3.20.19 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx\) [1919]

3.20.19.1 Optimal result

Integrand size = 24, antiderivative size = 140 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}-\frac {12279 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \]

output

-1/10*(1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2-12279/859375*arctanh(1/11*55^(1/2) 
*(1-2*x)^(1/2))*55^(1/2)+1404/3125*(2+3*x)^2*(1-2*x)^(1/2)+2643/1750*(2+3* 
x)^3*(1-2*x)^(1/2)-129/50*(2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)+9/31250*(32+1375 
*x)*(1-2*x)^(1/2)
 

3.20.19.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {-\frac {55 \sqrt {1-2 x} \left (96776-489445 x-2120880 x^2-496350 x^3+3267000 x^4+2025000 x^5\right )}{(3+5 x)^2}-171906 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{12031250} \]

input

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

output

((-55*Sqrt[1 - 2*x]*(96776 - 489445*x - 2120880*x^2 - 496350*x^3 + 3267000 
*x^4 + 2025000*x^5))/(3 + 5*x)^2 - 171906*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt 
[1 - 2*x]])/12031250
 

3.20.19.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {108, 27, 166, 170, 27, 170, 25, 164, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{10} \int \frac {3 (2-11 x) \sqrt {1-2 x} (3 x+2)^3}{(5 x+3)^2}dx-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{10} \int \frac {(2-11 x) \sqrt {1-2 x} (3 x+2)^3}{(5 x+3)^2}dx-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \int \frac {(290-881 x) (3 x+2)^3}{\sqrt {1-2 x} (5 x+3)}dx-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3-\frac {1}{35} \int -\frac {7 (257-936 x) (3 x+2)^2}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {1}{5} \int \frac {(257-936 x) (3 x+2)^2}{\sqrt {1-2 x} (5 x+3)}dx+\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {936}{25} \sqrt {1-2 x} (3 x+2)^2-\frac {1}{25} \int -\frac {(1618-4125 x) (3 x+2)}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \int \frac {(1618-4125 x) (3 x+2)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {936}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {4093}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {3}{5} \sqrt {1-2 x} (1375 x+32)\right )+\frac {936}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {3}{5} \sqrt {1-2 x} (1375 x+32)-\frac {4093}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {936}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{10} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {3}{5} \sqrt {1-2 x} (1375 x+32)-\frac {8186 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}\right )+\frac {936}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {881}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {43 \sqrt {1-2 x} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}\)

input

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

output

-1/10*((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2 + (3*((-43*Sqrt[1 - 2*x]*( 
2 + 3*x)^4)/(5*(3 + 5*x)) + ((881*Sqrt[1 - 2*x]*(2 + 3*x)^3)/35 + ((936*Sq 
rt[1 - 2*x]*(2 + 3*x)^2)/25 + ((3*Sqrt[1 - 2*x]*(32 + 1375*x))/5 - (8186*A 
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]))/25)/5)/5))/10
 

3.20.19.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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])
 

3.20.19.4 Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.47

method	result	size
risch	\(\frac {4050000 x^{6}+4509000 x^{5}-4259700 x^{4}-3745410 x^{3}+1141990 x^{2}+682997 x -96776}{218750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {12279 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\)	\(66\)
pseudoelliptic	\(\frac {-171906 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}-55 \sqrt {1-2 x}\, \left (2025000 x^{5}+3267000 x^{4}-496350 x^{3}-2120880 x^{2}-489445 x +96776\right )}{12031250 \left (3+5 x \right )^{2}}\)	\(70\)
derivativedivides	\(\frac {81 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {1107 \left (1-2 x \right )^{\frac {5}{2}}}{6250}+\frac {36 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {228 \sqrt {1-2 x}}{3125}+\frac {\frac {259 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {2871 \sqrt {1-2 x}}{15625}}{\left (-6-10 x \right )^{2}}-\frac {12279 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\)	\(84\)
default	\(\frac {81 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {1107 \left (1-2 x \right )^{\frac {5}{2}}}{6250}+\frac {36 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {228 \sqrt {1-2 x}}{3125}+\frac {\frac {259 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {2871 \sqrt {1-2 x}}{15625}}{\left (-6-10 x \right )^{2}}-\frac {12279 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\)	\(84\)
trager	\(-\frac {\left (2025000 x^{5}+3267000 x^{4}-496350 x^{3}-2120880 x^{2}-489445 x +96776\right ) \sqrt {1-2 x}}{218750 \left (3+5 x \right )^{2}}+\frac {12279 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1718750}\)	\(87\)

input

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

output

1/218750*(4050000*x^6+4509000*x^5-4259700*x^4-3745410*x^3+1141990*x^2+6829 
97*x-96776)/(3+5*x)^2/(1-2*x)^(1/2)-12279/859375*arctanh(1/11*55^(1/2)*(1- 
2*x)^(1/2))*55^(1/2)
 

3.20.19.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {85953 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (2025000 \, x^{5} + 3267000 \, x^{4} - 496350 \, x^{3} - 2120880 \, x^{2} - 489445 \, x + 96776\right )} \sqrt {-2 \, x + 1}}{12031250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

input

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

output

1/12031250*(85953*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2 
*x + 1) - 8)/(5*x + 3)) - 55*(2025000*x^5 + 3267000*x^4 - 496350*x^3 - 212 
0880*x^2 - 489445*x + 96776)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
 

3.20.19.6 Sympy [A] (verification not implemented)

Time = 166.13 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.69 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {7}{2}}}{1750} - \frac {1107 \left (1 - 2 x\right )^{\frac {5}{2}}}{6250} + \frac {36 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {228 \sqrt {1 - 2 x}}{3125} + \frac {1202 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{171875} - \frac {5632 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} + \frac {968 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]

input

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

output

81*(1 - 2*x)**(7/2)/1750 - 1107*(1 - 2*x)**(5/2)/6250 + 36*(1 - 2*x)**(3/2 
)/3125 + 228*sqrt(1 - 2*x)/3125 + 1202*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt( 
55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/171875 - 5632*Piecewise((sqrt(55 
)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 
 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2* 
x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55 
)/5)))/15625 + 968*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 
1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 
 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(5 
5)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/665 
5, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/15625
 

3.20.19.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1107}{6250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {36}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {12279}{1718750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {228}{3125} \, \sqrt {-2 \, x + 1} + \frac {1295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2871 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

input

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

output

81/1750*(-2*x + 1)^(7/2) - 1107/6250*(-2*x + 1)^(5/2) + 36/3125*(-2*x + 1) 
^(3/2) + 12279/1718750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(5 
5) + 5*sqrt(-2*x + 1))) + 228/3125*sqrt(-2*x + 1) + 1/15625*(1295*(-2*x + 
1)^(3/2) - 2871*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)
 

3.20.19.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {81}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1107}{6250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {36}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {12279}{1718750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {228}{3125} \, \sqrt {-2 \, x + 1} + \frac {1295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2871 \, \sqrt {-2 \, x + 1}}{62500 \, {\left (5 \, x + 3\right )}^{2}} \]

input

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

output

-81/1750*(2*x - 1)^3*sqrt(-2*x + 1) - 1107/6250*(2*x - 1)^2*sqrt(-2*x + 1) 
 + 36/3125*(-2*x + 1)^(3/2) + 12279/1718750*sqrt(55)*log(1/2*abs(-2*sqrt(5 
5) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 228/3125*sqrt(-2* 
x + 1) + 1/62500*(1295*(-2*x + 1)^(3/2) - 2871*sqrt(-2*x + 1))/(5*x + 3)^2
 

3.20.19.9 Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {228\,\sqrt {1-2\,x}}{3125}+\frac {36\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {1107\,{\left (1-2\,x\right )}^{5/2}}{6250}+\frac {81\,{\left (1-2\,x\right )}^{7/2}}{1750}-\frac {\frac {2871\,\sqrt {1-2\,x}}{390625}-\frac {259\,{\left (1-2\,x\right )}^{3/2}}{78125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,12279{}\mathrm {i}}{859375} \]

input

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

output

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*12279i)/859375 + (228*(1 
- 2*x)^(1/2))/3125 + (36*(1 - 2*x)^(3/2))/3125 - (1107*(1 - 2*x)^(5/2))/62 
50 + (81*(1 - 2*x)^(7/2))/1750 - ((2871*(1 - 2*x)^(1/2))/390625 - (259*(1 
- 2*x)^(3/2))/78125)/((44*x)/5 + (2*x - 1)^2 + 11/25)