\(\int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx\) [1868]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \]

[Out]

1/84*(1-2*x)^(5/2)/(2+3*x)^4-137/756*(1-2*x)^(3/2)/(2+3*x)^3-137/111132*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21
^(1/2)+137/1512*(1-2*x)^(1/2)/(2+3*x)^2-137/10584*(1-2*x)^(1/2)/(2+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=-\frac {137 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}}+\frac {(1-2 x)^{5/2}}{84 (3 x+2)^4}-\frac {137 (1-2 x)^{3/2}}{756 (3 x+2)^3}-\frac {137 \sqrt {1-2 x}}{10584 (3 x+2)}+\frac {137 \sqrt {1-2 x}}{1512 (3 x+2)^2} \]

[In]

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

[Out]

(1 - 2*x)^(5/2)/(84*(2 + 3*x)^4) - (137*(1 - 2*x)^(3/2))/(756*(2 + 3*x)^3) + (137*Sqrt[1 - 2*x])/(1512*(2 + 3*
x)^2) - (137*Sqrt[1 - 2*x])/(10584*(2 + 3*x)) - (137*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(5292*Sqrt[21])

Rule 43

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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}& = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}+\frac {137}{84} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}-\frac {137}{252} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1512} \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}+\frac {137 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{10584} \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{10584} \\ & = \frac {(1-2 x)^{5/2}}{84 (2+3 x)^4}-\frac {137 (1-2 x)^{3/2}}{756 (2+3 x)^3}+\frac {137 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {137 \sqrt {1-2 x}}{10584 (2+3 x)}-\frac {137 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5292 \sqrt {21}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (970-7990 x-13245 x^2+3699 x^3\right )}{2 (2+3 x)^4}-137 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132} \]

[In]

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

[Out]

((-21*Sqrt[1 - 2*x]*(970 - 7990*x - 13245*x^2 + 3699*x^3))/(2*(2 + 3*x)^4) - 137*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/111132

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {7398 x^{4}-30189 x^{3}-2735 x^{2}+9930 x -970}{10584 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {137 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{111132}\)	\(56\)
pseudoelliptic	\(\frac {-274 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (3699 x^{3}-13245 x^{2}-7990 x +970\right )}{222264 \left (2+3 x \right )^{4}}\)	\(60\)
derivativedivides	\(-\frac {1296 \left (-\frac {137 \left (1-2 x \right )^{\frac {7}{2}}}{254016}-\frac {733 \left (1-2 x \right )^{\frac {5}{2}}}{326592}+\frac {1507 \left (1-2 x \right )^{\frac {3}{2}}}{139968}-\frac {959 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{4}}-\frac {137 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{111132}\)	\(66\)
default	\(-\frac {1296 \left (-\frac {137 \left (1-2 x \right )^{\frac {7}{2}}}{254016}-\frac {733 \left (1-2 x \right )^{\frac {5}{2}}}{326592}+\frac {1507 \left (1-2 x \right )^{\frac {3}{2}}}{139968}-\frac {959 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{4}}-\frac {137 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{111132}\)	\(66\)
trager	\(-\frac {\left (3699 x^{3}-13245 x^{2}-7990 x +970\right ) \sqrt {1-2 x}}{10584 \left (2+3 x \right )^{4}}+\frac {137 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{222264}\)	\(77\)

[In]

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

[Out]

1/10584*(7398*x^4-30189*x^3-2735*x^2+9930*x-970)/(2+3*x)^4/(1-2*x)^(1/2)-137/111132*arctanh(1/7*21^(1/2)*(1-2*
x)^(1/2))*21^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {137 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (3699 \, x^{3} - 13245 \, x^{2} - 7990 \, x + 970\right )} \sqrt {-2 \, x + 1}}{222264 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

[In]

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

[Out]

1/222264*(137*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x +
2)) - 21*(3699*x^3 - 13245*x^2 - 7990*x + 970)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {137}{222264} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3699 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 15393 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 46991 \, \sqrt {-2 \, x + 1}}{5292 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

[In]

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

[Out]

137/222264*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/5292*(3699*(-2*x + 1
)^(7/2) + 15393*(-2*x + 1)^(5/2) - 73843*(-2*x + 1)^(3/2) + 46991*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x -
1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {137}{222264} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3699 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 15393 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 73843 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 46991 \, \sqrt {-2 \, x + 1}}{84672 \, {\left (3 \, x + 2\right )}^{4}} \]

[In]

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

[Out]

137/222264*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/84672*(3699
*(2*x - 1)^3*sqrt(-2*x + 1) - 15393*(2*x - 1)^2*sqrt(-2*x + 1) + 73843*(-2*x + 1)^(3/2) - 46991*sqrt(-2*x + 1)
)/(3*x + 2)^4

Mupad [B] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {\frac {959\,\sqrt {1-2\,x}}{8748}-\frac {1507\,{\left (1-2\,x\right )}^{3/2}}{8748}+\frac {733\,{\left (1-2\,x\right )}^{5/2}}{20412}+\frac {137\,{\left (1-2\,x\right )}^{7/2}}{15876}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {137\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{111132} \]

[In]

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

[Out]

((959*(1 - 2*x)^(1/2))/8748 - (1507*(1 - 2*x)^(3/2))/8748 + (733*(1 - 2*x)^(5/2))/20412 + (137*(1 - 2*x)^(7/2)
)/15876)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81) - (137*21^(1/2)*atanh
((21^(1/2)*(1 - 2*x)^(1/2))/7))/111132