\(\int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx\) [2038]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]

[Out]

54/25*(1-2*x)^(3/2)-27/100*(1-2*x)^(5/2)-2/6875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-3897/500*(1-2*x)
^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {90, 65, 212} \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}}-\frac {27}{100} (1-2 x)^{5/2}+\frac {54}{25} (1-2 x)^{3/2}-\frac {3897}{500} \sqrt {1-2 x} \]

[In]

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

[Out]

(-3897*Sqrt[1 - 2*x])/500 + (54*(1 - 2*x)^(3/2))/25 - (27*(1 - 2*x)^(5/2))/100 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(125*Sqrt[55])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 90

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3897}{500 \sqrt {1-2 x}}-\frac {162}{25} \sqrt {1-2 x}+\frac {27}{20} (1-2 x)^{3/2}+\frac {1}{125 \sqrt {1-2 x} (3+5 x)}\right ) \, dx \\ & = -\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}+\frac {1}{125} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}-\frac {1}{125} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {3897}{500} \sqrt {1-2 x}+\frac {54}{25} (1-2 x)^{3/2}-\frac {27}{100} (1-2 x)^{5/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {9}{125} \sqrt {1-2 x} \left (82+45 x+15 x^2\right )-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]

[In]

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

[Out]

(-9*Sqrt[1 - 2*x]*(82 + 45*x + 15*x^2))/125 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}-\frac {9 \sqrt {1-2 x}\, \left (15 x^{2}+45 x +82\right )}{125}\)	\(39\)
risch	\(\frac {9 \left (15 x^{2}+45 x +82\right ) \left (-1+2 x \right )}{125 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\)	\(44\)
derivativedivides	\(\frac {54 \left (1-2 x \right )^{\frac {3}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{100}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}-\frac {3897 \sqrt {1-2 x}}{500}\)	\(47\)
default	\(\frac {54 \left (1-2 x \right )^{\frac {3}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{100}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}-\frac {3897 \sqrt {1-2 x}}{500}\)	\(47\)
trager	\(\left (-\frac {27}{25} x^{2}-\frac {81}{25} x -\frac {738}{125}\right ) \sqrt {1-2 x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{6875}\)	\(64\)

[In]

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

[Out]

-2/6875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-9/125*(1-2*x)^(1/2)*(15*x^2+45*x+82)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {9}{125} \, {\left (15 \, x^{2} + 45 \, x + 82\right )} \sqrt {-2 \, x + 1} + \frac {1}{6875} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]

[In]

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

[Out]

-9/125*(15*x^2 + 45*x + 82)*sqrt(-2*x + 1) + 1/6875*sqrt(55)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)
)

Sympy [A] (verification not implemented)

Time = 1.70 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=- \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{100} + \frac {54 \left (1 - 2 x\right )^{\frac {3}{2}}}{25} - \frac {3897 \sqrt {1 - 2 x}}{500} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{6875} \]

[In]

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

[Out]

-27*(1 - 2*x)**(5/2)/100 + 54*(1 - 2*x)**(3/2)/25 - 3897*sqrt(1 - 2*x)/500 + sqrt(55)*(log(sqrt(1 - 2*x) - sqr
t(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/6875

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {27}{100} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {54}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{6875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3897}{500} \, \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-27/100*(-2*x + 1)^(5/2) + 54/25*(-2*x + 1)^(3/2) + 1/6875*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(5
5) + 5*sqrt(-2*x + 1))) - 3897/500*sqrt(-2*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=-\frac {27}{100} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {54}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{6875} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3897}{500} \, \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-27/100*(2*x - 1)^2*sqrt(-2*x + 1) + 54/25*(-2*x + 1)^(3/2) + 1/6875*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqr
t(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3897/500*sqrt(-2*x + 1)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx=\frac {54\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {3897\,\sqrt {1-2\,x}}{500}-\frac {27\,{\left (1-2\,x\right )}^{5/2}}{100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{6875} \]

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*2i)/6875 - (3897*(1 - 2*x)^(1/2))/500 + (54*(1 - 2*x)^(3/2))/
25 - (27*(1 - 2*x)^(5/2))/100