3.2192 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^2}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{2133933 x+1287116}{2196150 \sqrt{1-2 x} (5 x+3)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}} \]

[Out]

(-73*(2 + 3*x)^2)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (12871
16 + 2133933*x)/(2196150*Sqrt[1 - 2*x]*(3 + 5*x)) - (14423*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(366025*Sqrt[55]
)

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Rubi [A]  time = 0.0305056, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 149, 144, 63, 206} \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^2}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{2133933 x+1287116}{2196150 \sqrt{1-2 x} (5 x+3)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^(5/2)*(3 + 5*x)^3),x]

[Out]

(-73*(2 + 3*x)^2)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (12871
16 + 2133933*x)/(2196150*Sqrt[1 - 2*x]*(3 + 5*x)) - (14423*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(366025*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 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + 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
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -1]

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)^4}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1}{33} \int \frac{(2+3 x)^2 (43+96 x)}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=-\frac{73 (2+3 x)^2}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{\int \frac{(2+3 x) (3056+6117 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630}\\ &=-\frac{73 (2+3 x)^2}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1287116+2133933 x}{2196150 \sqrt{1-2 x} (3+5 x)}+\frac{14423 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{732050}\\ &=-\frac{73 (2+3 x)^2}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1287116+2133933 x}{2196150 \sqrt{1-2 x} (3+5 x)}-\frac{14423 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{732050}\\ &=-\frac{73 (2+3 x)^2}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^3}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1287116+2133933 x}{2196150 \sqrt{1-2 x} (3+5 x)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0463362, size = 95, normalized size = 0.89 \[ -\frac{-4634 (5 x+3)^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+1548 (2 x-1) (5 x+3)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-33 \left (490050 x^3+567140 x^2+151043 x-7696\right )}{998250 (1-2 x)^{3/2} (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^(5/2)*(3 + 5*x)^3),x]

[Out]

-(-33*(-7696 + 151043*x + 567140*x^2 + 490050*x^3) - 4634*(3 + 5*x)^2*Hypergeometric2F1[-3/2, 1, -1/2, (5*(1 -
2*x))/11] + 1548*(-1 + 2*x)*(3 + 5*x)^2*Hypergeometric2F1[-1/2, 1, 1/2, (5*(1 - 2*x))/11])/(998250*(1 - 2*x)^
(3/2)*(3 + 5*x)^2)

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Maple [A]  time = 0.012, size = 66, normalized size = 0.6 \begin{align*}{\frac{2401}{7986} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{9261}{29282}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{100}{14641\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{11}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3047}{2500}\sqrt{1-2\,x}} \right ) }-{\frac{14423\,\sqrt{55}}{20131375}{\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)^4/(1-2*x)^(5/2)/(3+5*x)^3,x)

[Out]

2401/7986/(1-2*x)^(3/2)-9261/29282/(1-2*x)^(1/2)+100/14641*(11/20*(1-2*x)^(3/2)-3047/2500*(1-2*x)^(1/2))/(-10*
x-6)^2-14423/20131375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.2661, size = 124, normalized size = 1.16 \begin{align*} \frac{14423}{40262750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17356125 \,{\left (2 \, x - 1\right )}^{3} + 92891843 \,{\left (2 \, x - 1\right )}^{2} + 313347650 \, x - 76780550}{2196150 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="maxima")

[Out]

14423/40262750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1/2196150*(1735612
5*(2*x - 1)^3 + 92891843*(2*x - 1)^2 + 313347650*x - 76780550)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 1
21*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.09393, size = 309, normalized size = 2.89 \begin{align*} \frac{43269 \, \sqrt{55}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (34712250 \, x^{3} + 40823468 \, x^{2} + 11479257 \, x - 311208\right )} \sqrt{-2 \, x + 1}}{120788250 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/120788250*(43269*sqrt(55)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) + 55*(34712250*x^3 + 40823468*x^2 + 11479257*x - 311208)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6
*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)**4/(1-2*x)**(5/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.57501, size = 120, normalized size = 1.12 \begin{align*} \frac{14423}{40262750} \, \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{343 \,{\left (81 \, x - 2\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{125 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 277 \, \sqrt{-2 \, x + 1}}{133100 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="giac")

[Out]

14423/40262750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 343/4392
3*(81*x - 2)/((2*x - 1)*sqrt(-2*x + 1)) + 1/133100*(125*(-2*x + 1)^(3/2) - 277*sqrt(-2*x + 1))/(5*x + 3)^2