3.1965 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=161 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac{209 \sqrt{1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac{146971 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7938 \sqrt{21}} \]

[Out]

(-209*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(756*(2 + 3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(15*(2 + 3*x)^5) + (11*(1 -
2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^4) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)^3) - (11*Sqrt[1 - 2*x]
*(3911 + 6475*x))/(15876*(2 + 3*x)) - (146971*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7938*Sqrt[21])

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Rubi [A]  time = 0.0623411, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 12, 149, 146, 63, 206} \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac{209 \sqrt{1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac{146971 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7938 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-209*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(756*(2 + 3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(15*(2 + 3*x)^5) + (11*(1 -
2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^4) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)^3) - (11*Sqrt[1 - 2*x]
*(3911 + 6475*x))/(15876*(2 + 3*x)) - (146971*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7938*Sqrt[21])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

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)^3}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{1}{15} \int -\frac{55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}-\frac{11}{3} \int \frac{(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac{11}{36} \int \frac{\sqrt{1-2 x} (3+5 x)^2 (24+18 x)}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac{11}{324} \int \frac{(-162-72 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{209 \sqrt{1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac{11 \int \frac{(-9522-3330 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{13608}\\ &=-\frac{209 \sqrt{1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac{11 \sqrt{1-2 x} (3911+6475 x)}{15876 (2+3 x)}+\frac{146971 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{15876}\\ &=-\frac{209 \sqrt{1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac{11 \sqrt{1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac{146971 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{15876}\\ &=-\frac{209 \sqrt{1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac{11 \sqrt{1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac{146971 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7938 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0707331, size = 89, normalized size = 0.55 \[ \frac{-21 \left (52920000 x^6+226697490 x^5+288394965 x^4+106869513 x^3-43687652 x^2-40879074 x-7933096\right )-1469710 \sqrt{21-42 x} (3 x+2)^5 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1666980 \sqrt{1-2 x} (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*(-7933096 - 40879074*x - 43687652*x^2 + 106869513*x^3 + 288394965*x^4 + 226697490*x^5 + 52920000*x^6) - 1
469710*Sqrt[21 - 42*x]*(2 + 3*x)^5*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1666980*Sqrt[1 - 2*x]*(2 + 3*x)^5)

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Maple [A]  time = 0.01, size = 84, normalized size = 0.5 \begin{align*}{\frac{1000}{729}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{284287}{784} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{226727}{72} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{1383554}{135} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{9599737}{648} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{31200211}{3888}\sqrt{1-2\,x}} \right ) }-{\frac{146971\,\sqrt{21}}{166698}{\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)^3/(2+3*x)^6,x)

[Out]

1000/729*(1-2*x)^(1/2)+8/3*(-284287/784*(1-2*x)^(9/2)+226727/72*(1-2*x)^(7/2)-1383554/135*(1-2*x)^(5/2)+959973
7/648*(1-2*x)^(3/2)-31200211/3888*(1-2*x)^(1/2))/(-6*x-4)^5-146971/166698*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*
21^(1/2)

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Maxima [A]  time = 2.07559, size = 185, normalized size = 1.15 \begin{align*} \frac{146971}{333396} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1000}{729} \, \sqrt{-2 \, x + 1} + \frac{345408705 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2999598210 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 9762357024 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 14111613390 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 7644051695 \, \sqrt{-2 \, x + 1}}{357210 \,{\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)^3/(2+3*x)^6,x, algorithm="maxima")

[Out]

146971/333396*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1000/729*sqrt(-2*x
+ 1) + 1/357210*(345408705*(-2*x + 1)^(9/2) - 2999598210*(-2*x + 1)^(7/2) + 9762357024*(-2*x + 1)^(5/2) - 1411
1613390*(-2*x + 1)^(3/2) + 7644051695*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)

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Fricas [A]  time = 1.39306, size = 397, normalized size = 2.47 \begin{align*} \frac{734855 \, \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 (26460000 \, x^{5} + 126578745 \, x^{4} + 207486855 \, x^{3} + 157178184 \, x^{2} + 56745266 \, x + 7933096\right )} \sqrt{-2 \, x + 1}}{1666980 \,{\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)^3/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/1666980*(734855*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*(26460000*x^5 + 126578745*x^4 + 207486855*x^3 + 157178184*x^2 + 56745266*x + 793309
6)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**3/(2+3*x)**6,x)

[Out]

Timed out

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Giac [A]  time = 2.26974, size = 169, normalized size = 1.05 \begin{align*} \frac{146971}{333396} \, \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{1000}{729} \, \sqrt{-2 \, x + 1} + \frac{345408705 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 2999598210 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 9762357024 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 14111613390 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 7644051695 \, \sqrt{-2 \, x + 1}}{11430720 \,{\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)^3/(2+3*x)^6,x, algorithm="giac")

[Out]

146971/333396*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1000/729*s
qrt(-2*x + 1) + 1/11430720*(345408705*(2*x - 1)^4*sqrt(-2*x + 1) + 2999598210*(2*x - 1)^3*sqrt(-2*x + 1) + 976
2357024*(2*x - 1)^2*sqrt(-2*x + 1) - 14111613390*(-2*x + 1)^(3/2) + 7644051695*sqrt(-2*x + 1))/(3*x + 2)^5