3.1966 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=162 \[ \frac{11 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^5}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}+\frac{559625 \sqrt{1-2 x}}{1333584 (3 x+2)}-\frac{559625 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{33275 (1-2 x)^{3/2}}{95256 (3 x+2)^3}-\frac{121 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{559625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792 \sqrt{21}} \]

[Out]

(-121*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) + (33275*(1 - 2*x)^(3/2))/(95256*(2 + 3*x)^3) - (559625*Sqrt[1 - 2*x
])/(190512*(2 + 3*x)^2) + (559625*Sqrt[1 - 2*x])/(1333584*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(18*(2 +
3*x)^6) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(27*(2 + 3*x)^5) + (559625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(66679
2*Sqrt[21])

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Rubi [A]  time = 0.0676559, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {97, 12, 149, 89, 78, 47, 51, 63, 206} \[ \frac{11 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^5}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}+\frac{559625 \sqrt{1-2 x}}{1333584 (3 x+2)}-\frac{559625 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{33275 (1-2 x)^{3/2}}{95256 (3 x+2)^3}-\frac{121 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac{559625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

(-121*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) + (33275*(1 - 2*x)^(3/2))/(95256*(2 + 3*x)^3) - (559625*Sqrt[1 - 2*x
])/(190512*(2 + 3*x)^2) + (559625*Sqrt[1 - 2*x])/(1333584*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(18*(2 +
3*x)^6) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(27*(2 + 3*x)^5) + (559625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(66679
2*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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^3}{(2+3 x)^7} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{1}{18} \int -\frac{55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}-\frac{55}{18} \int \frac{(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac{11}{54} \int \frac{33 \sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac{121}{18} \int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac{121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac{121 \int \frac{\sqrt{1-2 x} (1125+2100 x)}{(2+3 x)^4} \, dx}{4536}\\ &=-\frac{121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac{559625 \int \frac{\sqrt{1-2 x}}{(2+3 x)^3} \, dx}{31752}\\ &=-\frac{121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac{559625 \sqrt{1-2 x}}{190512 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}-\frac{559625 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{190512}\\ &=-\frac{121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac{559625 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{559625 \sqrt{1-2 x}}{1333584 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}-\frac{559625 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1333584}\\ &=-\frac{121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac{559625 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{559625 \sqrt{1-2 x}}{1333584 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac{559625 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1333584}\\ &=-\frac{121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac{33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac{559625 \sqrt{1-2 x}}{190512 (2+3 x)^2}+\frac{559625 \sqrt{1-2 x}}{1333584 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac{559625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0306291, size = 52, normalized size = 0.32 \[ \frac{(1-2 x)^{7/2} \left (\frac{1764735 \left (110250 x^2+146875 x+48919\right )}{(3 x+2)^6}-161172000 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{4669488810} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

((1 - 2*x)^(7/2)*((1764735*(48919 + 146875*x + 110250*x^2))/(2 + 3*x)^6 - 161172000*Hypergeometric2F1[7/2, 5,
9/2, 3/7 - (6*x)/7]))/4669488810

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Maple [A]  time = 0.011, size = 84, normalized size = 0.5 \begin{align*} -11664\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ( -{\frac{3809125\, \left ( 1-2\,x \right ) ^{11/2}}{96018048}}+{\frac{47350325\, \left ( 1-2\,x \right ) ^{9/2}}{123451776}}-{\frac{4383467\, \left ( 1-2\,x \right ) ^{7/2}}{2939328}}+{\frac{1231175\, \left ( 1-2\,x \right ) ^{5/2}}{419904}}-{\frac{66595375\, \left ( 1-2\,x \right ) ^{3/2}}{22674816}}+{\frac{27421625\,\sqrt{1-2\,x}}{22674816}} \right ) }+{\frac{559625\,\sqrt{21}}{14002632}{\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)^7,x)

[Out]

-11664*(-3809125/96018048*(1-2*x)^(11/2)[phone]/123451776*(1-2*x)^(9/2)-4383467/2939328*(1-2*x)^(7/2)+123117
5/419904*(1-2*x)^(5/2)-66595375/22674816*(1-2*x)^(3/2)[phone]/22674816*(1-2*x)^(1/2))/(-6*x-4)^6+559625/1400
2632*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.66017, size = 197, normalized size = 1.22 \begin{align*} -\frac{559625}{28005264} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{308539125 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2983070475 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 11598653682 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 22803823350 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 22842213625 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 9405617375 \, \sqrt{-2 \, x + 1}}{666792 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\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)^7,x, algorithm="maxima")

[Out]

-559625/28005264*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/666792*(308539
125*(-2*x + 1)^(11/2) - 2983070475*(-2*x + 1)^(9/2) + 11598653682*(-2*x + 1)^(7/2) - 22803823350*(-2*x + 1)^(5
/2) + 22842213625*(-2*x + 1)^(3/2) - 9405617375*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(
2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]  time = 1.40188, size = 437, normalized size = 2.7 \begin{align*} \frac{559625 \, \sqrt{21}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (308539125 \, x^{5} + 720187425 \, x^{4} + 687940758 \, x^{3} + 352611738 \, x^{2} + 102558856 \, x + 13847024\right )} \sqrt{-2 \, x + 1}}{28005264 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="fricas")

[Out]

1/28005264*(559625*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((3*x - sqrt
(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(308539125*x^5 + 720187425*x^4 + 687940758*x^3 + 352611738*x^2 + 1025
58856*x + 13847024)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**7,x)

[Out]

Timed out

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Giac [A]  time = 2.49629, size = 178, normalized size = 1.1 \begin{align*} -\frac{559625}{28005264} \, \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{308539125 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 2983070475 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 11598653682 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 22803823350 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 22842213625 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 9405617375 \, \sqrt{-2 \, x + 1}}{42674688 \,{\left (3 \, x + 2\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

-559625/28005264*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/42674
688*(308539125*(2*x - 1)^5*sqrt(-2*x + 1) + 2983070475*(2*x - 1)^4*sqrt(-2*x + 1) + 11598653682*(2*x - 1)^3*sq
rt(-2*x + 1) + 22803823350*(2*x - 1)^2*sqrt(-2*x + 1) - 22842213625*(-2*x + 1)^(3/2) + 9405617375*sqrt(-2*x +
1))/(3*x + 2)^6