3.1813 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{7 (1-2 x)^{3/2}}{180 (3 x+2)^4}-\frac{(1-2 x)^{3/2}}{315 (3 x+2)^5}+\frac{31 \sqrt{1-2 x}}{3528 (3 x+2)}+\frac{31 \sqrt{1-2 x}}{1512 (3 x+2)^2}-\frac{31 \sqrt{1-2 x}}{108 (3 x+2)^3}+\frac{31 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1764 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(3/2)/(315*(2 + 3*x)^5) + (7*(1 - 2*x)^(3/2))/(180*(2 + 3*x)^4) - (31*Sqrt[1 - 2*x])/(108*(2 + 3*x)
^3) + (31*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2) + (31*Sqrt[1 - 2*x])/(3528*(2 + 3*x)) + (31*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/(1764*Sqrt[21])

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Rubi [A]  time = 0.037988, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{180 (3 x+2)^4}-\frac{(1-2 x)^{3/2}}{315 (3 x+2)^5}+\frac{31 \sqrt{1-2 x}}{3528 (3 x+2)}+\frac{31 \sqrt{1-2 x}}{1512 (3 x+2)^2}-\frac{31 \sqrt{1-2 x}}{108 (3 x+2)^3}+\frac{31 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1764 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-(1 - 2*x)^(3/2)/(315*(2 + 3*x)^5) + (7*(1 - 2*x)^(3/2))/(180*(2 + 3*x)^4) - (31*Sqrt[1 - 2*x])/(108*(2 + 3*x)
^3) + (31*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2) + (31*Sqrt[1 - 2*x])/(3528*(2 + 3*x)) + (31*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/(1764*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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{1}{315} \int \frac{\sqrt{1-2 x} (1407+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{7 (1-2 x)^{3/2}}{180 (2+3 x)^4}+\frac{31}{12} \int \frac{\sqrt{1-2 x}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac{31 \sqrt{1-2 x}}{108 (2+3 x)^3}-\frac{31}{108} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac{31 \sqrt{1-2 x}}{108 (2+3 x)^3}+\frac{31 \sqrt{1-2 x}}{1512 (2+3 x)^2}-\frac{31}{504} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac{31 \sqrt{1-2 x}}{108 (2+3 x)^3}+\frac{31 \sqrt{1-2 x}}{1512 (2+3 x)^2}+\frac{31 \sqrt{1-2 x}}{3528 (2+3 x)}-\frac{31 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{3528}\\ &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac{31 \sqrt{1-2 x}}{108 (2+3 x)^3}+\frac{31 \sqrt{1-2 x}}{1512 (2+3 x)^2}+\frac{31 \sqrt{1-2 x}}{3528 (2+3 x)}+\frac{31 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3528}\\ &=-\frac{(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac{7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac{31 \sqrt{1-2 x}}{108 (2+3 x)^3}+\frac{31 \sqrt{1-2 x}}{1512 (2+3 x)^2}+\frac{31 \sqrt{1-2 x}}{3528 (2+3 x)}+\frac{31 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1764 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0229491, size = 47, normalized size = 0.37 \[ \frac{(1-2 x)^{3/2} \left (\frac{343 (147 x+94)}{(3 x+2)^5}-2480 \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{432180} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

((1 - 2*x)^(3/2)*((343*(94 + 147*x))/(2 + 3*x)^5 - 2480*Hypergeometric2F1[3/2, 4, 5/2, 3/7 - (6*x)/7]))/432180

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Maple [A]  time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -3888\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{31\, \left ( 1-2\,x \right ) ^{9/2}}{84672}}-{\frac{31\, \left ( 1-2\,x \right ) ^{7/2}}{7776}}+{\frac{37\, \left ( 1-2\,x \right ) ^{5/2}}{3645}}-{\frac{983\, \left ( 1-2\,x \right ) ^{3/2}}{489888}}-{\frac{1519\,\sqrt{1-2\,x}}{139968}} \right ) }+{\frac{31\,\sqrt{21}}{37044}{\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)^6,x)

[Out]

-3888*(31/84672*(1-2*x)^(9/2)-31/7776*(1-2*x)^(7/2)+37/3645*(1-2*x)^(5/2)-983/489888*(1-2*x)^(3/2)-1519/139968
*(1-2*x)^(1/2))/(-6*x-4)^5+31/37044*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 3.25448, size = 173, normalized size = 1.35 \begin{align*} -\frac{31}{74088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12555 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 136710 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 348096 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 68810 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 372155 \, \sqrt{-2 \, x + 1}}{8820 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\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)^6,x, algorithm="maxima")

[Out]

-31/74088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/8820*(12555*(-2*x + 1
)^(9/2) - 136710*(-2*x + 1)^(7/2) + 348096*(-2*x + 1)^(5/2) - 68810*(-2*x + 1)^(3/2) - 372155*sqrt(-2*x + 1))/
(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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Fricas [A]  time = 1.59032, size = 347, normalized size = 2.71 \begin{align*} \frac{155 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (12555 \, x^{4} + 43245 \, x^{3} + 3324 \, x^{2} - 33434 \, x - 13564\right )} \sqrt{-2 \, x + 1}}{370440 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

1/370440*(155*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt(21)*sqrt(-2*x + 1
) - 5)/(3*x + 2)) + 21*(12555*x^4 + 43245*x^3 + 3324*x^2 - 33434*x - 13564)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.98302, size = 157, normalized size = 1.23 \begin{align*} -\frac{31}{74088} \, \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{12555 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 136710 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 348096 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 68810 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 372155 \, \sqrt{-2 \, x + 1}}{282240 \,{\left (3 \, x + 2\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-31/74088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/282240*(1255
5*(2*x - 1)^4*sqrt(-2*x + 1) + 136710*(2*x - 1)^3*sqrt(-2*x + 1) + 348096*(2*x - 1)^2*sqrt(-2*x + 1) - 68810*(
-2*x + 1)^(3/2) - 372155*sqrt(-2*x + 1))/(3*x + 2)^5