3.1964 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=154 \[ -\frac{55 \sqrt{1-2 x} (5 x+3)^3}{24 (3 x+2)^2}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{54 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}-\frac{2255 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)}+\frac{275 \sqrt{1-2 x} (4595 x+1123)}{13608}+\frac{645865 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6804 \sqrt{21}} \]

[Out]

(-2255*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (55*(1 -
2*x)^(3/2)*(3 + 5*x)^3)/(54*(2 + 3*x)^3) - (55*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(24*(2 + 3*x)^2) + (275*Sqrt[1 - 2*x
]*(1123 + 4595*x))/13608 + (645865*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6804*Sqrt[21])

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Rubi [A]  time = 0.06041, antiderivative size = 154, 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, 147, 63, 206} \[ -\frac{55 \sqrt{1-2 x} (5 x+3)^3}{24 (3 x+2)^2}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{54 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}-\frac{2255 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)}+\frac{275 \sqrt{1-2 x} (4595 x+1123)}{13608}+\frac{645865 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6804 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2255*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (55*(1 -
2*x)^(3/2)*(3 + 5*x)^3)/(54*(2 + 3*x)^3) - (55*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(24*(2 + 3*x)^2) + (275*Sqrt[1 - 2*x
]*(1123 + 4595*x))/13608 + (645865*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6804*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 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(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^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && 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)^5} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{1}{12} \int -\frac{55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}-\frac{55}{12} \int \frac{(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}+\frac{55}{108} \int \frac{\sqrt{1-2 x} (3+5 x)^2 (15+36 x)}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}-\frac{55 \sqrt{1-2 x} (3+5 x)^3}{24 (2+3 x)^2}-\frac{55}{648} \int \frac{(3+5 x)^2 (-126+549 x)}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{2255 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}-\frac{55 \sqrt{1-2 x} (3+5 x)^3}{24 (2+3 x)^2}-\frac{55 \int \frac{(3+5 x) (-7659+41355 x)}{\sqrt{1-2 x} (2+3 x)} \, dx}{13608}\\ &=-\frac{2255 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}-\frac{55 \sqrt{1-2 x} (3+5 x)^3}{24 (2+3 x)^2}+\frac{275 \sqrt{1-2 x} (1123+4595 x)}{13608}-\frac{645865 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{13608}\\ &=-\frac{2255 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}-\frac{55 \sqrt{1-2 x} (3+5 x)^3}{24 (2+3 x)^2}+\frac{275 \sqrt{1-2 x} (1123+4595 x)}{13608}+\frac{645865 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{13608}\\ &=-\frac{2255 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}-\frac{55 \sqrt{1-2 x} (3+5 x)^3}{24 (2+3 x)^2}+\frac{275 \sqrt{1-2 x} (1123+4595 x)}{13608}+\frac{645865 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6804 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.021819, size = 59, normalized size = 0.38 \[ \frac{(1-2 x)^{7/2} \left (1033384 (3 x+2)^4 \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )-2401 \left (73500 x^2+98419 x+32939\right )\right )}{12706092 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(7/2)*(-2401*(32939 + 98419*x + 73500*x^2) + 1033384*(2 + 3*x)^4*Hypergeometric2F1[3, 7/2, 9/2, 3/7
- (6*x)/7]))/(12706092*(2 + 3*x)^4)

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Maple [A]  time = 0.011, size = 84, normalized size = 0.6 \begin{align*} -{\frac{500}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{7600}{729}\sqrt{1-2\,x}}-{\frac{4}{9\, \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{159975}{112} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{4220087}{432} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{28870415}{1296} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{21951755}{1296}\sqrt{1-2\,x}} \right ) }+{\frac{645865\,\sqrt{21}}{142884}{\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)^5,x)

[Out]

-500/729*(1-2*x)^(3/2)-7600/729*(1-2*x)^(1/2)-4/9*(-159975/112*(1-2*x)^(7/2)+4220087/432*(1-2*x)^(5/2)-2887041
5/1296*(1-2*x)^(3/2)[phone]/1296*(1-2*x)^(1/2))/(-6*x-4)^4+645865/142884*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
*21^(1/2)

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Maxima [A]  time = 2.7103, size = 173, normalized size = 1.12 \begin{align*} -\frac{500}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{645865}{285768} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7600}{729} \, \sqrt{-2 \, x + 1} + \frac{12957975 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 88621827 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 202092905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 153662285 \, \sqrt{-2 \, x + 1}}{20412 \,{\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)^(5/2)*(3+5*x)^3/(2+3*x)^5,x, algorithm="maxima")

[Out]

-500/729*(-2*x + 1)^(3/2) - 645865/285768*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 7600/729*sqrt(-2*x + 1) + 1/20412*(12957975*(-2*x + 1)^(7/2) - 88621827*(-2*x + 1)^(5/2) + 202092905*
(-2*x + 1)^(3/2) - 153662285*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1
715)

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Fricas [A]  time = 1.30321, size = 354, normalized size = 2.3 \begin{align*} \frac{645865 \, \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 (1512000 \, x^{5} - 8215200 \, x^{4} - 32946525 \, x^{3} - 39158517 \, x^{2} - 19526798 \, x - 3553918\right )} \sqrt{-2 \, x + 1}}{285768 \,{\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)^(5/2)*(3+5*x)^3/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/285768*(645865*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*(1512000*x^5 - 8215200*x^4 - 32946525*x^3 - 39158517*x^2 - 19526798*x - 3553918)*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)**(5/2)*(3+5*x)**3/(2+3*x)**5,x)

[Out]

Timed out

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Giac [A]  time = 2.53796, size = 159, normalized size = 1.03 \begin{align*} -\frac{500}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{645865}{285768} \, \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{7600}{729} \, \sqrt{-2 \, x + 1} - \frac{12957975 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 88621827 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 202092905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 153662285 \, \sqrt{-2 \, x + 1}}{326592 \,{\left (3 \, x + 2\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

-500/729*(-2*x + 1)^(3/2) - 645865/285768*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - 7600/729*sqrt(-2*x + 1) - 1/326592*(12957975*(2*x - 1)^3*sqrt(-2*x + 1) + 88621827*(2*x - 1)
^2*sqrt(-2*x + 1) - 202092905*(-2*x + 1)^(3/2) + 153662285*sqrt(-2*x + 1))/(3*x + 2)^4