3.1802 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x}}{1029 (3 x+2)}+\frac{\sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{2 \sqrt{1-2 x}}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

(1 - 2*x)^(3/2)/(105*(2 + 3*x)^5) - (2*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (2*Sqrt[1 - 2*x])/(315*(2 + 3*x)^3) +
Sqrt[1 - 2*x]/(441*(2 + 3*x)^2) + Sqrt[1 - 2*x]/(1029*(2 + 3*x)) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029
*Sqrt[21])

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Rubi [A]  time = 0.0375136, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 51, 63, 206} \[ \frac{(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x}}{1029 (3 x+2)}+\frac{\sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{2 \sqrt{1-2 x}}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(3/2)/(105*(2 + 3*x)^5) - (2*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (2*Sqrt[1 - 2*x])/(315*(2 + 3*x)^3) +
Sqrt[1 - 2*x]/(441*(2 + 3*x)^2) + Sqrt[1 - 2*x]/(1029*(2 + 3*x)) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029
*Sqrt[21])

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+3 x)^6} \, dx &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}+\frac{8}{5} \int \frac{\sqrt{1-2 x}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac{2 \sqrt{1-2 x}}{15 (2+3 x)^4}-\frac{2}{15} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac{2 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{2 \sqrt{1-2 x}}{315 (2+3 x)^3}-\frac{2}{63} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac{2 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{2 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{\sqrt{1-2 x}}{441 (2+3 x)^2}-\frac{1}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac{2 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{2 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{\sqrt{1-2 x}}{441 (2+3 x)^2}+\frac{\sqrt{1-2 x}}{1029 (2+3 x)}-\frac{\int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1029}\\ &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac{2 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{2 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{\sqrt{1-2 x}}{441 (2+3 x)^2}+\frac{\sqrt{1-2 x}}{1029 (2+3 x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1029}\\ &=\frac{(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac{2 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{2 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{\sqrt{1-2 x}}{441 (2+3 x)^2}+\frac{\sqrt{1-2 x}}{1029 (2+3 x)}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0147839, size = 42, normalized size = 0.33 \[ \frac{(1-2 x)^{3/2} \left (\frac{2401}{(3 x+2)^5}-256 \, _2F_1\left (\frac{3}{2},5;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{252105} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(3/2)*(2401/(2 + 3*x)^5 - 256*Hypergeometric2F1[3/2, 5, 5/2, 3/7 - (6*x)/7]))/252105

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Maple [A]  time = 0.012, size = 75, normalized size = 0.6 \begin{align*} 7776\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{9/2}}{49392}}+{\frac{ \left ( 1-2\,x \right ) ^{7/2}}{4536}}-{\frac{8\, \left ( 1-2\,x \right ) ^{5/2}}{8505}}+{\frac{13\, \left ( 1-2\,x \right ) ^{3/2}}{13608}}+{\frac{7\,\sqrt{1-2\,x}}{11664}} \right ) }+{\frac{2\,\sqrt{21}}{21609}{\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)*(1-2*x)^(1/2)/(2+3*x)^6,x)

[Out]

7776*(-1/49392*(1-2*x)^(9/2)+1/4536*(1-2*x)^(7/2)-8/8505*(1-2*x)^(5/2)+13/13608*(1-2*x)^(3/2)+7/11664*(1-2*x)^
(1/2))/(-6*x-4)^5+2/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.60916, size = 173, normalized size = 1.35 \begin{align*} -\frac{1}{21609} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (405 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4410 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 18816 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 19110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12005 \, \sqrt{-2 \, x + 1}\right )}}{5145 \,{\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)*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

-1/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/5145*(405*(-2*x + 1)^(
9/2) - 4410*(-2*x + 1)^(7/2) + 18816*(-2*x + 1)^(5/2) - 19110*(-2*x + 1)^(3/2) - 12005*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 = 1.61819, size = 336, normalized size = 2.62 \begin{align*} \frac{5 \, \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 (405 \, x^{4} + 1395 \, x^{3} + 2004 \, x^{2} - 864 \, x - 1019\right )} \sqrt{-2 \, x + 1}}{108045 \,{\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)*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/108045*(5*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*(405*x^4 + 1395*x^3 + 2004*x^2 - 864*x - 1019)*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)*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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Giac [A]  time = 2.37465, size = 157, normalized size = 1.23 \begin{align*} -\frac{1}{21609} \, \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{405 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 4410 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 18816 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 19110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12005 \, \sqrt{-2 \, x + 1}}{82320 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

-1/21609*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/82320*(405*(2
*x - 1)^4*sqrt(-2*x + 1) + 4410*(2*x - 1)^3*sqrt(-2*x + 1) + 18816*(2*x - 1)^2*sqrt(-2*x + 1) - 19110*(-2*x +
1)^(3/2) - 12005*sqrt(-2*x + 1))/(3*x + 2)^5