3.19.80 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx\) [1880]

3.19.80.1 Optimal result

Integrand size = 24, antiderivative size = 128 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {(1-2 x)^{5/2}}{315 (2+3 x)^5}+\frac {23 (1-2 x)^{5/2}}{588 (2+3 x)^4}-\frac {4693 (1-2 x)^{3/2}}{15876 (2+3 x)^3}+\frac {4693 \sqrt {1-2 x}}{31752 (2+3 x)^2}-\frac {4693 \sqrt {1-2 x}}{222264 (2+3 x)}-\frac {4693 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \]

output

-1/315*(1-2*x)^(5/2)/(2+3*x)^5+23/588*(1-2*x)^(5/2)/(2+3*x)^4-4693/15876*( 
1-2*x)^(3/2)/(2+3*x)^3-4693/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21 
^(1/2)+4693/31752*(1-2*x)^(1/2)/(2+3*x)^2-4693/222264*(1-2*x)^(1/2)/(2+3*x 
)
 

3.19.80.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (292028-2143262 x-8540988 x^2-5801265 x^3+1900665 x^4\right )}{2 (2+3 x)^5}-23465 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{11668860} \]

input

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^2)/(2 + 3*x)^6,x]
 

output

((-21*Sqrt[1 - 2*x]*(292028 - 2143262*x - 8540988*x^2 - 5801265*x^3 + 1900 
665*x^4))/(2*(2 + 3*x)^5) - 23465*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x] 
])/11668860
 

3.19.80.3 Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {100, 27, 87, 51, 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.

\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^2}{(3 x+2)^6} \, dx\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{315} \int \frac {5 (1-2 x)^{3/2} (525 x+281)}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{63} \int \frac {(1-2 x)^{3/2} (525 x+281)}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{63} \left (\frac {4693}{28} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^4}dx+\frac {69 (1-2 x)^{5/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{63} \left (\frac {4693}{28} \left (-\frac {1}{3} \int \frac {\sqrt {1-2 x}}{(3 x+2)^3}dx-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )+\frac {69 (1-2 x)^{5/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{63} \left (\frac {4693}{28} \left (\frac {1}{3} \left (\frac {1}{6} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )+\frac {69 (1-2 x)^{5/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{63} \left (\frac {4693}{28} \left (\frac {1}{3} \left (\frac {1}{6} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )+\frac {69 (1-2 x)^{5/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{63} \left (\frac {4693}{28} \left (\frac {1}{3} \left (\frac {1}{6} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )+\frac {69 (1-2 x)^{5/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{63} \left (\frac {4693}{28} \left (\frac {1}{3} \left (\frac {1}{6} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )+\frac {69 (1-2 x)^{5/2}}{28 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{315 (3 x+2)^5}\)

input

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^2)/(2 + 3*x)^6,x]
 

output

-1/315*(1 - 2*x)^(5/2)/(2 + 3*x)^5 + ((69*(1 - 2*x)^(5/2))/(28*(2 + 3*x)^4 
) + (4693*(-1/9*(1 - 2*x)^(3/2)/(2 + 3*x)^3 + (Sqrt[1 - 2*x]/(6*(2 + 3*x)^ 
2) + (-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/ 
(7*Sqrt[21]))/6)/3))/28)/63
 

3.19.80.3.1 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])
 

3.19.80.4 Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48

method	result	size
risch	\(\frac {3801330 x^{5}-13503195 x^{4}-11280711 x^{3}+4254464 x^{2}+2727318 x -292028}{1111320 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {4693 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\)	\(61\)
pseudoelliptic	\(\frac {-46930 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-21 \sqrt {1-2 x}\, \left (1900665 x^{4}-5801265 x^{3}-8540988 x^{2}-2143262 x +292028\right )}{23337720 \left (2+3 x \right )^{5}}\)	\(65\)
derivativedivides	\(-\frac {3888 \left (-\frac {4693 \left (1-2 x \right )^{\frac {9}{2}}}{5334336}-\frac {907 \left (1-2 x \right )^{\frac {7}{2}}}{489888}+\frac {6119 \left (1-2 x \right )^{\frac {5}{2}}}{229635}-\frac {32851 \left (1-2 x \right )^{\frac {3}{2}}}{629856}+\frac {32851 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}-\frac {4693 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\)	\(75\)
default	\(-\frac {3888 \left (-\frac {4693 \left (1-2 x \right )^{\frac {9}{2}}}{5334336}-\frac {907 \left (1-2 x \right )^{\frac {7}{2}}}{489888}+\frac {6119 \left (1-2 x \right )^{\frac {5}{2}}}{229635}-\frac {32851 \left (1-2 x \right )^{\frac {3}{2}}}{629856}+\frac {32851 \sqrt {1-2 x}}{1259712}\right )}{\left (-4-6 x \right )^{5}}-\frac {4693 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\)	\(75\)
trager	\(-\frac {\left (1900665 x^{4}-5801265 x^{3}-8540988 x^{2}-2143262 x +292028\right ) \sqrt {1-2 x}}{1111320 \left (2+3 x \right )^{5}}+\frac {4693 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4667544}\)	\(82\)

input

int((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x)^6,x,method=_RETURNVERBOSE)
 

output

1/1111320*(3801330*x^5-13503195*x^4-11280711*x^3+4254464*x^2+2727318*x-292 
028)/(2+3*x)^5/(1-2*x)^(1/2)-4693/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/ 
2))*21^(1/2)
 

3.19.80.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {23465 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1900665 \, x^{4} - 5801265 \, x^{3} - 8540988 \, x^{2} - 2143262 \, x + 292028\right )} \sqrt {-2 \, x + 1}}{23337720 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

input

integrate((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x)^6,x, algorithm="fricas")
 

output

1/23337720*(23465*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x 
 + 32)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(1900665*x^ 
4 - 5801265*x^3 - 8540988*x^2 - 2143262*x + 292028)*sqrt(-2*x + 1))/(243*x 
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
 

3.19.80.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\text {Timed out} \]

input

integrate((1-2*x)**(3/2)*(3+5*x)**2/(2+3*x)**6,x)
 

output

Timed out
 

3.19.80.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {4693}{4667544} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1900665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 3999870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 57567552 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 112678930 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 56339465 \, \sqrt {-2 \, x + 1}}{555660 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

input

integrate((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x)^6,x, algorithm="maxima")
 

output

4693/4667544*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr 
t(-2*x + 1))) - 1/555660*(1900665*(-2*x + 1)^(9/2) + 3999870*(-2*x + 1)^(7 
/2) - 57567552*(-2*x + 1)^(5/2) + 112678930*(-2*x + 1)^(3/2) - 56339465*sq 
rt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30 
870*(2*x - 1)^2 + 72030*x - 19208)
 

3.19.80.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {4693}{4667544} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {1900665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 3999870 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 57567552 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 112678930 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 56339465 \, \sqrt {-2 \, x + 1}}{17781120 \, {\left (3 \, x + 2\right )}^{5}} \]

input

integrate((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x)^6,x, algorithm="giac")
 

output

4693/4667544*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 
) + 3*sqrt(-2*x + 1))) - 1/17781120*(1900665*(2*x - 1)^4*sqrt(-2*x + 1) - 
3999870*(2*x - 1)^3*sqrt(-2*x + 1) - 57567552*(2*x - 1)^2*sqrt(-2*x + 1) + 
 112678930*(-2*x + 1)^(3/2) - 56339465*sqrt(-2*x + 1))/(3*x + 2)^5
 

3.19.80.9 Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {4693\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2333772}-\frac {\frac {32851\,{\left (1-2\,x\right )}^{3/2}}{39366}-\frac {32851\,\sqrt {1-2\,x}}{78732}-\frac {97904\,{\left (1-2\,x\right )}^{5/2}}{229635}+\frac {907\,{\left (1-2\,x\right )}^{7/2}}{30618}+\frac {4693\,{\left (1-2\,x\right )}^{9/2}}{333396}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]

input

int(((1 - 2*x)^(3/2)*(5*x + 3)^2)/(3*x + 2)^6,x)
 

output

- (4693*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2333772 - ((32851*(1 
 - 2*x)^(3/2))/39366 - (32851*(1 - 2*x)^(1/2))/78732 - (97904*(1 - 2*x)^(5 
/2))/229635 + (907*(1 - 2*x)^(7/2))/30618 + (4693*(1 - 2*x)^(9/2))/333396) 
/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 
1)^4)/3 + (2*x - 1)^5 - 19208/243)