3.22.55 \(\int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\) [2155]

3.22.55.1 Optimal result

Integrand size = 24, antiderivative size = 121 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {1415}{7203 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac {1091}{882 \sqrt {1-2 x} (2+3 x)^3}-\frac {283}{882 \sqrt {1-2 x} (2+3 x)^2}-\frac {1415}{6174 \sqrt {1-2 x} (2+3 x)}-\frac {1415 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401 \sqrt {21}} \]

output

121/42/(1-2*x)^(3/2)/(2+3*x)^3-1415/50421*arctanh(1/7*21^(1/2)*(1-2*x)^(1/ 
2))*21^(1/2)+1415/7203/(1-2*x)^(1/2)-1091/882/(2+3*x)^3/(1-2*x)^(1/2)-283/ 
882/(2+3*x)^2/(1-2*x)^(1/2)-1415/6174/(2+3*x)/(1-2*x)^(1/2)
 

3.22.55.2 Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {-\frac {7 \left (-23872-83655 x-26319 x^2+169800 x^3+152820 x^4\right )}{2 (1-2 x)^{3/2} (2+3 x)^3}-1415 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{50421} \]

input

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

output

((-7*(-23872 - 83655*x - 26319*x^2 + 169800*x^3 + 152820*x^4))/(2*(1 - 2*x 
)^(3/2)*(2 + 3*x)^3) - 1415*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/504 
21
 

3.22.55.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {100, 27, 87, 52, 52, 61, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}-\frac {1}{42} \int -\frac {3 (247-175 x)}{(1-2 x)^{3/2} (3 x+2)^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \int \frac {247-175 x}{(1-2 x)^{3/2} (3 x+2)^4}dx+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{14} \left (\frac {566}{9} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3}dx-\frac {1091}{63 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{14} \left (\frac {566}{9} \left (\frac {5}{14} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2}dx-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1091}{63 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{14} \left (\frac {566}{9} \left (\frac {5}{14} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1091}{63 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{14} \left (\frac {566}{9} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1091}{63 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{14} \left (\frac {566}{9} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1091}{63 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{14} \left (\frac {566}{9} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1091}{63 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^3}\)

input

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

output

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (-1091/(63*Sqrt[1 - 2*x]*(2 + 3*x)^ 
3) + (566*(-1/14*1/(Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*(-1/7*1/(Sqrt[1 - 2*x] 
*(2 + 3*x)) + (3*(2/(7*Sqrt[1 - 2*x]) - (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqr 
t[1 - 2*x]])/7))/7))/14))/9)/14
 

3.22.55.3.1 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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, 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 87

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]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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])
 

3.22.55.4 Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {152820 x^{4}+169800 x^{3}-26319 x^{2}-83655 x -23872}{14406 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {1415 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{50421}\)	\(63\)
pseudoelliptic	\(\frac {\frac {1415 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{3} \sqrt {21}}{50421}-\frac {25470 x^{4}}{2401}-\frac {28300 x^{3}}{2401}+\frac {8773 x^{2}}{4802}+\frac {27885 x}{4802}+\frac {11936}{7203}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{3}}\)	\(74\)
derivativedivides	\(\frac {\frac {15489 \left (1-2 x \right )^{\frac {5}{2}}}{16807}-\frac {1420 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {1595 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{3}}-\frac {1415 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{50421}+\frac {484}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2728}{16807 \sqrt {1-2 x}}\)	\(75\)
default	\(\frac {\frac {15489 \left (1-2 x \right )^{\frac {5}{2}}}{16807}-\frac {1420 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {1595 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{3}}-\frac {1415 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{50421}+\frac {484}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2728}{16807 \sqrt {1-2 x}}\)	\(75\)
trager	\(-\frac {\left (152820 x^{4}+169800 x^{3}-26319 x^{2}-83655 x -23872\right ) \sqrt {1-2 x}}{14406 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}+\frac {1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{100842}\)	\(89\)

input

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

output

1/14406*(152820*x^4+169800*x^3-26319*x^2-83655*x-23872)/(2+3*x)^3/(1-2*x)^ 
(1/2)/(-1+2*x)-1415/50421*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
 

3.22.55.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {1415 \, \sqrt {21} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 7 \, {\left (152820 \, x^{4} + 169800 \, x^{3} - 26319 \, x^{2} - 83655 \, x - 23872\right )} \sqrt {-2 \, x + 1}}{100842 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

input

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

output

1/100842*(1415*sqrt(21)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*lo 
g((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 7*(152820*x^4 + 169800* 
x^3 - 26319*x^2 - 83655*x - 23872)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45 
*x^3 - 58*x^2 + 4*x + 8)
 

3.22.55.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.22.55.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {1415}{100842} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {38205 \, {\left (2 \, x - 1\right )}^{4} + 237720 \, {\left (2 \, x - 1\right )}^{3} + 457611 \, {\left (2 \, x - 1\right )}^{2} + 375144 \, x - 353584}{7203 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

input

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

output

1415/100842*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt 
(-2*x + 1))) + 1/7203*(38205*(2*x - 1)^4 + 237720*(2*x - 1)^3 + 457611*(2* 
x - 1)^2 + 375144*x - 353584)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) 
+ 441*(-2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))
 

3.22.55.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {1415}{100842} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {38205 \, {\left (2 \, x - 1\right )}^{4} + 237720 \, {\left (2 \, x - 1\right )}^{3} + 457611 \, {\left (2 \, x - 1\right )}^{2} + 375144 \, x - 353584}{7203 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \]

input

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

output

1415/100842*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) 
 + 3*sqrt(-2*x + 1))) + 1/7203*(38205*(2*x - 1)^4 + 237720*(2*x - 1)^3 + 4 
57611*(2*x - 1)^2 + 375144*x - 353584)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 
 1))^3
 

3.22.55.9 Mupad [B] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {\frac {2552\,x}{1323}+\frac {3113\,{\left (2\,x-1\right )}^2}{1323}+\frac {11320\,{\left (2\,x-1\right )}^3}{9261}+\frac {1415\,{\left (2\,x-1\right )}^4}{7203}-\frac {7216}{3969}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}-\frac {1415\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{50421} \]

input

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

output

- ((2552*x)/1323 + (3113*(2*x - 1)^2)/1323 + (11320*(2*x - 1)^3)/9261 + (1 
415*(2*x - 1)^4)/7203 - 7216/3969)/((343*(1 - 2*x)^(3/2))/27 - (49*(1 - 2* 
x)^(5/2))/3 + 7*(1 - 2*x)^(7/2) - (1 - 2*x)^(9/2)) - (1415*21^(1/2)*atanh( 
(21^(1/2)*(1 - 2*x)^(1/2))/7))/50421