3.1940 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{(1-2 x)^{7/2}}{105 (3 x+2)^5}-\frac{43 (1-2 x)^{5/2}}{315 (3 x+2)^4}+\frac{43 (1-2 x)^{3/2}}{567 (3 x+2)^3}+\frac{43 \sqrt{1-2 x}}{7938 (3 x+2)}-\frac{43 \sqrt{1-2 x}}{1134 (3 x+2)^2}+\frac{43 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]

[Out]

(1 - 2*x)^(7/2)/(105*(2 + 3*x)^5) - (43*(1 - 2*x)^(5/2))/(315*(2 + 3*x)^4) + (43*(1 - 2*x)^(3/2))/(567*(2 + 3*
x)^3) - (43*Sqrt[1 - 2*x])/(1134*(2 + 3*x)^2) + (43*Sqrt[1 - 2*x])/(7938*(2 + 3*x)) + (43*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/(3969*Sqrt[21])

________________________________________________________________________________________

Rubi [A]  time = 0.0362533, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac{(1-2 x)^{7/2}}{105 (3 x+2)^5}-\frac{43 (1-2 x)^{5/2}}{315 (3 x+2)^4}+\frac{43 (1-2 x)^{3/2}}{567 (3 x+2)^3}+\frac{43 \sqrt{1-2 x}}{7938 (3 x+2)}-\frac{43 \sqrt{1-2 x}}{1134 (3 x+2)^2}+\frac{43 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

(1 - 2*x)^(7/2)/(105*(2 + 3*x)^5) - (43*(1 - 2*x)^(5/2))/(315*(2 + 3*x)^4) + (43*(1 - 2*x)^(3/2))/(567*(2 + 3*
x)^3) - (43*Sqrt[1 - 2*x])/(1134*(2 + 3*x)^2) + (43*Sqrt[1 - 2*x])/(7938*(2 + 3*x)) + (43*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/(3969*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 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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)^6} \, dx &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}+\frac{172}{105} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac{43 (1-2 x)^{5/2}}{315 (2+3 x)^4}-\frac{43}{63} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac{43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac{43 (1-2 x)^{3/2}}{567 (2+3 x)^3}+\frac{43}{189} \int \frac{\sqrt{1-2 x}}{(2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac{43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac{43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac{43 \sqrt{1-2 x}}{1134 (2+3 x)^2}-\frac{43 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{1134}\\ &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac{43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac{43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac{43 \sqrt{1-2 x}}{1134 (2+3 x)^2}+\frac{43 \sqrt{1-2 x}}{7938 (2+3 x)}-\frac{43 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{7938}\\ &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac{43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac{43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac{43 \sqrt{1-2 x}}{1134 (2+3 x)^2}+\frac{43 \sqrt{1-2 x}}{7938 (2+3 x)}+\frac{43 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{7938}\\ &=\frac{(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac{43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac{43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac{43 \sqrt{1-2 x}}{1134 (2+3 x)^2}+\frac{43 \sqrt{1-2 x}}{7938 (2+3 x)}+\frac{43 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.016138, size = 42, normalized size = 0.33 \[ \frac{(1-2 x)^{7/2} \left (\frac{117649}{(3 x+2)^5}-5504 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{12353145} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

((1 - 2*x)^(7/2)*(117649/(2 + 3*x)^5 - 5504*Hypergeometric2F1[7/2, 5, 9/2, 3/7 - (6*x)/7]))/12353145

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 75, normalized size = 0.6 \begin{align*} 7776\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{43\, \left ( 1-2\,x \right ) ^{9/2}}{381024}}-{\frac{37\, \left ( 1-2\,x \right ) ^{7/2}}{34992}}+{\frac{172\, \left ( 1-2\,x \right ) ^{5/2}}{32805}}-{\frac{2107\, \left ( 1-2\,x \right ) ^{3/2}}{314928}}+{\frac{2107\,\sqrt{1-2\,x}}{629856}} \right ) }+{\frac{43\,\sqrt{21}}{83349}{\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)^6,x)

[Out]

7776*(-43/381024*(1-2*x)^(9/2)-37/34992*(1-2*x)^(7/2)+172/32805*(1-2*x)^(5/2)-2107/314928*(1-2*x)^(3/2)+2107/6
29856*(1-2*x)^(1/2))/(-6*x-4)^5+43/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 2.25892, size = 173, normalized size = 1.35 \begin{align*} -\frac{43}{166698} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17415 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 163170 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 809088 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 1032430 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 516215 \, \sqrt{-2 \, x + 1}}{19845 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\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)^6,x, algorithm="maxima")

[Out]

-43/166698*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/19845*(17415*(-2*x +
1)^(9/2) + 163170*(-2*x + 1)^(7/2) - 809088*(-2*x + 1)^(5/2) + 1032430*(-2*x + 1)^(3/2) - 516215*sqrt(-2*x +
1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

________________________________________________________________________________________

Fricas [A]  time = 1.59561, size = 347, normalized size = 2.71 \begin{align*} \frac{215 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (17415 \, x^{4} - 116415 \, x^{3} - 53772 \, x^{2} + 3322 \, x - 7018\right )} \sqrt{-2 \, x + 1}}{833490 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

1/833490*(215*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt(21)*sqrt(-2*x + 1
) - 5)/(3*x + 2)) + 21*(17415*x^4 - 116415*x^3 - 53772*x^2 + 3322*x - 7018)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32)

________________________________________________________________________________________

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)**6,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.45913, size = 157, normalized size = 1.23 \begin{align*} -\frac{43}{166698} \, \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{17415 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 163170 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 809088 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 1032430 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 516215 \, \sqrt{-2 \, x + 1}}{635040 \,{\left (3 \, x + 2\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-43/166698*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/635040*(174
15*(2*x - 1)^4*sqrt(-2*x + 1) - 163170*(2*x - 1)^3*sqrt(-2*x + 1) - 809088*(2*x - 1)^2*sqrt(-2*x + 1) + 103243
0*(-2*x + 1)^(3/2) - 516215*sqrt(-2*x + 1))/(3*x + 2)^5