3.1801 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=110 \[ \frac{(1-2 x)^{3/2}}{84 (3 x+2)^4}+\frac{15 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{5 \sqrt{1-2 x}}{392 (3 x+2)^2}-\frac{5 \sqrt{1-2 x}}{28 (3 x+2)^3}+\frac{5 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372} \]

[Out]

(1 - 2*x)^(3/2)/(84*(2 + 3*x)^4) - (5*Sqrt[1 - 2*x])/(28*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x])/(392*(2 + 3*x)^2) +
(15*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (5*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1372

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Rubi [A]  time = 0.030174, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{(1-2 x)^{3/2}}{84 (3 x+2)^4}+\frac{15 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{5 \sqrt{1-2 x}}{392 (3 x+2)^2}-\frac{5 \sqrt{1-2 x}}{28 (3 x+2)^3}+\frac{5 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(3/2)/(84*(2 + 3*x)^4) - (5*Sqrt[1 - 2*x])/(28*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x])/(392*(2 + 3*x)^2) +
(15*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (5*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1372

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)^5} \, dx &=\frac{(1-2 x)^{3/2}}{84 (2+3 x)^4}+\frac{45}{28} \int \frac{\sqrt{1-2 x}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac{5 \sqrt{1-2 x}}{28 (2+3 x)^3}-\frac{5}{28} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac{5 \sqrt{1-2 x}}{28 (2+3 x)^3}+\frac{5 \sqrt{1-2 x}}{392 (2+3 x)^2}-\frac{15}{392} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac{5 \sqrt{1-2 x}}{28 (2+3 x)^3}+\frac{5 \sqrt{1-2 x}}{392 (2+3 x)^2}+\frac{15 \sqrt{1-2 x}}{2744 (2+3 x)}-\frac{15 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2744}\\ &=\frac{(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac{5 \sqrt{1-2 x}}{28 (2+3 x)^3}+\frac{5 \sqrt{1-2 x}}{392 (2+3 x)^2}+\frac{15 \sqrt{1-2 x}}{2744 (2+3 x)}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2744}\\ &=\frac{(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac{5 \sqrt{1-2 x}}{28 (2+3 x)^3}+\frac{5 \sqrt{1-2 x}}{392 (2+3 x)^2}+\frac{15 \sqrt{1-2 x}}{2744 (2+3 x)}+\frac{5 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372}\\ \end{align*}

Mathematica [C]  time = 0.0146884, size = 42, normalized size = 0.38 \[ \frac{(1-2 x)^{3/2} \left (\frac{2401}{(3 x+2)^4}-720 \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{201684} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(3/2)*(2401/(2 + 3*x)^4 - 720*Hypergeometric2F1[3/2, 4, 5/2, 3/7 - (6*x)/7]))/201684

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Maple [A]  time = 0.01, size = 66, normalized size = 0.6 \begin{align*} -1296\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{5\, \left ( 1-2\,x \right ) ^{7/2}}{21952}}-{\frac{55\, \left ( 1-2\,x \right ) ^{5/2}}{28224}}+{\frac{209\, \left ( 1-2\,x \right ) ^{3/2}}{108864}}+{\frac{5\,\sqrt{1-2\,x}}{1728}} \right ) }+{\frac{5\,\sqrt{21}}{9604}{\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)^5,x)

[Out]

-1296*(5/21952*(1-2*x)^(7/2)-55/28224*(1-2*x)^(5/2)+209/108864*(1-2*x)^(3/2)+5/1728*(1-2*x)^(1/2))/(-6*x-4)^4+
5/9604*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 3.28501, size = 149, normalized size = 1.35 \begin{align*} -\frac{5}{19208} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1215 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 10395 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 10241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 15435 \, \sqrt{-2 \, x + 1}}{4116 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\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)^5,x, algorithm="maxima")

[Out]

-5/19208*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/4116*(1215*(-2*x + 1)^
(7/2) - 10395*(-2*x + 1)^(5/2) + 10241*(-2*x + 1)^(3/2) + 15435*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1
)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.55162, size = 308, normalized size = 2.8 \begin{align*} \frac{15 \, \sqrt{7} \sqrt{3}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 7 \,{\left (1215 \, x^{3} + 3375 \, x^{2} - 1726 \, x - 2062\right )} \sqrt{-2 \, x + 1}}{57624 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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)^5,x, algorithm="fricas")

[Out]

1/57624*(15*sqrt(7)*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*
x + 5)/(3*x + 2)) + 7*(1215*x^3 + 3375*x^2 - 1726*x - 2062)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x
+ 16)

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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)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.94612, size = 135, normalized size = 1.23 \begin{align*} -\frac{5}{19208} \, \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{1215 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 10395 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 10241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 15435 \, \sqrt{-2 \, x + 1}}{65856 \,{\left (3 \, x + 2\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

-5/19208*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/65856*(1215*(
2*x - 1)^3*sqrt(-2*x + 1) + 10395*(2*x - 1)^2*sqrt(-2*x + 1) - 10241*(-2*x + 1)^(3/2) - 15435*sqrt(-2*x + 1))/
(3*x + 2)^4