\(\int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx\) [627]

Optimal result

Integrand size = 22, antiderivative size = 101 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {40}{81} \sqrt {1-2 x}+\frac {(1-2 x)^{5/2}}{27 (2+3 x)^3}-\frac {55 (1-2 x)^{3/2}}{81 (2+3 x)^2}+\frac {125 \sqrt {1-2 x}}{81 (2+3 x)}-\frac {530 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \] Output:

40/81*(1-2*x)^(1/2)+1/27*(1-2*x)^(5/2)/(2+3*x)^3-55/81*(1-2*x)^(3/2)/(2+3* 
x)^2+125*(1-2*x)^(1/2)/(162+243*x)-530/1701*21^(1/2)*arctanh(1/7*21^(1/2)* 
(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {\sqrt {1-2 x} \left (713+2983 x+3627 x^2+1080 x^3\right )}{81 (2+3 x)^3}-\frac {530 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{81 \sqrt {21}} \] Input:

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

Output:

(Sqrt[1 - 2*x]*(713 + 2983*x + 3627*x^2 + 1080*x^3))/(81*(2 + 3*x)^3) - (5 
30*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 51, 51, 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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {106}{63} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^3}dx+\frac {(1-2 x)^{7/2}}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {106}{63} \left (-\frac {5}{6} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^2}dx-\frac {(1-2 x)^{5/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{7/2}}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {106}{63} \left (-\frac {5}{6} \left (-\int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}\right )-\frac {(1-2 x)^{5/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{7/2}}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {106}{63} \left (-\frac {5}{6} \left (-\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{5/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{7/2}}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {106}{63} \left (-\frac {5}{6} \left (\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{5/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{7/2}}{63 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {106}{63} \left (-\frac {5}{6} \left (\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{5/2}}{6 (3 x+2)^2}\right )+\frac {(1-2 x)^{7/2}}{63 (3 x+2)^3}\)

Input:

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

Output:

(1 - 2*x)^(7/2)/(63*(2 + 3*x)^3) + (106*(-1/6*(1 - 2*x)^(5/2)/(2 + 3*x)^2 
- (5*((-2*Sqrt[1 - 2*x])/3 - (1 - 2*x)^(3/2)/(3*(2 + 3*x)) + (2*Sqrt[7/3]* 
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/6))/63
 

Defintions of rubi rules used

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 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 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 = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55

method	result	size
risch	\(-\frac {2160 x^{4}+6174 x^{3}+2339 x^{2}-1557 x -713}{81 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {530 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1701}\)	\(56\)
pseudoelliptic	\(\frac {-530 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (1080 x^{3}+3627 x^{2}+2983 x +713\right )}{1701 \left (2+3 x \right )^{3}}\)	\(60\)
derivativedivides	\(\frac {40 \sqrt {1-2 x}}{81}+\frac {-\frac {326 \left (1-2 x \right )^{\frac {5}{2}}}{9}+\frac {12040 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {12250 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {530 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1701}\)	\(66\)
default	\(\frac {40 \sqrt {1-2 x}}{81}+\frac {-\frac {326 \left (1-2 x \right )^{\frac {5}{2}}}{9}+\frac {12040 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {12250 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {530 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1701}\)	\(66\)
trager	\(\frac {\left (1080 x^{3}+3627 x^{2}+2983 x +713\right ) \sqrt {1-2 x}}{81 \left (2+3 x \right )^{3}}+\frac {265 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1701}\)	\(77\)

Input:

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

Output:

-1/81*(2160*x^4+6174*x^3+2339*x^2-1557*x-713)/(2+3*x)^3/(1-2*x)^(1/2)-530/ 
1701*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {265 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1080 \, x^{3} + 3627 \, x^{2} + 2983 \, x + 713\right )} \sqrt {-2 \, x + 1}}{1701 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

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

Output:

1/1701*(265*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt 
(-2*x + 1) - 5)/(3*x + 2)) + 21*(1080*x^3 + 3627*x^2 + 2983*x + 713)*sqrt( 
-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [A] (verification not implemented)

Time = 165.58 (sec) , antiderivative size = 554, normalized size of antiderivative = 5.49 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

40*sqrt(1 - 2*x)/81 + 428*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log( 
sqrt(1 - 2*x) + sqrt(21)/3))/1701 + 2072*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*(sqr 
t(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)))/27 + 16072*P 
iecewise((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)))/81 + 5488*Piecewise((sqrt(21)*(-5* 
log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1) 
/32 - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x 
)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)* 
sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*( 
sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & ( 
sqrt(1 - 2*x) < sqrt(21)/3)))/81
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {265}{1701} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {40}{81} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (1467 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 6020 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 6125 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:

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

Output:

265/1701*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 
*x + 1))) + 40/81*sqrt(-2*x + 1) + 2/81*(1467*(-2*x + 1)^(5/2) - 6020*(-2* 
x + 1)^(3/2) + 6125*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 88 
2*x - 98)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {265}{1701} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {40}{81} \, \sqrt {-2 \, x + 1} + \frac {1467 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 6020 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 6125 \, \sqrt {-2 \, x + 1}}{324 \, {\left (3 \, x + 2\right )}^{3}} \] Input:

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

Output:

265/1701*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 
3*sqrt(-2*x + 1))) + 40/81*sqrt(-2*x + 1) + 1/324*(1467*(2*x - 1)^2*sqrt(- 
2*x + 1) - 6020*(-2*x + 1)^(3/2) + 6125*sqrt(-2*x + 1))/(3*x + 2)^3
 

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {40\,\sqrt {1-2\,x}}{81}-\frac {530\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1701}+\frac {\frac {12250\,\sqrt {1-2\,x}}{2187}-\frac {12040\,{\left (1-2\,x\right )}^{3/2}}{2187}+\frac {326\,{\left (1-2\,x\right )}^{5/2}}{243}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \] Input:

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

Output:

(40*(1 - 2*x)^(1/2))/81 - (530*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7 
))/1701 + ((12250*(1 - 2*x)^(1/2))/2187 - (12040*(1 - 2*x)^(3/2))/2187 + ( 
326*(1 - 2*x)^(5/2))/243)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.06 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx=\frac {22680 \sqrt {-2 x +1}\, x^{3}+76167 \sqrt {-2 x +1}\, x^{2}+62643 \sqrt {-2 x +1}\, x +14973 \sqrt {-2 x +1}+7155 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}+14310 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+9540 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +2120 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-7155 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}-14310 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-9540 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -2120 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{45927 x^{3}+91854 x^{2}+61236 x +13608} \] Input:

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

Output:

(22680*sqrt( - 2*x + 1)*x**3 + 76167*sqrt( - 2*x + 1)*x**2 + 62643*sqrt( - 
 2*x + 1)*x + 14973*sqrt( - 2*x + 1) + 7155*sqrt(21)*log(3*sqrt( - 2*x + 1 
) - sqrt(21))*x**3 + 14310*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x** 
2 + 9540*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 2120*sqrt(21)*log 
(3*sqrt( - 2*x + 1) - sqrt(21)) - 7155*sqrt(21)*log(3*sqrt( - 2*x + 1) + s 
qrt(21))*x**3 - 14310*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 - 9 
540*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 2120*sqrt(21)*log(3*sq 
rt( - 2*x + 1) + sqrt(21)))/(1701*(27*x**3 + 54*x**2 + 36*x + 8))