3.20.21 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^3} \, dx\) [1921]

Optimal. Leaf size=94 \[ \frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]

[Out]

119/3025*(1-2*x)^(3/2)-1/550*(1-2*x)^(5/2)/(3+5*x)^2-131/6050*(1-2*x)^(5/2)/(3+5*x)-357/6875*arctanh(1/11*55^(
1/2)*(1-2*x)^(1/2))*55^(1/2)+357/1375*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 94, 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, 52, 65, 212} \begin {gather*} -\frac {131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac {119 (1-2 x)^{3/2}}{3025}+\frac {357 \sqrt {1-2 x}}{1375}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(357*Sqrt[1 - 2*x])/1375 + (119*(1 - 2*x)^(3/2))/3025 - (1 - 2*x)^(5/2)/(550*(3 + 5*x)^2) - (131*(1 - 2*x)^(5/
2))/(6050*(3 + 5*x)) - (357*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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} (2+3 x)^2}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}+\frac {1}{550} \int \frac {(1-2 x)^{3/2} (725+990 x)}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{1210}\\ &=\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357}{550} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357}{250} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357}{250} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 63, normalized size = 0.67 \begin {gather*} \frac {\sqrt {1-2 x} \left (656+2105 x+1320 x^2-600 x^3\right )}{250 (3+5 x)^2}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(656 + 2105*x + 1320*x^2 - 600*x^3))/(250*(3 + 5*x)^2) - (357*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/(125*Sqrt[55])

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Maple [A]
time = 0.11, size = 66, normalized size = 0.70

method	result	size
risch	\(\frac {1200 x^{4}-3240 x^{3}-2890 x^{2}+793 x +656}{250 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {357 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\)	\(56\)
derivativedivides	\(\frac {6 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {174 \sqrt {1-2 x}}{625}+\frac {\frac {127 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {1419 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {357 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\)	\(66\)
default	\(\frac {6 \left (1-2 x \right )^{\frac {3}{2}}}{125}+\frac {174 \sqrt {1-2 x}}{625}+\frac {\frac {127 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {1419 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {357 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6875}\)	\(66\)
trager	\(-\frac {\left (600 x^{3}-1320 x^{2}-2105 x -656\right ) \sqrt {1-2 x}}{250 \left (3+5 x \right )^{2}}+\frac {357 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{13750}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(1-2*x)^(3/2)+174/625*(1-2*x)^(1/2)+2/25*(127/10*(1-2*x)^(3/2)-1419/50*(1-2*x)^(1/2))/(-6-10*x)^2-357/68
75*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]
time = 0.53, size = 92, normalized size = 0.98 \begin {gather*} \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {357}{13750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {174}{625} \, \sqrt {-2 \, x + 1} + \frac {635 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1419 \, \sqrt {-2 \, x + 1}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(-2*x + 1)^(3/2) + 357/13750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 174/625*sqrt(-2*x + 1) + 1/625*(635*(-2*x + 1)^(3/2) - 1419*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]
time = 0.60, size = 79, normalized size = 0.84 \begin {gather*} \frac {357 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (600 \, x^{3} - 1320 \, x^{2} - 2105 \, x - 656\right )} \sqrt {-2 \, x + 1}}{13750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13750*(357*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(600*x^3 - 1
320*x^2 - 2105*x - 656)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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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)*(2+3*x)**2/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.52, size = 86, normalized size = 0.91 \begin {gather*} \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {357}{13750} \, \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 {174}{625} \, \sqrt {-2 \, x + 1} + \frac {635 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1419 \, \sqrt {-2 \, x + 1}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(-2*x + 1)^(3/2) + 357/13750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 174/625*sqrt(-2*x + 1) + 1/2500*(635*(-2*x + 1)^(3/2) - 1419*sqrt(-2*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 1.20, size = 74, normalized size = 0.79 \begin {gather*} \frac {174\,\sqrt {1-2\,x}}{625}+\frac {6\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {\frac {1419\,\sqrt {1-2\,x}}{15625}-\frac {127\,{\left (1-2\,x\right )}^{3/2}}{3125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,357{}\mathrm {i}}{6875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*357i)/6875 + (174*(1 - 2*x)^(1/2))/625 + (6*(1 - 2*x)^(3/2))/
125 - ((1419*(1 - 2*x)^(1/2))/15625 - (127*(1 - 2*x)^(3/2))/3125)/((44*x)/5 + (2*x - 1)^2 + 11/25)

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