3.1810 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=81 \[ \frac{139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac{(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac{863}{441} \sqrt{1-2 x}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

[Out]

(863*Sqrt[1 - 2*x])/441 - (1 - 2*x)^(3/2)/(126*(2 + 3*x)^2) + (139*(1 - 2*x)^(3/2))/(882*(2 + 3*x)) - (863*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

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Rubi [A]  time = 0.0196552, antiderivative size = 81, 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 = {89, 78, 50, 63, 206} \[ \frac{139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac{(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac{863}{441} \sqrt{1-2 x}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(863*Sqrt[1 - 2*x])/441 - (1 - 2*x)^(3/2)/(126*(2 + 3*x)^2) + (139*(1 - 2*x)^(3/2))/(882*(2 + 3*x)) - (863*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac{1}{126} \int \frac{\sqrt{1-2 x} (561+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac{139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac{863}{294} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{863}{441} \sqrt{1-2 x}-\frac{(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac{139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac{863}{126} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{863}{441} \sqrt{1-2 x}-\frac{(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac{139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac{863}{126} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{863}{441} \sqrt{1-2 x}-\frac{(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac{139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0330618, size = 58, normalized size = 0.72 \[ \frac{\sqrt{1-2 x} \left (2100 x^2+2941 x+1025\right )}{126 (3 x+2)^2}-\frac{863 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(1025 + 2941*x + 2100*x^2))/(126*(2 + 3*x)^2) - (863*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt
[21])

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Maple [A]  time = 0.009, size = 57, normalized size = 0.7 \begin{align*}{\frac{50}{27}\sqrt{1-2\,x}}+{\frac{2}{3\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{47}{14} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{139}{18}\sqrt{1-2\,x}} \right ) }-{\frac{863\,\sqrt{21}}{1323}{\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*(1-2*x)^(1/2)/(2+3*x)^3,x)

[Out]

50/27*(1-2*x)^(1/2)+2/3*(-47/14*(1-2*x)^(3/2)+139/18*(1-2*x)^(1/2))/(-6*x-4)^2-863/1323*arctanh(1/7*21^(1/2)*(
1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.82983, size = 112, normalized size = 1.38 \begin{align*} \frac{863}{2646} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{50}{27} \, \sqrt{-2 \, x + 1} - \frac{423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 973 \, \sqrt{-2 \, x + 1}}{189 \,{\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*(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="maxima")

[Out]

863/2646*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 50/27*sqrt(-2*x + 1) - 1
/189*(423*(-2*x + 1)^(3/2) - 973*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 1.6265, size = 215, normalized size = 2.65 \begin{align*} \frac{863 \, \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 ) + 21 \,{\left (2100 \, x^{2} + 2941 \, x + 1025\right )} \sqrt{-2 \, x + 1}}{2646 \,{\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*(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="fricas")

[Out]

1/2646*(863*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(2100*x^2 + 29
41*x + 1025)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 161.623, size = 326, normalized size = 4.02 \begin{align*} \frac{50 \sqrt{1 - 2 x}}{27} + \frac{32 \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 )}{3} + \frac{56 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{27} + \frac{130 \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 )}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50*sqrt(1 - 2*x)/27 + 32*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)))/3 + 56*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7
+ 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*
sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3)))/27 + 130*Pie
cewise((-sqrt(21)*acoth(sqrt(21)*sqrt(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))/9

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Giac [A]  time = 1.73833, size = 104, normalized size = 1.28 \begin{align*} \frac{863}{2646} \, \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{50}{27} \, \sqrt{-2 \, x + 1} - \frac{423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 973 \, \sqrt{-2 \, x + 1}}{756 \,{\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)^(1/2)/(2+3*x)^3,x, algorithm="giac")

[Out]

863/2646*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 50/27*sqrt(-2*x
+ 1) - 1/756*(423*(-2*x + 1)^(3/2) - 973*sqrt(-2*x + 1))/(3*x + 2)^2