\(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\) [524]

Optimal result

Integrand size = 24, antiderivative size = 81 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50}{27} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{54 (2+3 x)^2}+\frac {47 \sqrt {1-2 x}}{126 (2+3 x)}-\frac {863 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \] Output:

50/27*(1-2*x)^(1/2)-1/54*(1-2*x)^(1/2)/(2+3*x)^2+47*(1-2*x)^(1/2)/(252+378 
*x)-863/1323*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (1025+2941 x+2100 x^2\right )}{126 (2+3 x)^2}-\frac {863 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \] Input:

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

Output:

(Sqrt[1 - 2*x]*(1025 + 2941*x + 2100*x^2))/(126*(2 + 3*x)^2) - (863*ArcTan 
h[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 27, 87, 60, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{126} \int \frac {3 \sqrt {1-2 x} (350 x+187)}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{42} \int \frac {\sqrt {1-2 x} (350 x+187)}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{42} \left (\frac {863}{7} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {139 (1-2 x)^{3/2}}{21 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{42} \left (\frac {863}{7} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {139 (1-2 x)^{3/2}}{21 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{42} \left (\frac {863}{7} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {139 (1-2 x)^{3/2}}{21 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{42} \left (\frac {863}{7} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {139 (1-2 x)^{3/2}}{21 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}\)

Input:

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

Output:

-1/126*(1 - 2*x)^(3/2)/(2 + 3*x)^2 + ((139*(1 - 2*x)^(3/2))/(21*(2 + 3*x)) 
 + (863*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x 
]])/3))/7)/42
 

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 60

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {4200 x^{3}+3782 x^{2}-891 x -1025}{126 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {863 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1323}\)	\(51\)
pseudoelliptic	\(\frac {-1726 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+21 \sqrt {1-2 x}\, \left (2100 x^{2}+2941 x +1025\right )}{2646 \left (2+3 x \right )^{2}}\)	\(55\)
derivativedivides	\(\frac {50 \sqrt {1-2 x}}{27}+\frac {-\frac {47 \left (1-2 x \right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{2}}-\frac {863 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1323}\)	\(57\)
default	\(\frac {50 \sqrt {1-2 x}}{27}+\frac {-\frac {47 \left (1-2 x \right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{2}}-\frac {863 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1323}\)	\(57\)
trager	\(\frac {\left (2100 x^{2}+2941 x +1025\right ) \sqrt {1-2 x}}{126 \left (2+3 x \right )^{2}}-\frac {863 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2646}\)	\(72\)

Input:

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

Output:

-1/126*(4200*x^3+3782*x^2-891*x-1025)/(2+3*x)^2/(1-2*x)^(1/2)-863/1323*21^ 
(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {863 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (2100 \, x^{2} + 2941 \, x + 1025\right )} \sqrt {-2 \, x + 1}}{2646 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/2646*(863*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) 
 - 5)/(3*x + 2)) + 21*(2100*x^2 + 2941*x + 1025)*sqrt(-2*x + 1))/(9*x^2 + 
12*x + 4)
 

Sympy [A] (verification not implemented)

Time = 128.42 (sec) , antiderivative size = 342, normalized size of antiderivative = 4.22 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50 \sqrt {1 - 2 x}}{27} + \frac {65 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{189} + \frac {32 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} + \frac {56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} \] Input:

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

Output:

50*sqrt(1 - 2*x)/27 + 65*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(s 
qrt(1 - 2*x) + sqrt(21)/3))/189 + 32*Piecewise((sqrt(21)*(-log(sqrt(21)*sq 
rt(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)))/3 + 56*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)))/27
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {863}{2646} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{189 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \] Input:

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

Output:

863/2646*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 
*x + 1))) + 50/27*sqrt(-2*x + 1) - 1/189*(423*(-2*x + 1)^(3/2) - 973*sqrt( 
-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {863}{2646} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{756 \, {\left (3 \, x + 2\right )}^{2}} \] Input:

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

Output:

863/2646*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 
3*sqrt(-2*x + 1))) + 50/27*sqrt(-2*x + 1) - 1/756*(423*(-2*x + 1)^(3/2) - 
973*sqrt(-2*x + 1))/(3*x + 2)^2
 

Mupad [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50\,\sqrt {1-2\,x}}{27}-\frac {863\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1323}+\frac {\frac {139\,\sqrt {1-2\,x}}{243}-\frac {47\,{\left (1-2\,x\right )}^{3/2}}{189}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \] Input:

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

Output:

(50*(1 - 2*x)^(1/2))/27 - (863*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7 
))/1323 + ((139*(1 - 2*x)^(1/2))/243 - (47*(1 - 2*x)^(3/2))/189)/((28*x)/3 
 + (2*x - 1)^2 + 7/9)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {44100 \sqrt {-2 x +1}\, x^{2}+61761 \sqrt {-2 x +1}\, x +21525 \sqrt {-2 x +1}+7767 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+10356 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +3452 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-7767 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-10356 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -3452 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{23814 x^{2}+31752 x +10584} \] Input:

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

Output:

(44100*sqrt( - 2*x + 1)*x**2 + 61761*sqrt( - 2*x + 1)*x + 21525*sqrt( - 2* 
x + 1) + 7767*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 + 10356*sqr 
t(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 3452*sqrt(21)*log(3*sqrt( - 2 
*x + 1) - sqrt(21)) - 7767*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x** 
2 - 10356*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 3452*sqrt(21)*lo 
g(3*sqrt( - 2*x + 1) + sqrt(21)))/(2646*(9*x**2 + 12*x + 4))