3.22.54 \(\int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\) [2154]

3.22.54.1 Optimal result

Integrand size = 24, antiderivative size = 103 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {85}{343 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {26}{21 \sqrt {1-2 x} (2+3 x)^2}-\frac {85}{294 \sqrt {1-2 x} (2+3 x)}-\frac {85}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]

output

121/42/(1-2*x)^(3/2)/(2+3*x)^2-85/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2)) 
*21^(1/2)+85/343/(1-2*x)^(1/2)-26/21/(2+3*x)^2/(1-2*x)^(1/2)-85/294/(2+3*x 
)/(1-2*x)^(1/2)
 

3.22.54.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.63 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {-\frac {7 \left (-4231-7731 x+4080 x^2+9180 x^3\right )}{2 (1-2 x)^{3/2} (2+3 x)^2}-255 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203} \]

input

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

output

((-7*(-4231 - 7731*x + 4080*x^2 + 9180*x^3))/(2*(1 - 2*x)^(3/2)*(2 + 3*x)^ 
2) - 255*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7203
 

3.22.54.3 Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 27, 87, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac {1}{42} \int -\frac {21 (18-25 x)}{(1-2 x)^{3/2} (3 x+2)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {18-25 x}{(1-2 x)^{3/2} (3 x+2)^3}dx+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{2} \left (\frac {85}{21} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2}dx-\frac {52}{21 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{2} \left (\frac {85}{21} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {52}{21 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{2} \left (\frac {85}{21} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {52}{21 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (\frac {85}{21} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {52}{21 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {85}{21} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {52}{21 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^2}\)

input

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

output

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (-52/(21*Sqrt[1 - 2*x]*(2 + 3*x)^2) 
 + (85*(-1/7*1/(Sqrt[1 - 2*x]*(2 + 3*x)) + (3*(2/(7*Sqrt[1 - 2*x]) - (2*Sq 
rt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7))/7))/21)/2
 

3.22.54.3.1 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])
 

3.22.54.4 Maple [A] (verified)

Time = 3.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {9180 x^{3}+4080 x^{2}-7731 x -4231}{2058 \sqrt {1-2 x}\, \left (-1+2 x \right ) \left (2+3 x \right )^{2}}-\frac {85 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(58\)
derivativedivides	\(\frac {-\frac {387 \left (1-2 x \right )^{\frac {3}{2}}}{2401}+\frac {127 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {85 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {242}{1029 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {638}{2401 \sqrt {1-2 x}}\)	\(66\)
default	\(\frac {-\frac {387 \left (1-2 x \right )^{\frac {3}{2}}}{2401}+\frac {127 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {85 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {242}{1029 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {638}{2401 \sqrt {1-2 x}}\)	\(66\)
pseudoelliptic	\(\frac {\frac {85 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{2} \sqrt {21}}{2401}-\frac {1530 x^{3}}{343}-\frac {680 x^{2}}{343}+\frac {2577 x}{686}+\frac {4231}{2058}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2}}\)	\(69\)
trager	\(-\frac {\left (9180 x^{3}+4080 x^{2}-7731 x -4231\right ) \sqrt {1-2 x}}{2058 \left (6 x^{2}+x -2\right )^{2}}-\frac {85 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{4802}\)	\(80\)

input

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

output

1/2058*(9180*x^3+4080*x^2-7731*x-4231)/(1-2*x)^(1/2)/(-1+2*x)/(2+3*x)^2-85 
/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
 

3.22.54.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {255 \, \sqrt {7} \sqrt {3} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (9180 \, x^{3} + 4080 \, x^{2} - 7731 \, x - 4231\right )} \sqrt {-2 \, x + 1}}{14406 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

input

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

output

1/14406*(255*sqrt(7)*sqrt(3)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((sqr 
t(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(9180*x^3 + 4080*x^2 
 - 7731*x - 4231)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
 

3.22.54.6 Sympy [A] (verification not implemented)

Time = 107.45 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.44 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {319 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{16807} + \frac {264 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{343} + \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{49} + \frac {638}{2401 \sqrt {1 - 2 x}} + \frac {242}{1029 \left (1 - 2 x\right )^{\frac {3}{2}}} \]

input

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

output

319*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 
1)/3))/16807 + 264*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)))/343 + 8*Piecewise((sqrt(21)*(3*log 
(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 
 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 
 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 
 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < 
 sqrt(21)/3)))/49 + 638/(2401*sqrt(1 - 2*x)) + 242/(1029*(1 - 2*x)**(3/2))
 

3.22.54.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {85}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2295 \, {\left (2 \, x - 1\right )}^{3} + 8925 \, {\left (2 \, x - 1\right )}^{2} + 6468 \, x - 15092}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

input

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

output

85/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* 
x + 1))) - 1/1029*(2295*(2*x - 1)^3 + 8925*(2*x - 1)^2 + 6468*x - 15092)/( 
9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))
 

3.22.54.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {85}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {44 \, {\left (87 \, x - 82\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {387 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 889 \, \sqrt {-2 \, x + 1}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \]

input

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

output

85/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 
*sqrt(-2*x + 1))) + 44/7203*(87*x - 82)/((2*x - 1)*sqrt(-2*x + 1)) - 1/960 
4*(387*(-2*x + 1)^(3/2) - 889*sqrt(-2*x + 1))/(3*x + 2)^2
 

3.22.54.9 Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {85\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {44\,x}{63}+\frac {425\,{\left (2\,x-1\right )}^2}{441}+\frac {85\,{\left (2\,x-1\right )}^3}{343}-\frac {44}{27}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]

input

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

output

- (85*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - ((44*x)/63 + (4 
25*(2*x - 1)^2)/441 + (85*(2*x - 1)^3)/343 - 44/27)/((49*(1 - 2*x)^(3/2))/ 
9 - (14*(1 - 2*x)^(5/2))/3 + (1 - 2*x)^(7/2))