3.1867 \(\int \frac{(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=88 \[ \frac{(1-2 x)^{5/2}}{63 (3 x+2)^3}-\frac{52 (1-2 x)^{3/2}}{189 (3 x+2)^2}+\frac{52 \sqrt{1-2 x}}{189 (3 x+2)}-\frac{104 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]

[Out]

(1 - 2*x)^(5/2)/(63*(2 + 3*x)^3) - (52*(1 - 2*x)^(3/2))/(189*(2 + 3*x)^2) + (52*Sqrt[1 - 2*x])/(189*(2 + 3*x))
- (104*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqrt[21])

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Rubi [A]  time = 0.0220452, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 47, 63, 206} \[ \frac{(1-2 x)^{5/2}}{63 (3 x+2)^3}-\frac{52 (1-2 x)^{3/2}}{189 (3 x+2)^2}+\frac{52 \sqrt{1-2 x}}{189 (3 x+2)}-\frac{104 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(1 - 2*x)^(5/2)/(63*(2 + 3*x)^3) - (52*(1 - 2*x)^(3/2))/(189*(2 + 3*x)^2) + (52*Sqrt[1 - 2*x])/(189*(2 + 3*x))
- (104*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(189*Sqrt[21])

Rule 78

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] || LtQ
[p, n]))))

Rule 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 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+3 x)^4} \, dx &=\frac{(1-2 x)^{5/2}}{63 (2+3 x)^3}+\frac{104}{63} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac{52 (1-2 x)^{3/2}}{189 (2+3 x)^2}-\frac{52}{63} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac{52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac{52 \sqrt{1-2 x}}{189 (2+3 x)}+\frac{52}{189} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac{52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac{52 \sqrt{1-2 x}}{189 (2+3 x)}-\frac{52}{189} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{(1-2 x)^{5/2}}{63 (2+3 x)^3}-\frac{52 (1-2 x)^{3/2}}{189 (2+3 x)^2}+\frac{52 \sqrt{1-2 x}}{189 (2+3 x)}-\frac{104 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0489451, size = 74, normalized size = 0.84 \[ \frac{-21 \left (1584 x^3+536 x^2-450 x-107\right )-104 \sqrt{21-42 x} (3 x+2)^3 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(-21*(-107 - 450*x + 536*x^2 + 1584*x^3) - 104*Sqrt[21 - 42*x]*(2 + 3*x)^3*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
3969*Sqrt[1 - 2*x]*(2 + 3*x)^3)

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Maple [A]  time = 0.01, size = 57, normalized size = 0.7 \begin{align*} 216\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{22\, \left ( 1-2\,x \right ) ^{5/2}}{567}}+{\frac{104\, \left ( 1-2\,x \right ) ^{3/2}}{729}}-{\frac{91\,\sqrt{1-2\,x}}{729}} \right ) }-{\frac{104\,\sqrt{21}}{3969}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x)

[Out]

216*(-22/567*(1-2*x)^(5/2)+104/729*(1-2*x)^(3/2)-91/729*(1-2*x)^(1/2))/(-6*x-4)^3-104/3969*arctanh(1/7*21^(1/2
)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.52227, size = 124, normalized size = 1.41 \begin{align*} \frac{52}{3969} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8 \,{\left (198 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 728 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 637 \, \sqrt{-2 \, x + 1}\right )}}{189 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x, algorithm="maxima")

[Out]

52/3969*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 8/189*(198*(-2*x + 1)^(5/
2) - 728*(-2*x + 1)^(3/2) + 637*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 1.4264, size = 236, normalized size = 2.68 \begin{align*} \frac{52 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (792 \, x^{2} + 664 \, x + 107\right )} \sqrt{-2 \, x + 1}}{3969 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/3969*(52*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(792*
x^2 + 664*x + 107)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(3+5*x)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [A]  time = 2.52077, size = 113, normalized size = 1.28 \begin{align*} \frac{52}{3969} \, \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{198 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 728 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 637 \, \sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^4,x, algorithm="giac")

[Out]

52/3969*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/189*(198*(2*x
- 1)^2*sqrt(-2*x + 1) - 728*(-2*x + 1)^(3/2) + 637*sqrt(-2*x + 1))/(3*x + 2)^3