3.2092 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=108 \[ -\frac{905 \sqrt{1-2 x}}{2058 (3 x+2)}-\frac{905 \sqrt{1-2 x}}{882 (3 x+2)^2}-\frac{467 \sqrt{1-2 x}}{126 (3 x+2)^3}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^3}-\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (467*Sqrt[1 - 2*x])/(126*(2 + 3*x)^3) - (905*Sqrt[1 - 2*x])/(882*(2 + 3*x
)^2) - (905*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) - (905*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi [A]  time = 0.0295001, antiderivative size = 108, 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 = {89, 78, 51, 63, 206} \[ -\frac{905 \sqrt{1-2 x}}{2058 (3 x+2)}-\frac{905 \sqrt{1-2 x}}{882 (3 x+2)^2}-\frac{467 \sqrt{1-2 x}}{126 (3 x+2)^3}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^3}-\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (467*Sqrt[1 - 2*x])/(126*(2 + 3*x)^3) - (905*Sqrt[1 - 2*x])/(882*(2 + 3*x
)^2) - (905*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) - (905*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*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 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)^{3/2} (2+3 x)^4} \, dx &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^3}-\frac{1}{14} \int \frac{-973+175 x}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^3}-\frac{467 \sqrt{1-2 x}}{126 (2+3 x)^3}+\frac{905}{63} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^3}-\frac{467 \sqrt{1-2 x}}{126 (2+3 x)^3}-\frac{905 \sqrt{1-2 x}}{882 (2+3 x)^2}+\frac{905}{294} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^3}-\frac{467 \sqrt{1-2 x}}{126 (2+3 x)^3}-\frac{905 \sqrt{1-2 x}}{882 (2+3 x)^2}-\frac{905 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{905 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^3}-\frac{467 \sqrt{1-2 x}}{126 (2+3 x)^3}-\frac{905 \sqrt{1-2 x}}{882 (2+3 x)^2}-\frac{905 \sqrt{1-2 x}}{2058 (2+3 x)}-\frac{905 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2058}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^3}-\frac{467 \sqrt{1-2 x}}{126 (2+3 x)^3}-\frac{905 \sqrt{1-2 x}}{882 (2+3 x)^2}-\frac{905 \sqrt{1-2 x}}{2058 (2+3 x)}-\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0188587, size = 59, normalized size = 0.55 \[ \frac{7240 (2 x-1) (3 x+2)^3 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )+343 (467 x+311)}{21609 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(343*(311 + 467*x) + 7240*(-1 + 2*x)*(2 + 3*x)^3*Hypergeometric2F1[1/2, 3, 3/2, 3/7 - (6*x)/7])/(21609*Sqrt[1
- 2*x]*(2 + 3*x)^3)

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Maple [A]  time = 0.011, size = 66, normalized size = 0.6 \begin{align*}{\frac{108}{2401\, \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1979}{36} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{20083}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{90601}{324}\sqrt{1-2\,x}} \right ) }-{\frac{905\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{484}{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)^(3/2)/(2+3*x)^4,x)

[Out]

108/2401*(1979/36*(1-2*x)^(5/2)-20083/81*(1-2*x)^(3/2)+90601/324*(1-2*x)^(1/2))/(-6*x-4)^3-905/21609*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+484/2401/(1-2*x)^(1/2)

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Maxima [A]  time = 3.12828, size = 136, normalized size = 1.26 \begin{align*} \frac{905}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8145 \,{\left (2 \, x - 1\right )}^{3} + 50680 \,{\left (2 \, x - 1\right )}^{2} + 208838 \, x - 33271}{1029 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

905/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/1029*(8145*(2*x - 1)^
3 + 50680*(2*x - 1)^2 + 208838*x - 33271)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) -
343*sqrt(-2*x + 1))

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Fricas [A]  time = 1.63727, size = 286, normalized size = 2.65 \begin{align*} \frac{905 \, \sqrt{21}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (16290 \, x^{3} + 26245 \, x^{2} + 13747 \, x + 2316\right )} \sqrt{-2 \, x + 1}}{43218 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/43218*(905*sqrt(21)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2))
- 21*(16290*x^3 + 26245*x^2 + 13747*x + 2316)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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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)**(3/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.19894, size = 126, normalized size = 1.17 \begin{align*} \frac{905}{43218} \, \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{484}{2401 \, \sqrt{-2 \, x + 1}} - \frac{17811 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 80332 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 90601 \, \sqrt{-2 \, x + 1}}{57624 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

905/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 484/2401/sqrt(
-2*x + 1) - 1/57624*(17811*(2*x - 1)^2*sqrt(-2*x + 1) - 80332*(-2*x + 1)^(3/2) + 90601*sqrt(-2*x + 1))/(3*x +
2)^3