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

Optimal. Leaf size=140 \[ -\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}-\frac {117 \sqrt {1-2 x} (3 x+2)^4}{3025 (5 x+3)}-\frac {927 \sqrt {1-2 x} (3 x+2)^3}{211750}-\frac {56556 \sqrt {1-2 x} (3 x+2)^2}{378125}-\frac {9 \sqrt {1-2 x} (934875 x+2815648)}{3781250}-\frac {33069 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1890625 \sqrt {55}} \]

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Rubi [A]  time = 0.05, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 153, 147, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (3 x+2)^5}{110 (5 x+3)^2}-\frac {117 \sqrt {1-2 x} (3 x+2)^4}{3025 (5 x+3)}-\frac {927 \sqrt {1-2 x} (3 x+2)^3}{211750}-\frac {56556 \sqrt {1-2 x} (3 x+2)^2}{378125}-\frac {9 \sqrt {1-2 x} (934875 x+2815648)}{3781250}-\frac {33069 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1890625 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-56556*Sqrt[1 - 2*x]*(2 + 3*x)^2)/378125 - (927*Sqrt[1 - 2*x]*(2 + 3*x)^3)/211750 - (Sqrt[1 - 2*x]*(2 + 3*x)^
5)/(110*(3 + 5*x)^2) - (117*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(3025*(3 + 5*x)) - (9*Sqrt[1 - 2*x]*(2815648 + 934875*x
))/3781250 - (33069*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1890625*Sqrt[55])

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 98

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

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 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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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 {(2+3 x)^6}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {1}{110} \int \frac {(-153-177 x) (2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}-\frac {\int \frac {(-7170-927 x) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx}{6050}\\ &=-\frac {927 \sqrt {1-2 x} (2+3 x)^3}{211750}-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}+\frac {\int \frac {(2+3 x)^2 (521367+791784 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{211750}\\ &=-\frac {56556 \sqrt {1-2 x} (2+3 x)^2}{378125}-\frac {927 \sqrt {1-2 x} (2+3 x)^3}{211750}-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}-\frac {\int \frac {(-35569758-58897125 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{5293750}\\ &=-\frac {56556 \sqrt {1-2 x} (2+3 x)^2}{378125}-\frac {927 \sqrt {1-2 x} (2+3 x)^3}{211750}-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}-\frac {9 \sqrt {1-2 x} (2815648+934875 x)}{3781250}+\frac {33069 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3781250}\\ &=-\frac {56556 \sqrt {1-2 x} (2+3 x)^2}{378125}-\frac {927 \sqrt {1-2 x} (2+3 x)^3}{211750}-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}-\frac {9 \sqrt {1-2 x} (2815648+934875 x)}{3781250}-\frac {33069 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3781250}\\ &=-\frac {56556 \sqrt {1-2 x} (2+3 x)^2}{378125}-\frac {927 \sqrt {1-2 x} (2+3 x)^3}{211750}-\frac {\sqrt {1-2 x} (2+3 x)^5}{110 (3+5 x)^2}-\frac {117 \sqrt {1-2 x} (2+3 x)^4}{3025 (3+5 x)}-\frac {9 \sqrt {1-2 x} (2815648+934875 x)}{3781250}-\frac {33069 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1890625 \sqrt {55}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 73, normalized size = 0.52 \begin {gather*} \frac {-\frac {55 \sqrt {1-2 x} \left (551306250 x^5+2690374500 x^4+6078090150 x^3+9876010320 x^2+7254126105 x+1804176536\right )}{(5 x+3)^2}-462966 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1455781250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*Sqrt[1 - 2*x]*(1804176536 + 7254126105*x + 9876010320*x^2 + 6078090150*x^3 + 2690374500*x^4 + 551306250*
x^5))/(3 + 5*x)^2 - 462966*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1455781250

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IntegrateAlgebraic [A]  time = 0.19, size = 97, normalized size = 0.69 \begin {gather*} \frac {\left (275653125 (1-2 x)^5-4068640125 (1-2 x)^4+25674209550 (1-2 x)^3-94871360430 (1-2 x)^2+185649395925 (1-2 x)-141526082621\right ) \sqrt {1-2 x}}{105875000 (5 (1-2 x)-11)^2}-\frac {33069 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1890625 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-141526082621 + 185649395925*(1 - 2*x) - 94871360430*(1 - 2*x)^2 + 25674209550*(1 - 2*x)^3 - 4068640125*(1 -
2*x)^4 + 275653125*(1 - 2*x)^5)*Sqrt[1 - 2*x])/(105875000*(-11 + 5*(1 - 2*x))^2) - (33069*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(1890625*Sqrt[55])

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fricas [A]  time = 1.38, size = 89, normalized size = 0.64 \begin {gather*} \frac {231483 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (551306250 \, x^{5} + 2690374500 \, x^{4} + 6078090150 \, x^{3} + 9876010320 \, x^{2} + 7254126105 \, x + 1804176536\right )} \sqrt {-2 \, x + 1}}{1455781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1455781250*(231483*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(551
306250*x^5 + 2690374500*x^4 + 6078090150*x^3 + 9876010320*x^2 + 7254126105*x + 1804176536)*sqrt(-2*x + 1))/(25
*x^2 + 30*x + 9)

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giac [A]  time = 1.34, size = 118, normalized size = 0.84 \begin {gather*} -\frac {729}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {26973}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {111213}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33069}{207968750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {276183}{25000} \, \sqrt {-2 \, x + 1} + \frac {1995 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4411 \, \sqrt {-2 \, x + 1}}{7562500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/7000*(2*x - 1)^3*sqrt(-2*x + 1) - 26973/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 111213/25000*(-2*x + 1)^(3/2)
+ 33069/207968750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27618
3/25000*sqrt(-2*x + 1) + 1/7562500*(1995*(-2*x + 1)^(3/2) - 4411*sqrt(-2*x + 1))/(5*x + 3)^2

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maple [A]  time = 0.01, size = 84, normalized size = 0.60 \begin {gather*} -\frac {33069 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{103984375}+\frac {729 \left (-2 x +1\right )^{\frac {7}{2}}}{7000}-\frac {26973 \left (-2 x +1\right )^{\frac {5}{2}}}{25000}+\frac {111213 \left (-2 x +1\right )^{\frac {3}{2}}}{25000}-\frac {276183 \sqrt {-2 x +1}}{25000}+\frac {\frac {399 \left (-2 x +1\right )^{\frac {3}{2}}}{378125}-\frac {401 \sqrt {-2 x +1}}{171875}}{\left (-10 x -6\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^6/(5*x+3)^3/(-2*x+1)^(1/2),x)

[Out]

729/7000*(-2*x+1)^(7/2)-26973/25000*(-2*x+1)^(5/2)+111213/25000*(-2*x+1)^(3/2)-276183/25000*(-2*x+1)^(1/2)+2/1
25*(399/6050*(-2*x+1)^(3/2)-401/2750*(-2*x+1)^(1/2))/(-10*x-6)^2-33069/103984375*arctanh(1/11*55^(1/2)*(-2*x+1
)^(1/2))*55^(1/2)

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maxima [A]  time = 1.20, size = 110, normalized size = 0.79 \begin {gather*} \frac {729}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {26973}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {111213}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33069}{207968750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {276183}{25000} \, \sqrt {-2 \, x + 1} + \frac {1995 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4411 \, \sqrt {-2 \, x + 1}}{1890625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/7000*(-2*x + 1)^(7/2) - 26973/25000*(-2*x + 1)^(5/2) + 111213/25000*(-2*x + 1)^(3/2) + 33069/207968750*sqr
t(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 276183/25000*sqrt(-2*x + 1) + 1/1890
625*(1995*(-2*x + 1)^(3/2) - 4411*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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mupad [B]  time = 0.06, size = 92, normalized size = 0.66 \begin {gather*} \frac {111213\,{\left (1-2\,x\right )}^{3/2}}{25000}-\frac {276183\,\sqrt {1-2\,x}}{25000}-\frac {26973\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {729\,{\left (1-2\,x\right )}^{7/2}}{7000}-\frac {\frac {401\,\sqrt {1-2\,x}}{4296875}-\frac {399\,{\left (1-2\,x\right )}^{3/2}}{9453125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,33069{}\mathrm {i}}{103984375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^6/((1 - 2*x)^(1/2)*(5*x + 3)^3),x)

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*33069i)/103984375 - (276183*(1 - 2*x)^(1/2))/25000 + (111213*
(1 - 2*x)^(3/2))/25000 - (26973*(1 - 2*x)^(5/2))/25000 + (729*(1 - 2*x)^(7/2))/7000 - ((401*(1 - 2*x)^(1/2))/4
296875 - (399*(1 - 2*x)^(3/2))/9453125)/((44*x)/5 + (2*x - 1)^2 + 11/25)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Timed out

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