3.1891 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=131 \[ \frac{\sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}+\frac{13 \sqrt{1-2 x} (5 x+3)^2}{56 (3 x+2)^2}-\frac{\sqrt{1-2 x} (26775 x+18187)}{1176 (3 x+2)}+\frac{13243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}} \]

[Out]

(13*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(56*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (Sqrt[1 - 2
*x]*(3 + 5*x)^3)/(2 + 3*x)^3 - (Sqrt[1 - 2*x]*(18187 + 26775*x))/(1176*(2 + 3*x)) + (13243*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(588*Sqrt[21])

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Rubi [A]  time = 0.0403063, antiderivative size = 131, 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, 146, 63, 206} \[ \frac{\sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}+\frac{13 \sqrt{1-2 x} (5 x+3)^2}{56 (3 x+2)^2}-\frac{\sqrt{1-2 x} (26775 x+18187)}{1176 (3 x+2)}+\frac{13243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(56*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (Sqrt[1 - 2
*x]*(3 + 5*x)^3)/(2 + 3*x)^3 - (Sqrt[1 - 2*x]*(18187 + 26775*x))/(1176*(2 + 3*x)) + (13243*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(588*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 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)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{(6-45 x) \sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac{1}{108} \int \frac{(-189-810 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{13 \sqrt{1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac{\int \frac{(-14013-61965 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{4536}\\ &=\frac{13 \sqrt{1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac{\sqrt{1-2 x} (18187+26775 x)}{1176 (2+3 x)}-\frac{13243 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1176}\\ &=\frac{13 \sqrt{1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac{\sqrt{1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac{13243 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1176}\\ &=\frac{13 \sqrt{1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac{\sqrt{1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac{13243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0604671, size = 84, normalized size = 0.64 \[ \frac{21 \left (392000 x^5+1127278 x^4+915191 x^3+15285 x^2-252230 x-74810\right )+26486 \sqrt{21-42 x} (3 x+2)^4 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*(-74810 - 252230*x + 15285*x^2 + 915191*x^3 + 1127278*x^4 + 392000*x^5) + 26486*Sqrt[21 - 42*x]*(2 + 3*x)^
4*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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Maple [A]  time = 0.011, size = 75, normalized size = 0.6 \begin{align*} -{\frac{500}{243}\sqrt{1-2\,x}}-{\frac{4}{3\, \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{416917}{2352} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{406463}{336} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{1189171}{432} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2706781}{1296}\sqrt{1-2\,x}} \right ) }+{\frac{13243\,\sqrt{21}}{12348}{\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)^5,x)

[Out]

-500/243*(1-2*x)^(1/2)-4/3*(-416917/2352*(1-2*x)^(7/2)+406463/336*(1-2*x)^(5/2)-1189171/432*(1-2*x)^(3/2)+2706
781/1296*(1-2*x)^(1/2))/(-6*x-4)^4+13243/12348*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.59017, size = 161, normalized size = 1.23 \begin{align*} -\frac{13243}{24696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{500}{243} \, \sqrt{-2 \, x + 1} + \frac{11256759 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 76821507 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 174808137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 132632269 \, \sqrt{-2 \, x + 1}}{47628 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\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)^5,x, algorithm="maxima")

[Out]

-13243/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 500/243*sqrt(-2*x +
1) + 1/47628*(11256759*(-2*x + 1)^(7/2) - 76821507*(-2*x + 1)^(5/2) + 174808137*(-2*x + 1)^(3/2) - 132632269*s
qrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.38416, size = 320, normalized size = 2.44 \begin{align*} \frac{13243 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (196000 \, x^{4} + 661639 \, x^{3} + 788415 \, x^{2} + 401850 \, x + 74810\right )} \sqrt{-2 \, x + 1}}{24696 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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)^5,x, algorithm="fricas")

[Out]

1/24696*(13243*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x
+ 2)) - 21*(196000*x^4 + 661639*x^3 + 788415*x^2 + 401850*x + 74810)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x
^2 + 96*x + 16)

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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)**5,x)

[Out]

Timed out

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Giac [A]  time = 2.83585, size = 147, normalized size = 1.12 \begin{align*} -\frac{13243}{24696} \, \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{500}{243} \, \sqrt{-2 \, x + 1} - \frac{11256759 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 76821507 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 174808137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 132632269 \, \sqrt{-2 \, x + 1}}{762048 \,{\left (3 \, x + 2\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

-13243/24696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 500/243*sqr
t(-2*x + 1) - 1/762048*(11256759*(2*x - 1)^3*sqrt(-2*x + 1) + 76821507*(2*x - 1)^2*sqrt(-2*x + 1) - 174808137*
(-2*x + 1)^(3/2) + 132632269*sqrt(-2*x + 1))/(3*x + 2)^4