3.2058 \(\int \frac{(2+3 x)^6}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

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

[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])

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Rubi [A]  time = 0.0490658, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{110 (5 x+3)^2}-\frac{117 \sqrt{1-2 x} (3 x+2)^4}{3025 (5 x+3)}-\frac{927 \sqrt{1-2 x} (3 x+2)^3}{211750}-\frac{56556 \sqrt{1-2 x} (3 x+2)^2}{378125}-\frac{9 \sqrt{1-2 x} (934875 x+2815648)}{3781250}-\frac{33069 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1890625 \sqrt{55}} \]

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 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 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 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{(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*}

Mathematica [A]  time = 0.0730523, size = 73, normalized size = 0.52 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (551306250 x^5+2690374500 x^4+6078090150 x^3+9876010320 x^2+7254126105 x+1804176536\right )}{(5 x+3)^2}-462966 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1455781250} \]

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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Maple [A]  time = 0.01, size = 84, normalized size = 0.6 \begin{align*}{\frac{729}{7000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{26973}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{111213}{25000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{276183}{25000}\sqrt{1-2\,x}}+{\frac{2}{125\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{399}{6050} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{401}{2750}\sqrt{1-2\,x}} \right ) }-{\frac{33069\,\sqrt{55}}{103984375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/7000*(1-2*x)^(7/2)-26973/25000*(1-2*x)^(5/2)+111213/25000*(1-2*x)^(3/2)-276183/25000*(1-2*x)^(1/2)+2/125*(
399/6050*(1-2*x)^(3/2)-401/2750*(1-2*x)^(1/2))/(-10*x-6)^2-33069/103984375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2)
)*55^(1/2)

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Maxima [A]  time = 2.62211, size = 149, normalized size = 1.06 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.71181, size = 321, normalized size = 2.29 \begin{align*} \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{align*}

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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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [A]  time = 2.05186, size = 159, normalized size = 1.14 \begin{align*} -\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{align*}

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