\(\int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx\) [579]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {100}{81} \sqrt {1-2 x}-\frac {7 \sqrt {1-2 x}}{243 (2+3 x)^3}+\frac {217 \sqrt {1-2 x}}{486 (2+3 x)^2}-\frac {2801 \sqrt {1-2 x}}{1134 (2+3 x)}+\frac {7559 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \] Output:

-100/81*(1-2*x)^(1/2)-7/243*(1-2*x)^(1/2)/(2+3*x)^3+217/486*(1-2*x)^(1/2)/ 
(2+3*x)^2-2801*(1-2*x)^(1/2)/(2268+3402*x)+7559/11907*21^(1/2)*arctanh(1/7 
*21^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {\sqrt {1-2 x} \left (21424+82493 x+100809 x^2+37800 x^3\right )}{1134 (2+3 x)^3}+\frac {7559 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \] Input:

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

Output:

-1/1134*(Sqrt[1 - 2*x]*(21424 + 82493*x + 100809*x^2 + 37800*x^3))/(2 + 3* 
x)^3 + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 87, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{189} \int \frac {(1-2 x)^{3/2} (1575 x+841)}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{189} \left (\frac {7559}{14} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^2}dx+\frac {209 (1-2 x)^{5/2}}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{189} \left (\frac {7559}{14} \left (-\int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}\right )+\frac {209 (1-2 x)^{5/2}}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{189} \left (\frac {7559}{14} \left (-\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )+\frac {209 (1-2 x)^{5/2}}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{189} \left (\frac {7559}{14} \left (\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )+\frac {209 (1-2 x)^{5/2}}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{189} \left (\frac {7559}{14} \left (\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )+\frac {209 (1-2 x)^{5/2}}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{189 (3 x+2)^3}\)

Input:

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

Output:

-1/189*(1 - 2*x)^(5/2)/(2 + 3*x)^3 + ((209*(1 - 2*x)^(5/2))/(14*(2 + 3*x)^ 
2) + (7559*((-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))/14)/189
 

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 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 = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {75600 x^{4}+163818 x^{3}+64177 x^{2}-39645 x -21424}{1134 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {7559 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{11907}\)	\(56\)
pseudoelliptic	\(\frac {15118 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-21 \sqrt {1-2 x}\, \left (37800 x^{3}+100809 x^{2}+82493 x +21424\right )}{23814 \left (2+3 x \right )^{3}}\)	\(60\)
derivativedivides	\(-\frac {100 \sqrt {1-2 x}}{81}-\frac {4 \left (-\frac {2801 \left (1-2 x \right )^{\frac {5}{2}}}{84}+\frac {4093 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {18613 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{3}}+\frac {7559 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{11907}\)	\(66\)
default	\(-\frac {100 \sqrt {1-2 x}}{81}-\frac {4 \left (-\frac {2801 \left (1-2 x \right )^{\frac {5}{2}}}{84}+\frac {4093 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {18613 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{3}}+\frac {7559 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{11907}\)	\(66\)
trager	\(-\frac {\left (37800 x^{3}+100809 x^{2}+82493 x +21424\right ) \sqrt {1-2 x}}{1134 \left (2+3 x \right )^{3}}+\frac {7559 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{23814}\)	\(77\)

Input:

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

Output:

1/1134*(75600*x^4+163818*x^3+64177*x^2-39645*x-21424)/(2+3*x)^3/(1-2*x)^(1 
/2)+7559/11907*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 = 90, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {7559 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (37800 \, x^{3} + 100809 \, x^{2} + 82493 \, x + 21424\right )} \sqrt {-2 \, x + 1}}{23814 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

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

Output:

1/23814*(7559*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x - sqrt(21)*sq 
rt(-2*x + 1) - 5)/(3*x + 2)) - 21*(37800*x^3 + 100809*x^2 + 82493*x + 2142 
4)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {7559}{23814} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {100}{81} \, \sqrt {-2 \, x + 1} - \frac {25209 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 114604 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 130291 \, \sqrt {-2 \, x + 1}}{567 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:

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

Output:

-7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt 
(-2*x + 1))) - 100/81*sqrt(-2*x + 1) - 1/567*(25209*(-2*x + 1)^(5/2) - 114 
604*(-2*x + 1)^(3/2) + 130291*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 
 1)^2 + 882*x - 98)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {7559}{23814} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {100}{81} \, \sqrt {-2 \, x + 1} - \frac {25209 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 114604 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 130291 \, \sqrt {-2 \, x + 1}}{4536 \, {\left (3 \, x + 2\right )}^{3}} \] Input:

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

Output:

-7559/23814*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) 
 + 3*sqrt(-2*x + 1))) - 100/81*sqrt(-2*x + 1) - 1/4536*(25209*(2*x - 1)^2* 
sqrt(-2*x + 1) - 114604*(-2*x + 1)^(3/2) + 130291*sqrt(-2*x + 1))/(3*x + 2 
)^3
 

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {7559\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{11907}-\frac {100\,\sqrt {1-2\,x}}{81}-\frac {\frac {18613\,\sqrt {1-2\,x}}{2187}-\frac {16372\,{\left (1-2\,x\right )}^{3/2}}{2187}+\frac {2801\,{\left (1-2\,x\right )}^{5/2}}{1701}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \] Input:

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

Output:

(7559*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/11907 - (100*(1 - 2*x) 
^(1/2))/81 - ((18613*(1 - 2*x)^(1/2))/2187 - (16372*(1 - 2*x)^(3/2))/2187 
+ (2801*(1 - 2*x)^(5/2))/1701)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 9 
8/27)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.06 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {-793800 \sqrt {-2 x +1}\, x^{3}-2116989 \sqrt {-2 x +1}\, x^{2}-1732353 \sqrt {-2 x +1}\, x -449904 \sqrt {-2 x +1}-204093 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-408186 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-272124 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -60472 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+204093 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+408186 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+272124 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +60472 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{642978 x^{3}+1285956 x^{2}+857304 x +190512} \] Input:

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

Output:

( - 793800*sqrt( - 2*x + 1)*x**3 - 2116989*sqrt( - 2*x + 1)*x**2 - 1732353 
*sqrt( - 2*x + 1)*x - 449904*sqrt( - 2*x + 1) - 204093*sqrt(21)*log(3*sqrt 
( - 2*x + 1) - sqrt(21))*x**3 - 408186*sqrt(21)*log(3*sqrt( - 2*x + 1) - s 
qrt(21))*x**2 - 272124*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 604 
72*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 204093*sqrt(21)*log(3*sqr 
t( - 2*x + 1) + sqrt(21))*x**3 + 408186*sqrt(21)*log(3*sqrt( - 2*x + 1) + 
sqrt(21))*x**2 + 272124*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 60 
472*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(23814*(27*x**3 + 54*x**2 
 + 36*x + 8))