\(\int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx\) [2172]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \]

[Out]

343/132/(1-2*x)^(3/2)-2/33275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-784/121/(1-2*x)^(1/2)-27/20*(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 = {89, 65, 212} \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}}-\frac {27}{20} \sqrt {1-2 x}-\frac {784}{121 \sqrt {1-2 x}}+\frac {343}{132 (1-2 x)^{3/2}} \]

[In]

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

[Out]

343/(132*(1 - 2*x)^(3/2)) - 784/(121*Sqrt[1 - 2*x]) - (27*Sqrt[1 - 2*x])/20 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(605*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 89

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], (c + d*x)^n*((e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

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 {343}{44 (1-2 x)^{5/2}}-\frac {784}{121 (1-2 x)^{3/2}}+\frac {27}{20 \sqrt {1-2 x}}+\frac {1}{605 \sqrt {1-2 x} (3+5 x)}\right ) \, dx \\ & = \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}+\frac {1}{605} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {1}{605} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {343}{132 (1-2 x)^{3/2}}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27}{20} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {-\frac {55 \left (9494-33321 x+9801 x^2\right )}{(1-2 x)^{3/2}}-6 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{99825} \]

[In]

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

[Out]

((-55*(9494 - 33321*x + 9801*x^2))/(1 - 2*x)^(3/2) - 6*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/99825

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {343}{132 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{33275}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27 \sqrt {1-2 x}}{20}\)	\(47\)
default	\(\frac {343}{132 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{33275}-\frac {784}{121 \sqrt {1-2 x}}-\frac {27 \sqrt {1-2 x}}{20}\)	\(47\)
pseudoelliptic	\(\frac {-\frac {9494}{1815}+\frac {2 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \sqrt {55}}{33275}-\frac {27 x^{2}}{5}+\frac {11107 x}{605}}{\left (1-2 x \right )^{\frac {3}{2}}}\)	\(49\)
trager	\(-\frac {\left (9801 x^{2}-33321 x +9494\right ) \sqrt {1-2 x}}{1815 \left (-1+2 x \right )^{2}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{33275}\)	\(72\)

[In]

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

[Out]

343/132/(1-2*x)^(3/2)-2/33275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-784/121/(1-2*x)^(1/2)-27/20*(1-2*x
)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {3 \, \sqrt {55} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (9801 \, x^{2} - 33321 \, x + 9494\right )} \sqrt {-2 \, x + 1}}{99825 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

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

[Out]

1/99825*(3*sqrt(55)*(4*x^2 - 4*x + 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(9801*x^2 - 3332
1*x + 9494)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

Sympy [A] (verification not implemented)

Time = 2.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=- \frac {27 \sqrt {1 - 2 x}}{20} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{33275} - \frac {784}{121 \sqrt {1 - 2 x}} + \frac {343}{132 \left (1 - 2 x\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

-27*sqrt(1 - 2*x)/20 + sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/33275 - 78
4/(121*sqrt(1 - 2*x)) + 343/(132*(1 - 2*x)**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {1}{33275} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{20} \, \sqrt {-2 \, x + 1} + \frac {49 \, {\left (384 \, x - 115\right )}}{1452 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

1/33275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/20*sqrt(-2*x + 1) + 49
/1452*(384*x - 115)/(-2*x + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {1}{33275} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {27}{20} \, \sqrt {-2 \, x + 1} - \frac {49 \, {\left (384 \, x - 115\right )}}{1452 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]

[In]

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

[Out]

1/33275*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/20*sqrt(-2*x
+ 1) - 49/1452*(384*x - 115)/((2*x - 1)*sqrt(-2*x + 1))

Mupad [B] (verification not implemented)

Time = 1.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx=\frac {\frac {1568\,x}{121}-\frac {5635}{1452}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{33275}-\frac {27\,\sqrt {1-2\,x}}{20} \]

[In]

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

[Out]

((1568*x)/121 - 5635/1452)/(1 - 2*x)^(3/2) - (2*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/33275 - (27*(1
- 2*x)^(1/2))/20