3.2100 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=80 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{1-2 x} (2975 x+1978)}{147 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (2*Sqrt[1 - 2*x]*(1978 + 2975*x))/(147*(2 + 3*x)) - (68*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi [A]  time = 0.0198577, antiderivative size = 80, 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 = {98, 146, 63, 206} \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{1-2 x} (2975 x+1978)}{147 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (2*Sqrt[1 - 2*x]*(1978 + 2975*x))/(147*(2 + 3*x)) - (68*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

Rule 98

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

Rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

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)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)}-\frac{1}{7} \int \frac{(3+5 x) (124+170 x)}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)}+\frac{2 \sqrt{1-2 x} (1978+2975 x)}{147 (2+3 x)}+\frac{34}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)}+\frac{2 \sqrt{1-2 x} (1978+2975 x)}{147 (2+3 x)}-\frac{34}{147} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)}+\frac{2 \sqrt{1-2 x} (1978+2975 x)}{147 (2+3 x)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0325078, size = 67, normalized size = 0.84 \[ \frac{-21 \left (6125 x^2-4968 x-6035\right )-68 \sqrt{21-42 x} (3 x+2) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*(-6035 - 4968*x + 6125*x^2) - 68*Sqrt[21 - 42*x]*(2 + 3*x)*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3087*Sqrt[1
- 2*x]*(2 + 3*x))

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Maple [A]  time = 0.01, size = 54, normalized size = 0.7 \begin{align*}{\frac{125}{18}\sqrt{1-2\,x}}-{\frac{2}{1323}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{68\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1331}{98}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/18*(1-2*x)^(1/2)-2/1323*(1-2*x)^(1/2)/(-2*x-4/3)-68/3087*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1331
/98/(1-2*x)^(1/2)

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Maxima [A]  time = 3.82333, size = 100, normalized size = 1.25 \begin{align*} \frac{34}{3087} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{125}{18} \, \sqrt{-2 \, x + 1} - \frac{35933 \, x + 23960}{441 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 125/18*sqrt(-2*x + 1) - 1
/441*(35933*x + 23960)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))

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Fricas [A]  time = 1.57414, size = 205, normalized size = 2.56 \begin{align*} \frac{34 \, \sqrt{21}{\left (6 \, x^{2} + x - 2\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (6125 \, x^{2} - 4968 \, x - 6035\right )} \sqrt{-2 \, x + 1}}{3087 \,{\left (6 \, x^{2} + x - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3087*(34*sqrt(21)*(6*x^2 + x - 2)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(6125*x^2 - 4968*x
- 6035)*sqrt(-2*x + 1))/(6*x^2 + x - 2)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.46236, size = 104, normalized size = 1.3 \begin{align*} \frac{34}{3087} \, \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{125}{18} \, \sqrt{-2 \, x + 1} - \frac{35933 \, x + 23960}{441 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/3087*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 125/18*sqrt(-2*x
+ 1) - 1/441*(35933*x + 23960)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))