3.19.79 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx\) [1879]

Optimal. Leaf size=108 \[ -\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]

[Out]

-1/252*(1-2*x)^(5/2)/(2+3*x)^4+277/5292*(1-2*x)^(5/2)/(2+3*x)^3-14423/31752*(1-2*x)^(3/2)/(2+3*x)^2-14423/3333
96*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+14423/31752*(1-2*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 43, 65, 212} \begin {gather*} \frac {277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac {14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (3 x+2)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/252*(1 - 2*x)^(5/2)/(2 + 3*x)^4 + (277*(1 - 2*x)^(5/2))/(5292*(2 + 3*x)^3) - (14423*(1 - 2*x)^(3/2))/(31752
*(2 + 3*x)^2) + (14423*Sqrt[1 - 2*x])/(31752*(2 + 3*x)) - (14423*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(15876*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 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 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 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*} \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {(1-2 x)^{3/2} (1123+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}+\frac {14423 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {14423 \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {14423 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{31752}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {14423 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31752}\\ &=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {14423 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 65, normalized size = 0.60 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (60890+453730 x+988035 x^2+668979 x^3\right )}{2 (2+3 x)^4}-14423 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{333396} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*Sqrt[1 - 2*x]*(60890 + 453730*x + 988035*x^2 + 668979*x^3))/(2*(2 + 3*x)^4) - 14423*Sqrt[21]*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/333396

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 66, normalized size = 0.61

method	result	size
risch	\(-\frac {1337958 x^{4}+1307091 x^{3}-80575 x^{2}-331950 x -60890}{31752 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {14423 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\)	\(56\)
derivativedivides	\(\frac {-\frac {8259 \left (1-2 x \right )^{\frac {7}{2}}}{196}+\frac {189667 \left (1-2 x \right )^{\frac {5}{2}}}{756}-\frac {158653 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {100961 \sqrt {1-2 x}}{324}}{\left (-4-6 x \right )^{4}}-\frac {14423 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\)	\(66\)
default	\(\frac {-\frac {8259 \left (1-2 x \right )^{\frac {7}{2}}}{196}+\frac {189667 \left (1-2 x \right )^{\frac {5}{2}}}{756}-\frac {158653 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {100961 \sqrt {1-2 x}}{324}}{\left (-4-6 x \right )^{4}}-\frac {14423 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\)	\(66\)
trager	\(\frac {\left (668979 x^{3}+988035 x^{2}+453730 x +60890\right ) \sqrt {1-2 x}}{31752 \left (2+3 x \right )^{4}}-\frac {14423 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{666792}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

648*(-2753/42336*(1-2*x)^(7/2)+189667/489888*(1-2*x)^(5/2)-158653/209952*(1-2*x)^(3/2)+100961/209952*(1-2*x)^(
1/2))/(-4-6*x)^4-14423/333396*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 110, normalized size = 1.02 \begin {gather*} \frac {14423}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {668979 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 3983007 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 7773997 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4947089 \, \sqrt {-2 \, x + 1}}{15876 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14423/666792*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/15876*(668979*(-2*
x + 1)^(7/2) - 3983007*(-2*x + 1)^(5/2) + 7773997*(-2*x + 1)^(3/2) - 4947089*sqrt(-2*x + 1))/(81*(2*x - 1)^4 +
756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

________________________________________________________________________________________

Fricas [A]
time = 0.96, size = 99, normalized size = 0.92 \begin {gather*} \frac {14423 \, \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 (668979 \, x^{3} + 988035 \, x^{2} + 453730 \, x + 60890\right )} \sqrt {-2 \, x + 1}}{666792 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/666792*(14423*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*(668979*x^3 + 988035*x^2 + 453730*x + 60890)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x +
16)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.96, size = 100, normalized size = 0.93 \begin {gather*} \frac {14423}{666792} \, \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 {668979 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 3983007 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 7773997 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 4947089 \, \sqrt {-2 \, x + 1}}{254016 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14423/666792*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/254016*(6
68979*(2*x - 1)^3*sqrt(-2*x + 1) + 3983007*(2*x - 1)^2*sqrt(-2*x + 1) - 7773997*(-2*x + 1)^(3/2) + 4947089*sqr
t(-2*x + 1))/(3*x + 2)^4

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 89, normalized size = 0.82 \begin {gather*} \frac {\frac {100961\,\sqrt {1-2\,x}}{26244}-\frac {158653\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {189667\,{\left (1-2\,x\right )}^{5/2}}{61236}-\frac {2753\,{\left (1-2\,x\right )}^{7/2}}{5292}}{\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 {14423\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((100961*(1 - 2*x)^(1/2))/26244 - (158653*(1 - 2*x)^(3/2))/26244 + (189667*(1 - 2*x)^(5/2))/61236 - (2753*(1 -
2*x)^(7/2))/5292)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81) - (14423*21
^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/333396

________________________________________________________________________________________