3.1892 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=134 \[ \frac{3 \sqrt{1-2 x} (5 x+3)^3}{5 (3 x+2)^4}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac{67 \sqrt{1-2 x} (5 x+3)^2}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x} (15074 x+9529)}{9261 (3 x+2)^2}-\frac{13892 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}} \]

[Out]

(-67*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(315*(2 + 3*x)^3) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(15*(2 + 3*x)^5) + (3*Sqrt[1
- 2*x]*(3 + 5*x)^3)/(5*(2 + 3*x)^4) - (2*Sqrt[1 - 2*x]*(9529 + 15074*x))/(9261*(2 + 3*x)^2) - (13892*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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Rubi [A]  time = 0.042094, antiderivative size = 134, normalized size of antiderivative = 1., 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 = {97, 149, 145, 63, 206} \[ \frac{3 \sqrt{1-2 x} (5 x+3)^3}{5 (3 x+2)^4}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac{67 \sqrt{1-2 x} (5 x+3)^2}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x} (15074 x+9529)}{9261 (3 x+2)^2}-\frac{13892 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-67*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(315*(2 + 3*x)^3) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(15*(2 + 3*x)^5) + (3*Sqrt[1
- 2*x]*(3 + 5*x)^3)/(5*(2 + 3*x)^4) - (2*Sqrt[1 - 2*x]*(9529 + 15074*x))/(9261*(2 + 3*x)^2) - (13892*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*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 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 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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)^3}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{(6-45 x) \sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{3 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac{1}{180} \int \frac{(3+5 x)^2 (-684+180 x)}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{67 \sqrt{1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{3 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac{\int \frac{(3+5 x) (-48960+6840 x)}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{11340}\\ &=-\frac{67 \sqrt{1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{3 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac{2 \sqrt{1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}+\frac{6946 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{9261}\\ &=-\frac{67 \sqrt{1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{3 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac{2 \sqrt{1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac{6946 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{9261}\\ &=-\frac{67 \sqrt{1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac{3 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac{2 \sqrt{1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac{13892 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0324204, size = 52, normalized size = 0.39 \[ \frac{(1-2 x)^{5/2} \left (\frac{86436 \left (30625 x^2+40790 x+13583\right )}{(3 x+2)^5}-8001792 \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{63530460} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(5/2)*((86436*(13583 + 40790*x + 30625*x^2))/(2 + 3*x)^5 - 8001792*Hypergeometric2F1[5/2, 4, 7/2, 3
/7 - (6*x)/7]))/63530460

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Maple [A]  time = 0.011, size = 75, normalized size = 0.6 \begin{align*} 1944\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{54493\, \left ( 1-2\,x \right ) ^{9/2}}{500094}}+{\frac{4577\, \left ( 1-2\,x \right ) ^{7/2}}{5103}}-{\frac{210293\, \left ( 1-2\,x \right ) ^{5/2}}{76545}}+{\frac{24311\, \left ( 1-2\,x \right ) ^{3/2}}{6561}}-{\frac{24311\,\sqrt{1-2\,x}}{13122}} \right ) }-{\frac{13892\,\sqrt{21}}{194481}{\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)^3/(2+3*x)^6,x)

[Out]

1944*(-54493/500094*(1-2*x)^(9/2)+4577/5103*(1-2*x)^(7/2)-210293/76545*(1-2*x)^(5/2)+24311/6561*(1-2*x)^(3/2)-
24311/13122*(1-2*x)^(1/2))/(-6*x-4)^5-13892/194481*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 4.37924, size = 173, normalized size = 1.29 \begin{align*} \frac{6946}{194481} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (2452185 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 20184570 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 61826142 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 83386730 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 41693365 \, \sqrt{-2 \, x + 1}\right )}}{46305 \,{\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)^(3/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="maxima")

[Out]

6946/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/46305*(2452185*(-2*
x + 1)^(9/2) - 20184570*(-2*x + 1)^(7/2) + 61826142*(-2*x + 1)^(5/2) - 83386730*(-2*x + 1)^(3/2) + 41693365*sq
rt(-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 = 1.48709, size = 365, normalized size = 2.72 \begin{align*} \frac{34730 \, \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 (4904370 \, x^{4} + 10375830 \, x^{3} + 7992771 \, x^{2} + 2619854 \, x + 300049\right )} \sqrt{-2 \, x + 1}}{972405 \,{\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)^(3/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/972405*(34730*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*(4904370*x^4 + 10375830*x^3 + 7992771*x^2 + 2619854*x + 300049)*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)**(3/2)*(3+5*x)**3/(2+3*x)**6,x)

[Out]

Timed out

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Giac [A]  time = 2.18657, size = 157, normalized size = 1.17 \begin{align*} \frac{6946}{194481} \, \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{2452185 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 20184570 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 61826142 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 83386730 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 41693365 \, \sqrt{-2 \, x + 1}}{370440 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="giac")

[Out]

6946/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/370440*(24
52185*(2*x - 1)^4*sqrt(-2*x + 1) + 20184570*(2*x - 1)^3*sqrt(-2*x + 1) + 61826142*(2*x - 1)^2*sqrt(-2*x + 1) -
83386730*(-2*x + 1)^(3/2) + 41693365*sqrt(-2*x + 1))/(3*x + 2)^5