3.2154 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=103 \[ \frac{85}{343 \sqrt{1-2 x}}-\frac{85}{294 \sqrt{1-2 x} (3 x+2)}-\frac{26}{21 \sqrt{1-2 x} (3 x+2)^2}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{85}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

85/(343*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - 26/(21*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 85/(294*Sq
rt[1 - 2*x]*(2 + 3*x)) - (85*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

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Rubi [A]  time = 0.0309631, antiderivative size = 110, normalized size of antiderivative = 1.07, 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 = {89, 78, 51, 63, 206} \[ -\frac{255 \sqrt{1-2 x}}{686 (3 x+2)}+\frac{85}{147 \sqrt{1-2 x} (3 x+2)}-\frac{26}{21 \sqrt{1-2 x} (3 x+2)^2}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{85}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - 26/(21*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 85/(147*Sqrt[1 - 2*x]*(2 + 3*x)) -
(255*Sqrt[1 - 2*x])/(686*(2 + 3*x)) - (85*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

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 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 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}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{1}{42} \int \frac{-378+525 x}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{26}{21 \sqrt{1-2 x} (2+3 x)^2}+\frac{85}{42} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{26}{21 \sqrt{1-2 x} (2+3 x)^2}+\frac{85}{147 \sqrt{1-2 x} (2+3 x)}+\frac{255}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{26}{21 \sqrt{1-2 x} (2+3 x)^2}+\frac{85}{147 \sqrt{1-2 x} (2+3 x)}-\frac{255 \sqrt{1-2 x}}{686 (2+3 x)}+\frac{255}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{26}{21 \sqrt{1-2 x} (2+3 x)^2}+\frac{85}{147 \sqrt{1-2 x} (2+3 x)}-\frac{255 \sqrt{1-2 x}}{686 (2+3 x)}-\frac{255}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac{26}{21 \sqrt{1-2 x} (2+3 x)^2}+\frac{85}{147 \sqrt{1-2 x} (2+3 x)}-\frac{255 \sqrt{1-2 x}}{686 (2+3 x)}-\frac{85}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0202273, size = 59, normalized size = 0.57 \[ -\frac{340 (2 x-1) (3 x+2)^2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-49 (104 x+69)}{2058 (1-2 x)^{3/2} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-49*(69 + 104*x) + 340*(-1 + 2*x)*(2 + 3*x)^2*Hypergeometric2F1[-1/2, 2, 1/2, 3/7 - (6*x)/7])/(2058*(1 - 2*x
)^(3/2)*(2 + 3*x)^2)

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Maple [A]  time = 0.013, size = 66, normalized size = 0.6 \begin{align*}{\frac{18}{2401\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{43}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{889}{18}\sqrt{1-2\,x}} \right ) }-{\frac{85\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{1029} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{638}{2401}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18/2401*(-43/2*(1-2*x)^(3/2)+889/18*(1-2*x)^(1/2))/(-6*x-4)^2-85/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(
1/2)+242/1029/(1-2*x)^(3/2)+638/2401/(1-2*x)^(1/2)

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Maxima [A]  time = 2.65212, size = 124, normalized size = 1.2 \begin{align*} \frac{85}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2295 \,{\left (2 \, x - 1\right )}^{3} + 8925 \,{\left (2 \, x - 1\right )}^{2} + 6468 \, x - 15092}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

85/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/1029*(2295*(2*x - 1)^3
+ 8925*(2*x - 1)^2 + 6468*x - 15092)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.84895, size = 297, normalized size = 2.88 \begin{align*} \frac{255 \, \sqrt{7} \sqrt{3}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \,{\left (9180 \, x^{3} + 4080 \, x^{2} - 7731 \, x - 4231\right )} \sqrt{-2 \, x + 1}}{14406 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14406*(255*sqrt(7)*sqrt(3)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x -
5)/(3*x + 2)) - 7*(9180*x^3 + 4080*x^2 - 7731*x - 4231)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*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/(1-2*x)**(5/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.46069, size = 120, normalized size = 1.17 \begin{align*} \frac{85}{4802} \, \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{44 \,{\left (87 \, x - 82\right )}}{7203 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{387 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 889 \, \sqrt{-2 \, x + 1}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

85/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 44/7203*(87*x -
82)/((2*x - 1)*sqrt(-2*x + 1)) - 1/9604*(387*(-2*x + 1)^(3/2) - 889*sqrt(-2*x + 1))/(3*x + 2)^2