3.2142 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac{20}{147 (1-2 x)^{3/2}}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

20/(147*(1 - 2*x)^(3/2)) + 60/(343*Sqrt[1 - 2*x]) + 1/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (60*Sqrt[3/7]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/343

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Rubi [A]  time = 0.0194147, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ \frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac{20}{147 (1-2 x)^{3/2}}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

20/(147*(1 - 2*x)^(3/2)) + 60/(343*Sqrt[1 - 2*x]) + 1/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (60*Sqrt[3/7]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/343

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 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{3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx &=\frac{1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac{10}{7} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)} \, dx\\ &=\frac{20}{147 (1-2 x)^{3/2}}+\frac{1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac{30}{49} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac{20}{147 (1-2 x)^{3/2}}+\frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac{90}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{20}{147 (1-2 x)^{3/2}}+\frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{90}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{20}{147 (1-2 x)^{3/2}}+\frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0135683, size = 46, normalized size = 0.61 \[ -\frac{-20 (3 x+2) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-7}{147 (1-2 x)^{3/2} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-7 - 20*(2 + 3*x)*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7])/(147*(1 - 2*x)^(3/2)*(2 + 3*x))

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Maple [A]  time = 0.012, size = 54, normalized size = 0.7 \begin{align*} -{\frac{2}{343}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{60\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{22}{147} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{62}{343}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/343*(1-2*x)^(1/2)/(-2*x-4/3)-60/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+22/147/(1-2*x)^(3/2)+62/3
43/(1-2*x)^(1/2)

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Maxima [A]  time = 1.94715, size = 100, normalized size = 1.32 \begin{align*} \frac{30}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (270 \,{\left (2 \, x - 1\right )}^{2} + 840 \, x - 959\right )}}{1029 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/1029*(270*(2*x - 1)^2 +
840*x - 959)/(3*(-2*x + 1)^(5/2) - 7*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.59678, size = 250, normalized size = 3.29 \begin{align*} \frac{90 \, \sqrt{7} \sqrt{3}{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \,{\left (1080 \, x^{2} - 240 \, x - 689\right )} \sqrt{-2 \, x + 1}}{7203 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7203*(90*sqrt(7)*sqrt(3)*(12*x^3 - 4*x^2 - 5*x + 2)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)
) - 7*(1080*x^2 - 240*x - 689)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.07537, size = 104, normalized size = 1.37 \begin{align*} \frac{30}{2401} \, \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{4 \,{\left (93 \, x - 85\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{3 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/1029*(93*x - 8
5)/((2*x - 1)*sqrt(-2*x + 1)) + 3/343*sqrt(-2*x + 1)/(3*x + 2)