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]

7/33*(2+3*x)^5/(1-2*x)^(3/2)/(3+5*x)^2-6937/100656875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-256172/366
025*(2+3*x)^2/(1-2*x)^(1/2)-73/3630*(2+3*x)^4/(3+5*x)^2/(1-2*x)^(1/2)-3269/199650*(2+3*x)^3/(3+5*x)/(1-2*x)^(1
/2)-21/3660250*(2211616+736875*x)*(1-2*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 147, 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, 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 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 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 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*}

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Mathematica [C]  time = 0.11, 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]

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

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fricas [A]  time = 0.95, size = 109, normalized size = 0.74 \[ \frac {20811 \, \sqrt {55} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (533664450 \, x^{5} + 5763576060 \, x^{4} - 6510290070 \, x^{3} - 9509366452 \, x^{2} + 253794537 \, x + 1463964312\right )} \sqrt {-2 \, x + 1}}{603941250 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

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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giac [A]  time = 1.40, size = 107, normalized size = 0.73 \[ \frac {243}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6937}{201313750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {26973}{5000} \, \sqrt {-2 \, x + 1} - \frac {16807 \, {\left (279 \, x - 101\right )}}{175692 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {185 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 409 \, \sqrt {-2 \, x + 1}}{3327500 \, {\left (5 \, x + 3\right )}^{2}} \]

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

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maple [A]  time = 0.02, size = 84, normalized size = 0.57 \[ -\frac {6937 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{100656875}+\frac {243 \left (-2 x +1\right )^{\frac {3}{2}}}{1000}-\frac {26973 \sqrt {-2 x +1}}{5000}+\frac {117649}{31944 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {1563051}{117128 \sqrt {-2 x +1}}+\frac {\frac {37 \left (-2 x +1\right )^{\frac {3}{2}}}{166375}-\frac {409 \sqrt {-2 x +1}}{831875}}{\left (-10 x -6\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/1000*(-2*x+1)^(3/2)-26973/5000*(-2*x+1)^(1/2)+117649/31944/(-2*x+1)^(3/2)-1563051/117128/(-2*x+1)^(1/2)+2/
366025*(407/10*(-2*x+1)^(3/2)-4499/50*(-2*x+1)^(1/2))/(-10*x-6)^2-6937/100656875*arctanh(1/11*55^(1/2)*(-2*x+1
)^(1/2))*55^(1/2)

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maxima [A]  time = 1.16, size = 110, normalized size = 0.75 \[ \frac {243}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6937}{201313750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {26973}{5000} \, \sqrt {-2 \, x + 1} + \frac {73267966785 \, {\left (2 \, x - 1\right )}^{3} + 342600082649 \, {\left (2 \, x - 1\right )}^{2} + 887178503750 \, x - 345719990000}{219615000 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

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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mupad [B]  time = 1.25, size = 91, normalized size = 0.62 \[ \frac {\frac {5865643\,x}{36300}+\frac {31145462059\,{\left (2\,x-1\right )}^2}{499125000}+\frac {4884531119\,{\left (2\,x-1\right )}^3}{366025000}-\frac {571438}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}}-\frac {26973\,\sqrt {1-2\,x}}{5000}+\frac {243\,{\left (1-2\,x\right )}^{3/2}}{1000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,6937{}\mathrm {i}}{100656875} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5865643*x)/36300 + (31145462059*(2*x - 1)^2)/499125000 + (4884531119*(2*x - 1)^3)/366025000 - 571438/9075)/(
(121*(1 - 2*x)^(3/2))/25 - (22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2)) + (55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2
)*1i)/11)*6937i)/100656875 - (26973*(1 - 2*x)^(1/2))/5000 + (243*(1 - 2*x)^(3/2))/1000

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

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]

Timed out

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