3.2119 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{1227 \sqrt{1-2 x}}{1210 (5 x+3)}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)) - (1227*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (138*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(605*Sqrt[55])

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Rubi [A]  time = 0.0157186, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {89, 78, 63, 206} \[ -\frac{1227 \sqrt{1-2 x}}{1210 (5 x+3)}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)) - (1227*Sqrt[1 - 2*x])/(1210*(3 + 5*x)) - (138*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(605*Sqrt[55])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 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)^2}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)}-\frac{1}{22} \int \frac{-186+99 x}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)}-\frac{1227 \sqrt{1-2 x}}{1210 (3+5 x)}+\frac{69}{605} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)}-\frac{1227 \sqrt{1-2 x}}{1210 (3+5 x)}-\frac{69}{605} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)}-\frac{1227 \sqrt{1-2 x}}{1210 (3+5 x)}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0332373, size = 53, normalized size = 0.78 \[ \frac{1227 x+734}{605 \sqrt{1-2 x} (5 x+3)}-\frac{138 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(734 + 1227*x)/(605*Sqrt[1 - 2*x]*(3 + 5*x)) - (138*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(605*Sqrt[55])

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Maple [A]  time = 0.01, size = 45, normalized size = 0.7 \begin{align*}{\frac{49}{121}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{3025}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{138\,\sqrt{55}}{33275}{\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)^2/(1-2*x)^(3/2)/(3+5*x)^2,x)

[Out]

49/121/(1-2*x)^(1/2)+2/3025*(1-2*x)^(1/2)/(-2*x-6/5)-138/33275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.53454, size = 88, normalized size = 1.29 \begin{align*} \frac{69}{33275} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1227 \, x + 734\right )}}{605 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

69/33275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/605*(1227*x + 734)/(5*
(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 1.57457, size = 193, normalized size = 2.84 \begin{align*} \frac{69 \, \sqrt{55}{\left (10 \, x^{2} + x - 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (1227 \, x + 734\right )} \sqrt{-2 \, x + 1}}{33275 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33275*(69*sqrt(55)*(10*x^2 + x - 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(1227*x + 734)*s
qrt(-2*x + 1))/(10*x^2 + x - 3)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.62246, size = 92, normalized size = 1.35 \begin{align*} \frac{69}{33275} \, \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{2 \,{\left (1227 \, x + 734\right )}}{605 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

69/33275*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/605*(1227*x
+ 734)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))