3.2190 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^4}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{3269 (3 x+2)^3}{199650 \sqrt{1-2 x} (5 x+3)}-\frac{256172 (3 x+2)^2}{366025 \sqrt{1-2 x}}-\frac{21 \sqrt{1-2 x} (736875 x+2211616)}{3660250}-\frac{6937 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1830125 \sqrt{55}} \]

[Out]

(-256172*(2 + 3*x)^2)/(366025*Sqrt[1 - 2*x]) - (73*(2 + 3*x)^4)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 + 3*x
)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (3269*(2 + 3*x)^3)/(199650*Sqrt[1 - 2*x]*(3 + 5*x)) - (21*Sqrt[1 - 2*x
]*(2211616 + 736875*x))/3660250 - (6937*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1830125*Sqrt[55])

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Rubi [A]  time = 0.0515789, antiderivative size = 147, 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, 150, 147, 63, 206} \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^4}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{3269 (3 x+2)^3}{199650 \sqrt{1-2 x} (5 x+3)}-\frac{256172 (3 x+2)^2}{366025 \sqrt{1-2 x}}-\frac{21 \sqrt{1-2 x} (736875 x+2211616)}{3660250}-\frac{6937 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1830125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256172*(2 + 3*x)^2)/(366025*Sqrt[1 - 2*x]) - (73*(2 + 3*x)^4)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 + 3*x
)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (3269*(2 + 3*x)^3)/(199650*Sqrt[1 - 2*x]*(3 + 5*x)) - (21*Sqrt[1 - 2*x
]*(2211616 + 736875*x))/3660250 - (6937*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1830125*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 150

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] && IntegersQ[2*m, 2*n, 2*p]

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}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1}{33} \int \frac{(2+3 x)^4 (169+306 x)}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=-\frac{73 (2+3 x)^4}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{\int \frac{(2+3 x)^3 (11858+20853 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630}\\ &=-\frac{73 (2+3 x)^4}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{3269 (2+3 x)^3}{199650 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{(2+3 x)^2 (409731+717570 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{199650}\\ &=-\frac{256172 (2+3 x)^2}{366025 \sqrt{1-2 x}}-\frac{73 (2+3 x)^4}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{3269 (2+3 x)^3}{199650 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{(-27874686-46423125 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{2196150}\\ &=-\frac{256172 (2+3 x)^2}{366025 \sqrt{1-2 x}}-\frac{73 (2+3 x)^4}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{3269 (2+3 x)^3}{199650 \sqrt{1-2 x} (3+5 x)}-\frac{21 \sqrt{1-2 x} (2211616+736875 x)}{3660250}+\frac{6937 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{3660250}\\ &=-\frac{256172 (2+3 x)^2}{366025 \sqrt{1-2 x}}-\frac{73 (2+3 x)^4}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{3269 (2+3 x)^3}{199650 \sqrt{1-2 x} (3+5 x)}-\frac{21 \sqrt{1-2 x} (2211616+736875 x)}{3660250}-\frac{6937 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3660250}\\ &=-\frac{256172 (2+3 x)^2}{366025 \sqrt{1-2 x}}-\frac{73 (2+3 x)^4}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{3269 (2+3 x)^3}{199650 \sqrt{1-2 x} (3+5 x)}-\frac{21 \sqrt{1-2 x} (2211616+736875 x)}{3660250}-\frac{6937 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1830125 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0624443, size = 105, normalized size = 0.71 \[ -\frac{-1204 (5 x+3)^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+2142 (2 x-1) (5 x+3)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )+33 \left (7350750 x^5+79388100 x^4-89679150 x^3-130986110 x^2+3498263 x+20166158\right )}{4991250 (1-2 x)^{3/2} (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(33*(20166158 + 3498263*x - 130986110*x^2 - 89679150*x^3 + 79388100*x^4 + 7350750*x^5) - 1204*(3 + 5*x)^2*Hyp
ergeometric2F1[-3/2, 1, -1/2, (5*(1 - 2*x))/11] + 2142*(-1 + 2*x)*(3 + 5*x)^2*Hypergeometric2F1[-1/2, 1, 1/2,
(5*(1 - 2*x))/11])/(4991250*(1 - 2*x)^(3/2)*(3 + 5*x)^2)

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Maple [A]  time = 0.013, size = 84, normalized size = 0.6 \begin{align*}{\frac{243}{1000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{26973}{5000}\sqrt{1-2\,x}}+{\frac{117649}{31944} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{1563051}{117128}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{366025\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{407}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{4499}{50}\sqrt{1-2\,x}} \right ) }-{\frac{6937\,\sqrt{55}}{100656875}{\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/(1-2*x)^(5/2)/(3+5*x)^3,x)

[Out]

243/1000*(1-2*x)^(3/2)-26973/5000*(1-2*x)^(1/2)+117649/31944/(1-2*x)^(3/2)-1563051/117128/(1-2*x)^(1/2)+2/3660
25*(407/10*(1-2*x)^(3/2)-4499/50*(1-2*x)^(1/2))/(-10*x-6)^2-6937/100656875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2)
)*55^(1/2)

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Maxima [A]  time = 1.77366, size = 149, normalized size = 1.01 \begin{align*} \frac{243}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6937}{201313750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{26973}{5000} \, \sqrt{-2 \, x + 1} + \frac{73267966785 \,{\left (2 \, x - 1\right )}^{3} + 342600082649 \,{\left (2 \, x - 1\right )}^{2} + 887178503750 \, x - 345719990000}{219615000 \,{\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)^6/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="maxima")

[Out]

243/1000*(-2*x + 1)^(3/2) + 6937/201313750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 26973/5000*sqrt(-2*x + 1) + 1/219615000*(73267966785*(2*x - 1)^3 + 342600082649*(2*x - 1)^2 + 887178
503750*x - 345719990000)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.16075, size = 366, normalized size = 2.49 \begin{align*} \frac{20811 \, \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 (533664450 \, x^{5} + 5763576060 \, x^{4} - 6510290070 \, x^{3} - 9509366452 \, x^{2} + 253794537 \, x + 1463964312\right )} \sqrt{-2 \, x + 1}}{603941250 \,{\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)^6/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/603941250*(20811*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*(533664450*x^5 + 5763576060*x^4 - 6510290070*x^3 - 9509366452*x^2 + 253794537*x + 1463964312)*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)**6/(1-2*x)**(5/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.55033, size = 144, normalized size = 0.98 \begin{align*} \frac{243}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6937}{201313750} \, \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{26973}{5000} \, \sqrt{-2 \, x + 1} - \frac{16807 \,{\left (279 \, x - 101\right )}}{175692 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{185 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 409 \, \sqrt{-2 \, x + 1}}{3327500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/1000*(-2*x + 1)^(3/2) + 6937/201313750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5
*sqrt(-2*x + 1))) - 26973/5000*sqrt(-2*x + 1) - 16807/175692*(279*x - 101)/((2*x - 1)*sqrt(-2*x + 1)) + 1/3327
500*(185*(-2*x + 1)^(3/2) - 409*sqrt(-2*x + 1))/(5*x + 3)^2