3.2191 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=127 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^3}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{317 (3 x+2)^2}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{3 (544568-333311 x)}{732050 \sqrt{1-2 x}}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}} \]

[Out]

(-3*(544568 - 333311*x))/(732050*Sqrt[1 - 2*x]) - (73*(2 + 3*x)^3)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 +
3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (317*(2 + 3*x)^2)/(19965*Sqrt[1 - 2*x]*(3 + 5*x)) - (4693*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/(366025*Sqrt[55])

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Rubi [A]  time = 0.0399178, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 149, 146, 63, 206} \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{73 (3 x+2)^3}{3630 \sqrt{1-2 x} (5 x+3)^2}-\frac{317 (3 x+2)^2}{19965 \sqrt{1-2 x} (5 x+3)}-\frac{3 (544568-333311 x)}{732050 \sqrt{1-2 x}}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(544568 - 333311*x))/(732050*Sqrt[1 - 2*x]) - (73*(2 + 3*x)^3)/(3630*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (7*(2 +
3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (317*(2 + 3*x)^2)/(19965*Sqrt[1 - 2*x]*(3 + 5*x)) - (4693*ArcTanh[S
qrt[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 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(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*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && 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)^5}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1}{33} \int \frac{(2+3 x)^3 (106+201 x)}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=-\frac{73 (2+3 x)^3}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{\int \frac{(2+3 x)^2 (7457+13485 x)}{(1-2 x)^{3/2} (3+5 x)^2} \, dx}{3630}\\ &=-\frac{73 (2+3 x)^3}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{317 (2+3 x)^2}{19965 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{(2+3 x) (258630+454515 x)}{(1-2 x)^{3/2} (3+5 x)} \, dx}{199650}\\ &=-\frac{3 (544568-333311 x)}{732050 \sqrt{1-2 x}}-\frac{73 (2+3 x)^3}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{317 (2+3 x)^2}{19965 \sqrt{1-2 x} (3+5 x)}+\frac{4693 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{732050}\\ &=-\frac{3 (544568-333311 x)}{732050 \sqrt{1-2 x}}-\frac{73 (2+3 x)^3}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{317 (2+3 x)^2}{19965 \sqrt{1-2 x} (3+5 x)}-\frac{4693 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{732050}\\ &=-\frac{3 (544568-333311 x)}{732050 \sqrt{1-2 x}}-\frac{73 (2+3 x)^3}{3630 \sqrt{1-2 x} (3+5 x)^2}+\frac{7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{317 (2+3 x)^2}{19965 \sqrt{1-2 x} (3+5 x)}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{366025 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0530953, size = 100, normalized size = 0.79 \[ -\frac{-5327 (5 x+3)^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+5535 (2 x-1) (5 x+3)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )+33 \left (7350750 x^4-17151750 x^3-21347475 x^2-741695 x+2582641\right )}{4991250 (1-2 x)^{3/2} (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(33*(2582641 - 741695*x - 21347475*x^2 - 17151750*x^3 + 7350750*x^4) - 5327*(3 + 5*x)^2*Hypergeometric2F1[-3/
2, 1, -1/2, (5*(1 - 2*x))/11] + 5535*(-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 = 75, normalized size = 0.6 \begin{align*} -{\frac{243}{500}\sqrt{1-2\,x}}+{\frac{16807}{15972} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{36015}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{4}{73205\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{341}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3773}{100}\sqrt{1-2\,x}} \right ) }-{\frac{4693\,\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)^5/(1-2*x)^(5/2)/(3+5*x)^3,x)

[Out]

-243/500*(1-2*x)^(1/2)+16807/15972/(1-2*x)^(3/2)-36015/14641/(1-2*x)^(1/2)+4/73205*(341/20*(1-2*x)^(3/2)-3773/
100*(1-2*x)^(1/2))/(-10*x-6)^2-4693/20131375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.69433, size = 136, normalized size = 1.07 \begin{align*} \frac{4693}{40262750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{243}{500} \, \sqrt{-2 \, x + 1} + \frac{1350542040 \,{\left (2 \, x - 1\right )}^{3} + 6520170349 \,{\left (2 \, x - 1\right )}^{2} + 18157562500 \, x - 6282516625}{21961500 \,{\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)^5/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="maxima")

[Out]

4693/40262750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 243/500*sqrt(-2*x +
1) + 1/21961500*(1350542040*(2*x - 1)^3 + 6520170349*(2*x - 1)^2 + 18157562500*x - 6282516625)/(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 = 1.07082, size = 336, normalized size = 2.65 \begin{align*} \frac{14079 \, \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 (106732890 \, x^{4} - 248761830 \, x^{3} - 309826828 \, x^{2} - 10907307 \, x + 37428168\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)^5/(1-2*x)^(5/2)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/120788250*(14079*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*(106732890*x^4 - 248761830*x^3 - 309826828*x^2 - 10907307*x + 37428168)*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)**5/(1-2*x)**(5/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.08784, size = 132, normalized size = 1.04 \begin{align*} \frac{4693}{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{243}{500} \, \sqrt{-2 \, x + 1} - \frac{2401 \,{\left (360 \, x - 103\right )}}{175692 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{155 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}}{665500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4693/40262750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 243/500*s
qrt(-2*x + 1) - 2401/175692*(360*x - 103)/((2*x - 1)*sqrt(-2*x + 1)) + 1/665500*(155*(-2*x + 1)^(3/2) - 343*sq
rt(-2*x + 1))/(5*x + 3)^2