3.21.47 \(\int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx\) [2047]

3.21.47.1 Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {26352 \sqrt {1-2 x} (2+3 x)^2}{34375}-\frac {1717 \sqrt {1-2 x} (2+3 x)^3}{9625}-\frac {8}{275} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{55 (3+5 x)}-\frac {3 \sqrt {1-2 x} (1847824+615875 x)}{171875}-\frac {398 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{171875 \sqrt {55}} \]

output

-398/9453125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-26352/34375*(2+ 
3*x)^2*(1-2*x)^(1/2)-1717/9625*(2+3*x)^3*(1-2*x)^(1/2)-8/275*(2+3*x)^4*(1- 
2*x)^(1/2)-1/55*(2+3*x)^5*(1-2*x)^(1/2)/(3+5*x)-3/171875*(1847824+615875*x 
)*(1-2*x)^(1/2)
 

3.21.47.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=\frac {-\frac {55 \sqrt {1-2 x} \left (135011752+334366065 x+273540465 x^2+200942775 x^3+92998125 x^4+19490625 x^5\right )}{3+5 x}-2786 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{66171875} \]

input

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

output

((-55*Sqrt[1 - 2*x]*(135011752 + 334366065*x + 273540465*x^2 + 200942775*x 
^3 + 92998125*x^4 + 19490625*x^5))/(3 + 5*x) - 2786*Sqrt[55]*ArcTanh[Sqrt[ 
5/11]*Sqrt[1 - 2*x]])/66171875
 

3.21.47.3 Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {109, 25, 170, 27, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {1}{55} \int -\frac {(3 x+2)^4 (72 x+83)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{55} \int \frac {(3 x+2)^4 (72 x+83)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {1}{55} \left (-\frac {1}{45} \int -\frac {9 (3 x+2)^3 (1717 x+1070)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \int \frac {(3 x+2)^3 (1717 x+1070)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (-\frac {1}{35} \int -\frac {7 (3 x+2)^2 (26352 x+15851)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{5} \int \frac {(3 x+2)^2 (26352 x+15851)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{5} \left (-\frac {1}{25} \int -\frac {(3 x+2) (1847625 x+1108774)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {26352}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \int \frac {(3 x+2) (1847625 x+1108774)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {26352}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {199}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {3}{5} \sqrt {1-2 x} (615875 x+1847824)\right )-\frac {26352}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \left (-\frac {199}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {3}{5} \sqrt {1-2 x} (615875 x+1847824)\right )-\frac {26352}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{5} \left (\frac {1}{25} \left (-\frac {398 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {3}{5} \sqrt {1-2 x} (615875 x+1847824)\right )-\frac {26352}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {1717}{35} \sqrt {1-2 x} (3 x+2)^3\right )-\frac {8}{5} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{55 (5 x+3)}\)

input

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

output

-1/55*(Sqrt[1 - 2*x]*(2 + 3*x)^5)/(3 + 5*x) + ((-8*Sqrt[1 - 2*x]*(2 + 3*x) 
^4)/5 + ((-1717*Sqrt[1 - 2*x]*(2 + 3*x)^3)/35 + ((-26352*Sqrt[1 - 2*x]*(2 
+ 3*x)^2)/25 + ((-3*Sqrt[1 - 2*x]*(1847824 + 615875*x))/5 - (398*ArcTanh[S 
qrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]))/25)/5)/5)/55
 

3.21.47.3.1 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 109

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

3.21.47.4 Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50

method	result	size
risch	\(\frac {38981250 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-64342561 x -135011752}{1203125 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {398 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9453125}\)	\(66\)
pseudoelliptic	\(\frac {-2786 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-55 \sqrt {1-2 x}\, \left (19490625 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{198515625+330859375 x}\)	\(67\)
derivativedivides	\(-\frac {81 \left (1-2 x \right )^{\frac {9}{2}}}{400}+\frac {2187 \left (1-2 x \right )^{\frac {7}{2}}}{875}-\frac {315171 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {105228 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {607689 \sqrt {1-2 x}}{10000}+\frac {2 \sqrt {1-2 x}}{859375 \left (-\frac {6}{5}-2 x \right )}-\frac {398 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9453125}\)	\(81\)
default	\(-\frac {81 \left (1-2 x \right )^{\frac {9}{2}}}{400}+\frac {2187 \left (1-2 x \right )^{\frac {7}{2}}}{875}-\frac {315171 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {105228 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {607689 \sqrt {1-2 x}}{10000}+\frac {2 \sqrt {1-2 x}}{859375 \left (-\frac {6}{5}-2 x \right )}-\frac {398 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{9453125}\)	\(81\)
trager	\(-\frac {\left (19490625 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}}{1203125 \left (3+5 x \right )}-\frac {199 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{9453125}\)	\(87\)

input

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

output

1/1203125*(38981250*x^6+166505625*x^5+308887425*x^4+346138155*x^3+39519166 
5*x^2-64342561*x-135011752)/(3+5*x)/(1-2*x)^(1/2)-398/9453125*arctanh(1/11 
*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
 

3.21.47.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.59 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=\frac {1393 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (19490625 \, x^{5} + 92998125 \, x^{4} + 200942775 \, x^{3} + 273540465 \, x^{2} + 334366065 \, x + 135011752\right )} \sqrt {-2 \, x + 1}}{66171875 \, {\left (5 \, x + 3\right )}} \]

input

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

output

1/66171875*(1393*sqrt(55)*(5*x + 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8 
)/(5*x + 3)) - 55*(19490625*x^5 + 92998125*x^4 + 200942775*x^3 + 273540465 
*x^2 + 334366065*x + 135011752)*sqrt(-2*x + 1))/(5*x + 3)
 

3.21.47.6 Sympy [A] (verification not implemented)

Time = 56.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.66 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=- \frac {81 \left (1 - 2 x\right )^{\frac {9}{2}}}{400} + \frac {2187 \left (1 - 2 x\right )^{\frac {7}{2}}}{875} - \frac {315171 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} + \frac {105228 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} - \frac {607689 \sqrt {1 - 2 x}}{10000} + \frac {18 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{859375} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]

input

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

output

-81*(1 - 2*x)**(9/2)/400 + 2187*(1 - 2*x)**(7/2)/875 - 315171*(1 - 2*x)**( 
5/2)/25000 + 105228*(1 - 2*x)**(3/2)/3125 - 607689*sqrt(1 - 2*x)/10000 + 1 
8*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55) 
/5))/859375 - 4*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
 

3.21.47.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {81}{400} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {2187}{875} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {315171}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {105228}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {199}{9453125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {607689}{10000} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{171875 \, {\left (5 \, x + 3\right )}} \]

input

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

output

-81/400*(-2*x + 1)^(9/2) + 2187/875*(-2*x + 1)^(7/2) - 315171/25000*(-2*x 
+ 1)^(5/2) + 105228/3125*(-2*x + 1)^(3/2) + 199/9453125*sqrt(55)*log(-(sqr 
t(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 607689/10000*sq 
rt(-2*x + 1) - 1/171875*sqrt(-2*x + 1)/(5*x + 3)
 

3.21.47.8 Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=-\frac {81}{400} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {2187}{875} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {315171}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {105228}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {199}{9453125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {607689}{10000} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{171875 \, {\left (5 \, x + 3\right )}} \]

input

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

output

-81/400*(2*x - 1)^4*sqrt(-2*x + 1) - 2187/875*(2*x - 1)^3*sqrt(-2*x + 1) - 
 315171/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 105228/3125*(-2*x + 1)^(3/2) + 
199/9453125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55 
) + 5*sqrt(-2*x + 1))) - 607689/10000*sqrt(-2*x + 1) - 1/171875*sqrt(-2*x 
+ 1)/(5*x + 3)
 

3.21.47.9 Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^2} \, dx=\frac {105228\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {607689\,\sqrt {1-2\,x}}{10000}-\frac {2\,\sqrt {1-2\,x}}{859375\,\left (2\,x+\frac {6}{5}\right )}-\frac {315171\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {2187\,{\left (1-2\,x\right )}^{7/2}}{875}-\frac {81\,{\left (1-2\,x\right )}^{9/2}}{400}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,398{}\mathrm {i}}{9453125} \]

input

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

output

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*398i)/9453125 - (2*(1 - 2 
*x)^(1/2))/(859375*(2*x + 6/5)) - (607689*(1 - 2*x)^(1/2))/10000 + (105228 
*(1 - 2*x)^(3/2))/3125 - (315171*(1 - 2*x)^(5/2))/25000 + (2187*(1 - 2*x)^ 
(7/2))/875 - (81*(1 - 2*x)^(9/2))/400