3.1821 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=93 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{3 (3 x+2)}+\frac{7}{9} \sqrt{1-2 x} (5 x+3)^2-\frac{2}{81} \sqrt{1-2 x} (170 x+211)-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

[Out]

(7*Sqrt[1 - 2*x]*(3 + 5*x)^2)/9 - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(3*(2 + 3*x)) - (2*Sqrt[1 - 2*x]*(211 + 170*x))/
81 - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Rubi [A]  time = 0.0277882, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {97, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{3 (3 x+2)}+\frac{7}{9} \sqrt{1-2 x} (5 x+3)^2-\frac{2}{81} \sqrt{1-2 x} (170 x+211)-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[1 - 2*x]*(3 + 5*x)^2)/9 - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(3*(2 + 3*x)) - (2*Sqrt[1 - 2*x]*(211 + 170*x))/
81 - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

Rule 97

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(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^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && 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{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^2} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{3 (2+3 x)}+\frac{1}{3} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{7}{9} \sqrt{1-2 x} (3+5 x)^2-\frac{\sqrt{1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac{1}{45} \int \frac{(-50-340 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{7}{9} \sqrt{1-2 x} (3+5 x)^2-\frac{\sqrt{1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac{2}{81} \sqrt{1-2 x} (211+170 x)+\frac{106}{81} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{7}{9} \sqrt{1-2 x} (3+5 x)^2-\frac{\sqrt{1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac{2}{81} \sqrt{1-2 x} (211+170 x)-\frac{106}{81} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7}{9} \sqrt{1-2 x} (3+5 x)^2-\frac{\sqrt{1-2 x} (3+5 x)^3}{3 (2+3 x)}-\frac{2}{81} \sqrt{1-2 x} (211+170 x)-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0359963, size = 63, normalized size = 0.68 \[ \frac{\sqrt{1-2 x} \left (1350 x^3+1725 x^2-110 x-439\right )}{81 (3 x+2)}-\frac{212 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-439 - 110*x + 1725*x^2 + 1350*x^3))/(81*(2 + 3*x)) - (212*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
81*Sqrt[21])

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Maple [A]  time = 0.008, size = 63, normalized size = 0.7 \begin{align*}{\frac{25}{18} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{725}{162} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{10}{27}\sqrt{1-2\,x}}-{\frac{2}{243}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{212\,\sqrt{21}}{1701}{\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)^3*(1-2*x)^(1/2)/(2+3*x)^2,x)

[Out]

25/18*(1-2*x)^(5/2)-725/162*(1-2*x)^(3/2)+10/27*(1-2*x)^(1/2)-2/243*(1-2*x)^(1/2)/(-2*x-4/3)-212/1701*arctanh(
1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.52841, size = 108, normalized size = 1.16 \begin{align*} \frac{25}{18} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{725}{162} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{106}{1701} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{10}{27} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/18*(-2*x + 1)^(5/2) - 725/162*(-2*x + 1)^(3/2) + 106/1701*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) + 10/27*sqrt(-2*x + 1) + 1/81*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 1.60932, size = 203, normalized size = 2.18 \begin{align*} \frac{106 \, \sqrt{21}{\left (3 \, x + 2\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (1350 \, x^{3} + 1725 \, x^{2} - 110 \, x - 439\right )} \sqrt{-2 \, x + 1}}{1701 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1701*(106*sqrt(21)*(3*x + 2)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1350*x^3 + 1725*x^2 -
110*x - 439)*sqrt(-2*x + 1))/(3*x + 2)

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Sympy [A]  time = 84.2267, size = 202, normalized size = 2.17 \begin{align*} \frac{25 \left (1 - 2 x\right )^{\frac{5}{2}}}{18} - \frac{725 \left (1 - 2 x\right )^{\frac{3}{2}}}{162} + \frac{10 \sqrt{1 - 2 x}}{27} + \frac{28 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{81} + \frac{214 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*(1 - 2*x)**(5/2)/18 - 725*(1 - 2*x)**(3/2)/162 + 10*sqrt(1 - 2*x)/27 + 28*Piecewise((sqrt(21)*(-log(sqrt(21
)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(
sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/81 + 214*Piecewise((-sqrt(21)*acoth(sqrt(21)*sq
rt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3))/81

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Giac [A]  time = 2.4865, size = 122, normalized size = 1.31 \begin{align*} \frac{25}{18} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{725}{162} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{106}{1701} \, \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{10}{27} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/18*(2*x - 1)^2*sqrt(-2*x + 1) - 725/162*(-2*x + 1)^(3/2) + 106/1701*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq
rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 10/27*sqrt(-2*x + 1) + 1/81*sqrt(-2*x + 1)/(3*x + 2)