3.1937 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=96 \[ \frac{(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac{73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac{365}{567} (1-2 x)^{3/2}-\frac{365}{81} \sqrt{1-2 x}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-365*Sqrt[1 - 2*x])/81 - (365*(1 - 2*x)^(3/2))/567 + (1 - 2*x)^(7/2)/(42*(2 + 3*x)^2) - (73*(1 - 2*x)^(5/2))/
(126*(2 + 3*x)) + (365*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi [A]  time = 0.024988, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 206} \[ \frac{(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac{73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac{365}{567} (1-2 x)^{3/2}-\frac{365}{81} \sqrt{1-2 x}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-365*Sqrt[1 - 2*x])/81 - (365*(1 - 2*x)^(3/2))/567 + (1 - 2*x)^(7/2)/(42*(2 + 3*x)^2) - (73*(1 - 2*x)^(5/2))/
(126*(2 + 3*x)) + (365*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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 50

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 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)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx &=\frac{(1-2 x)^{7/2}}{42 (2+3 x)^2}+\frac{73}{42} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac{73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac{365}{126} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac{365}{567} (1-2 x)^{3/2}+\frac{(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac{73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac{365}{54} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{365}{81} \sqrt{1-2 x}-\frac{365}{567} (1-2 x)^{3/2}+\frac{(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac{73 (1-2 x)^{5/2}}{126 (2+3 x)}-\frac{2555}{162} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{365}{81} \sqrt{1-2 x}-\frac{365}{567} (1-2 x)^{3/2}+\frac{(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac{73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac{2555}{162} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{365}{81} \sqrt{1-2 x}-\frac{365}{567} (1-2 x)^{3/2}+\frac{(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac{73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0204062, size = 48, normalized size = 0.5 \[ \frac{(1-2 x)^{7/2} \left (343-292 (3 x+2)^2 \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{14406 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(7/2)*(343 - 292*(2 + 3*x)^2*Hypergeometric2F1[2, 7/2, 9/2, 3/7 - (6*x)/7]))/(14406*(2 + 3*x)^2)

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Maple [A]  time = 0.01, size = 66, normalized size = 0.7 \begin{align*} -{\frac{20}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{32}{9}\sqrt{1-2\,x}}-{\frac{28}{3\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{79}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{539}{108}\sqrt{1-2\,x}} \right ) }+{\frac{365\,\sqrt{21}}{243}{\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)^(5/2)*(3+5*x)/(2+3*x)^3,x)

[Out]

-20/81*(1-2*x)^(3/2)-32/9*(1-2*x)^(1/2)-28/3*(-79/36*(1-2*x)^(3/2)+539/108*(1-2*x)^(1/2))/(-6*x-4)^2+365/243*a
rctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.13475, size = 124, normalized size = 1.29 \begin{align*} -\frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{365}{486} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{32}{9} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (237 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20/81*(-2*x + 1)^(3/2) - 365/486*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) -
32/9*sqrt(-2*x + 1) + 7/81*(237*(-2*x + 1)^(3/2) - 539*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 1.58562, size = 246, normalized size = 2.56 \begin{align*} \frac{365 \, \sqrt{7} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 3 \,{\left (720 \, x^{3} - 4584 \, x^{2} - 8731 \, x - 3521\right )} \sqrt{-2 \, x + 1}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*(365*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 3*(
720*x^3 - 4584*x^2 - 8731*x - 3521)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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

[Out]

Timed out

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Giac [A]  time = 1.67163, size = 116, normalized size = 1.21 \begin{align*} -\frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{365}{486} \, \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{32}{9} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (237 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}}{324 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20/81*(-2*x + 1)^(3/2) - 365/486*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 32/9*sqrt(-2*x + 1) + 7/324*(237*(-2*x + 1)^(3/2) - 539*sqrt(-2*x + 1))/(3*x + 2)^2