\(\int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx\) [695]

Optimal result

Integrand size = 24, antiderivative size = 108 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {635 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {635 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {635 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \] Output:

-1/252*(1-2*x)^(1/2)/(2+3*x)^4+13/252*(1-2*x)^(1/2)/(2+3*x)^3-635/3528*(1- 
2*x)^(1/2)/(2+3*x)^2-635*(1-2*x)^(1/2)/(16464+24696*x)-635/86436*21^(1/2)* 
arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (10190+39366 x+47625 x^2+17145 x^3\right )}{2 (2+3 x)^4}-635 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{86436} \] Input:

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

Output:

((-21*Sqrt[1 - 2*x]*(10190 + 39366*x + 47625*x^2 + 17145*x^3))/(2*(2 + 3*x 
)^4) - 635*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/86436
 

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, 52, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{252} \int \frac {7 (300 x+161)}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{36} \int \frac {300 x+161}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\)

Input:

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

Output:

-1/252*Sqrt[1 - 2*x]/(2 + 3*x)^4 + ((13*Sqrt[1 - 2*x])/(7*(2 + 3*x)^3) + ( 
635*(-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - 
(2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14))/7)/36
 

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 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 = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {34290 x^{4}+78105 x^{3}+31107 x^{2}-18986 x -10190}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {635 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{86436}\)	\(56\)
pseudoelliptic	\(\frac {-1270 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (17145 x^{3}+47625 x^{2}+39366 x +10190\right )}{172872 \left (2+3 x \right )^{4}}\)	\(60\)
derivativedivides	\(\frac {\frac {5715 \left (1-2 x \right )^{\frac {7}{2}}}{1372}-\frac {6985 \left (1-2 x \right )^{\frac {5}{2}}}{196}+\frac {2717 \left (1-2 x \right )^{\frac {3}{2}}}{28}-\frac {7171 \sqrt {1-2 x}}{84}}{\left (-4-6 x \right )^{4}}-\frac {635 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{86436}\)	\(66\)
default	\(\frac {\frac {5715 \left (1-2 x \right )^{\frac {7}{2}}}{1372}-\frac {6985 \left (1-2 x \right )^{\frac {5}{2}}}{196}+\frac {2717 \left (1-2 x \right )^{\frac {3}{2}}}{28}-\frac {7171 \sqrt {1-2 x}}{84}}{\left (-4-6 x \right )^{4}}-\frac {635 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{86436}\)	\(66\)
trager	\(-\frac {\left (17145 x^{3}+47625 x^{2}+39366 x +10190\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}-\frac {635 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )-21 \sqrt {1-2 x}}{2+3 x}\right )}{172872}\)	\(78\)

Input:

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

Output:

1/8232*(34290*x^4+78105*x^3+31107*x^2-18986*x-10190)/(2+3*x)^4/(1-2*x)^(1/ 
2)-635/86436*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {635 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (17145 \, x^{3} + 47625 \, x^{2} + 39366 \, x + 10190\right )} \sqrt {-2 \, x + 1}}{172872 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:

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

Output:

1/172872*(635*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*(17145*x^3 + 47625*x^2 + 393 
66*x + 10190)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {635}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17145 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 146685 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 399399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 351379 \, \sqrt {-2 \, x + 1}}{4116 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \] Input:

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

Output:

635/172872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( 
-2*x + 1))) + 1/4116*(17145*(-2*x + 1)^(7/2) - 146685*(-2*x + 1)^(5/2) + 3 
99399*(-2*x + 1)^(3/2) - 351379*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x 
 - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {635}{172872} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {17145 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 146685 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 399399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 351379 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \] Input:

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

Output:

635/172872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) 
+ 3*sqrt(-2*x + 1))) - 1/65856*(17145*(2*x - 1)^3*sqrt(-2*x + 1) + 146685* 
(2*x - 1)^2*sqrt(-2*x + 1) - 399399*(-2*x + 1)^(3/2) + 351379*sqrt(-2*x + 
1))/(3*x + 2)^4
 

Mupad [B] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {635\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {7171\,\sqrt {1-2\,x}}{6804}-\frac {2717\,{\left (1-2\,x\right )}^{3/2}}{2268}+\frac {6985\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {635\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \] Input:

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

Output:

- (635*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/86436 - ((7171*(1 - 2 
*x)^(1/2))/6804 - (2717*(1 - 2*x)^(3/2))/2268 + (6985*(1 - 2*x)^(5/2))/158 
76 - (635*(1 - 2*x)^(7/2))/12348)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28* 
(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.34 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {-360045 \sqrt {-2 x +1}\, x^{3}-1000125 \sqrt {-2 x +1}\, x^{2}-826686 \sqrt {-2 x +1}\, x -213990 \sqrt {-2 x +1}+51435 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{4}+137160 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}+137160 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+60960 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +10160 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-51435 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{4}-137160 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}-137160 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-60960 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -10160 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{14002632 x^{4}[phone] x^{3}[phone] x^{2}[phone] x +2765952} \] Input:

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

Output:

( - 360045*sqrt( - 2*x + 1)*x**3 - 1000125*sqrt( - 2*x + 1)*x**2 - 826686* 
sqrt( - 2*x + 1)*x - 213990*sqrt( - 2*x + 1) + 51435*sqrt(21)*log(3*sqrt( 
- 2*x + 1) - sqrt(21))*x**4 + 137160*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqr 
t(21))*x**3 + 137160*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 + 60 
960*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 10160*sqrt(21)*log(3*s 
qrt( - 2*x + 1) - sqrt(21)) - 51435*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt 
(21))*x**4 - 137160*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 - 137 
160*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 - 60960*sqrt(21)*log( 
3*sqrt( - 2*x + 1) + sqrt(21))*x - 10160*sqrt(21)*log(3*sqrt( - 2*x + 1) + 
 sqrt(21)))/(172872*(81*x**4 + 216*x**3 + 216*x**2 + 96*x + 16))