1\right )}^{2} + 2310992332 \, x + 115533209}{3327500 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \]

input

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

output

-81/500*(-2*x + 1)^(3/2) + 4557/18301250*sqrt(55)*log(-(sqrt(55) - 5*sqrt( 
-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1539/625*sqrt(-2*x + 1) + 1/33 
27500*(262616115*(2*x - 1)^2 + 2310992332*x + 115533209)/(25*(-2*x + 1)^(5 
/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))
 

3.22.26.8 Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=-\frac {81}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4557}{18301250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1539}{625} \, \sqrt {-2 \, x + 1} + \frac {16807}{5324 \, \sqrt {-2 \, x + 1}} + \frac {1685 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3729 \, \sqrt {-2 \, x + 1}}{3327500 \, {\left (5 \, x + 3\right )}^{2}} \]

input

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

output

-81/500*(-2*x + 1)^(3/2) + 4557/18301250*sqrt(55)*log(1/2*abs(-2*sqrt(55) 
+ 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1539/625*sqrt(-2*x + 
 1) + 16807/5324/sqrt(-2*x + 1) + 1/3327500*(1685*(-2*x + 1)^(3/2) - 3729* 
sqrt(-2*x + 1))/(5*x + 3)^2
 

3.22.26.9 Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {1539\,\sqrt {1-2\,x}}{625}-\frac {81\,{\left (1-2\,x\right )}^{3/2}}{500}+\frac {\frac {52522553\,x}{1890625}+\frac {52523223\,{\left (2\,x-1\right )}^2}{16637500}+\frac {10503019}{7562500}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,4557{}\mathrm {i}}{9150625} \]

input

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

output

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*4557i)/9150625 + (1539*(1 
 - 2*x)^(1/2))/625 - (81*(1 - 2*x)^(3/2))/500 + ((52522553*x)/1890625 + (5 
2523223*(2*x - 1)^2)/16637500 + 10503019/7562500)/((121*(1 - 2*x)^(1/2))/2 
5 - (22*(1 - 2*x)^(3/2))/5 + (1 - 2*x)^(5/2))