3.2020 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{137 \sqrt{1-2 x}}{882 (3 x+2)}-\frac{\sqrt{1-2 x}}{126 (3 x+2)^2}-\frac{257 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

[Out]

-Sqrt[1 - 2*x]/(126*(2 + 3*x)^2) + (137*Sqrt[1 - 2*x])/(882*(2 + 3*x)) - (257*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/(49*Sqrt[21])

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Rubi [A]  time = 0.0149314, 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} \[ \frac{137 \sqrt{1-2 x}}{882 (3 x+2)}-\frac{\sqrt{1-2 x}}{126 (3 x+2)^2}-\frac{257 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[1 - 2*x]/(126*(2 + 3*x)^2) + (137*Sqrt[1 - 2*x])/(882*(2 + 3*x)) - (257*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/(49*Sqrt[21])

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{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^3} \, dx &=-\frac{\sqrt{1-2 x}}{126 (2+3 x)^2}+\frac{1}{126} \int \frac{563+1050 x}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x}}{126 (2+3 x)^2}+\frac{137 \sqrt{1-2 x}}{882 (2+3 x)}+\frac{257}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{\sqrt{1-2 x}}{126 (2+3 x)^2}+\frac{137 \sqrt{1-2 x}}{882 (2+3 x)}-\frac{257}{98} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{\sqrt{1-2 x}}{126 (2+3 x)^2}+\frac{137 \sqrt{1-2 x}}{882 (2+3 x)}-\frac{257 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.028407, size = 53, normalized size = 0.78 \[ \frac{\frac{7 \sqrt{1-2 x} (137 x+89)}{(3 x+2)^2}-514 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2058} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*(89 + 137*x))/(2 + 3*x)^2 - 514*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2058

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Maple [A]  time = 0.01, size = 48, normalized size = 0.7 \begin{align*} 18\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{137\, \left ( 1-2\,x \right ) ^{3/2}}{2646}}+{\frac{5\,\sqrt{1-2\,x}}{42}} \right ) }-{\frac{257\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*(-137/2646*(1-2*x)^(3/2)+5/42*(1-2*x)^(1/2))/(-6*x-4)^2-257/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/
2)

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Maxima [A]  time = 3.19209, size = 100, normalized size = 1.47 \begin{align*} \frac{257}{2058} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 315 \, \sqrt{-2 \, x + 1}}{147 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

257/2058*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/147*(137*(-2*x + 1)^(3
/2) - 315*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 1.60372, size = 194, normalized size = 2.85 \begin{align*} \frac{257 \, \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \,{\left (137 \, x + 89\right )} \sqrt{-2 \, x + 1}}{2058 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2058*(257*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 7*(137*x + 89)*sq
rt(-2*x + 1))/(9*x^2 + 12*x + 4)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.96769, size = 92, normalized size = 1.35 \begin{align*} \frac{257}{2058} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 315 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

257/2058*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/588*(137*(-2*
x + 1)^(3/2) - 315*sqrt(-2*x + 1))/(3*x + 2)^2