3.19.12 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\) [1812]

3.19.12.1 Optimal result

Integrand size = 24, antiderivative size = 108 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx=-\frac {(1-2 x)^{3/2}}{252 (2+3 x)^4}+\frac {275 (1-2 x)^{3/2}}{5292 (2+3 x)^3}-\frac {4625 \sqrt {1-2 x}}{10584 (2+3 x)^2}+\frac {4625 \sqrt {1-2 x}}{74088 (2+3 x)}+\frac {4625 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{37044 \sqrt {21}} \]

output

-1/252*(1-2*x)^(3/2)/(2+3*x)^4+275/5292*(1-2*x)^(3/2)/(2+3*x)^3+4625/77792 
4*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-4625/10584*(1-2*x)^(1/2)/(2 
+3*x)^2+4625/74088*(1-2*x)^(1/2)/(2+3*x)
 

3.19.12.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-85094-225262 x-64725 x^2+124875 x^3\right )}{2 (2+3 x)^4}+4625 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{777924} \]

input

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

output

((21*Sqrt[1 - 2*x]*(-85094 - 225262*x - 64725*x^2 + 124875*x^3))/(2*(2 + 3 
*x)^4) + 4625*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/777924
 

3.19.12.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, 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, 51, 52, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{252} \int \frac {75 \sqrt {1-2 x} (28 x+15)}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{84} \int \frac {\sqrt {1-2 x} (28 x+15)}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {25}{84} \left (\frac {185}{21} \int \frac {\sqrt {1-2 x}}{(3 x+2)^3}dx+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {25}{84} \left (\frac {185}{21} \left (-\frac {1}{6} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {25}{84} \left (\frac {185}{21} \left (\frac {1}{6} \left (\frac {\sqrt {1-2 x}}{7 (3 x+2)}-\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {25}{84} \left (\frac {185}{21} \left (\frac {1}{6} \left (\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {25}{84} \left (\frac {185}{21} \left (\frac {1}{6} \left (\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\)

input

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

output

-1/252*(1 - 2*x)^(3/2)/(2 + 3*x)^4 + (25*((11*(1 - 2*x)^(3/2))/(63*(2 + 3* 
x)^3) + (185*(-1/6*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (Sqrt[1 - 2*x]/(7*(2 + 3*x) 
) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21]))/6))/21))/84
 

3.19.12.3.1 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 51

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

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 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])
 

3.19.12.4 Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {249750 x^{4}-254325 x^{3}-385799 x^{2}+55074 x +85094}{74088 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {4625 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\)	\(56\)
pseudoelliptic	\(\frac {9250 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}+21 \sqrt {1-2 x}\, \left (124875 x^{3}-64725 x^{2}-225262 x -85094\right )}{1555848 \left (2+3 x \right )^{4}}\)	\(60\)
derivativedivides	\(\frac {-\frac {4625 \left (1-2 x \right )^{\frac {7}{2}}}{1372}+\frac {11675 \left (1-2 x \right )^{\frac {5}{2}}}{1764}+\frac {16027 \left (1-2 x \right )^{\frac {3}{2}}}{756}-\frac {4625 \sqrt {1-2 x}}{108}}{\left (-4-6 x \right )^{4}}+\frac {4625 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\)	\(66\)
default	\(\frac {-\frac {4625 \left (1-2 x \right )^{\frac {7}{2}}}{1372}+\frac {11675 \left (1-2 x \right )^{\frac {5}{2}}}{1764}+\frac {16027 \left (1-2 x \right )^{\frac {3}{2}}}{756}-\frac {4625 \sqrt {1-2 x}}{108}}{\left (-4-6 x \right )^{4}}+\frac {4625 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{777924}\)	\(66\)
trager	\(\frac {\left (124875 x^{3}-64725 x^{2}-225262 x -85094\right ) \sqrt {1-2 x}}{74088 \left (2+3 x \right )^{4}}+\frac {4625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1555848}\)	\(77\)

input

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

output

-1/74088*(249750*x^4-254325*x^3-385799*x^2+55074*x+85094)/(2+3*x)^4/(1-2*x 
)^(1/2)+4625/777924*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
 

3.19.12.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {4625 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (124875 \, x^{3} - 64725 \, x^{2} - 225262 \, x - 85094\right )} \sqrt {-2 \, x + 1}}{1555848 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

input

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

output

1/1555848*(4625*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x 
 - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(124875*x^3 - 64725*x^2 - 
225262*x - 85094)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

3.19.12.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.19.12.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx=-\frac {4625}{1555848} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {124875 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 245175 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 785323 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1586375 \, \sqrt {-2 \, x + 1}}{37044 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

input

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

output

-4625/1555848*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq 
rt(-2*x + 1))) - 1/37044*(124875*(-2*x + 1)^(7/2) - 245175*(-2*x + 1)^(5/2 
) - 785323*(-2*x + 1)^(3/2) + 1586375*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 75 
6*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
 

3.19.12.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx=-\frac {4625}{1555848} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {124875 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 245175 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 785323 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1586375 \, \sqrt {-2 \, x + 1}}{592704 \, {\left (3 \, x + 2\right )}^{4}} \]

input

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

output

-4625/1555848*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 
1) + 3*sqrt(-2*x + 1))) + 1/592704*(124875*(2*x - 1)^3*sqrt(-2*x + 1) + 24 
5175*(2*x - 1)^2*sqrt(-2*x + 1) + 785323*(-2*x + 1)^(3/2) - 1586375*sqrt(- 
2*x + 1))/(3*x + 2)^4
 

3.19.12.9 Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {4625\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{777924}-\frac {\frac {4625\,\sqrt {1-2\,x}}{8748}-\frac {16027\,{\left (1-2\,x\right )}^{3/2}}{61236}-\frac {11675\,{\left (1-2\,x\right )}^{5/2}}{142884}+\frac {4625\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]

input

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

output

(4625*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/777924 - ((4625*(1 - 2 
*x)^(1/2))/8748 - (16027*(1 - 2*x)^(3/2))/61236 - (11675*(1 - 2*x)^(5/2))/ 
142884 + (4625*(1 - 2*x)^(7/2))/111132)/((2744*x)/27 + (98*(2*x - 1)^2)/3 
+ (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)