3.2185 \(\int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac {76}{1815 (1-2 x)^{3/2}}-\frac {76 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]

[Out]

76/1815/(1-2*x)^(3/2)-1/55/(1-2*x)^(3/2)/(3+5*x)-76/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+76/133
1/(1-2*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ \frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac {76}{1815 (1-2 x)^{3/2}}-\frac {76 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

76/(1815*(1 - 2*x)^(3/2)) + 76/(1331*Sqrt[1 - 2*x]) - 1/(55*(1 - 2*x)^(3/2)*(3 + 5*x)) - (76*Sqrt[5/11]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac {38}{55} \int \frac {1}{(1-2 x)^{5/2} (3+5 x)} \, dx\\ &=\frac {76}{1815 (1-2 x)^{3/2}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac {38}{121} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac {190 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac {190 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1331}\\ &=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac {76 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 46, normalized size = 0.61 \[ -\frac {33-76 (5 x+3) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{1815 (1-2 x)^{3/2} (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1815*(33 - 76*(3 + 5*x)*Hypergeometric2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11])/((1 - 2*x)^(3/2)*(3 + 5*x))

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fricas [A]  time = 0.81, size = 90, normalized size = 1.18 \[ \frac {114 \, \sqrt {11} \sqrt {5} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (2280 \, x^{2} - 608 \, x - 1113\right )} \sqrt {-2 \, x + 1}}{43923 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43923*(114*sqrt(11)*sqrt(5)*(20*x^3 - 8*x^2 - 7*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x
+ 3)) - 11*(2280*x^2 - 608*x - 1113)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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giac [A]  time = 1.28, size = 77, normalized size = 1.01 \[ \frac {38}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (111 \, x - 94\right )}}{3993 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {5 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

38/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4/3993*(111*x
- 94)/((2*x - 1)*sqrt(-2*x + 1)) - 5/1331*sqrt(-2*x + 1)/(5*x + 3)

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maple [A]  time = 0.01, size = 54, normalized size = 0.71 \[ -\frac {76 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{14641}+\frac {14}{363 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {74}{1331 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{1331 \left (-2 x -\frac {6}{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14/363/(-2*x+1)^(3/2)+74/1331/(-2*x+1)^(1/2)+2/1331*(-2*x+1)^(1/2)/(-2*x-6/5)-76/14641*arctanh(1/11*55^(1/2)*(
-2*x+1)^(1/2))*55^(1/2)

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maxima [A]  time = 1.16, size = 74, normalized size = 0.97 \[ \frac {38}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (570 \, {\left (2 \, x - 1\right )}^{2} + 1672 \, x - 1683\right )}}{3993 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

38/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/3993*(570*(2*x - 1)^2
+ 1672*x - 1683)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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mupad [B]  time = 1.20, size = 56, normalized size = 0.74 \[ -\frac {\frac {304\,x}{1815}+\frac {76\,{\left (2\,x-1\right )}^2}{1331}-\frac {102}{605}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {76\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((304*x)/1815 + (76*(2*x - 1)^2)/1331 - 102/605)/((11*(1 - 2*x)^(3/2))/5 - (1 - 2*x)^(5/2)) - (76*55^(1/2)*a
tanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641

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

[Out]

Timed out

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