3.2141 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2}{49 \sqrt{1-2 x}}+\frac{11}{21 (1-2 x)^{3/2}}+\frac{2}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

11/(21*(1 - 2*x)^(3/2)) - 2/(49*Sqrt[1 - 2*x]) + (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49

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Rubi [A]  time = 0.0129129, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{2}{49 \sqrt{1-2 x}}+\frac{11}{21 (1-2 x)^{3/2}}+\frac{2}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

11/(21*(1 - 2*x)^(3/2)) - 2/(49*Sqrt[1 - 2*x]) + (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49

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)} \, dx &=\frac{11}{21 (1-2 x)^{3/2}}-\frac{1}{7} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac{11}{21 (1-2 x)^{3/2}}-\frac{2}{49 \sqrt{1-2 x}}-\frac{3}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{11}{21 (1-2 x)^{3/2}}-\frac{2}{49 \sqrt{1-2 x}}+\frac{3}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{11}{21 (1-2 x)^{3/2}}-\frac{2}{49 \sqrt{1-2 x}}+\frac{2}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0114623, size = 38, normalized size = 0.68 \[ -\frac{(6-12 x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-77}{147 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-77 + (6 - 12*x)*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7])/(147*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.008, size = 38, normalized size = 0.7 \begin{align*}{\frac{11}{21} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2}{49}{\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),x)

[Out]

11/21/(1-2*x)^(3/2)+2/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2/49/(1-2*x)^(1/2)

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Maxima [A]  time = 2.90141, size = 69, normalized size = 1.23 \begin{align*} -\frac{1}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12 \, x + 71}{147 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \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),x, algorithm="maxima")

[Out]

-1/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/147*(12*x + 71)/(-2*x +
1)^(3/2)

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Fricas [B]  time = 1.6084, size = 208, normalized size = 3.71 \begin{align*} \frac{3 \, \sqrt{7} \sqrt{3}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 7 \,{\left (12 \, x + 71\right )} \sqrt{-2 \, x + 1}}{1029 \,{\left (4 \, x^{2} - 4 \, x + 1\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),x, algorithm="fricas")

[Out]

1/1029*(3*sqrt(7)*sqrt(3)*(4*x^2 - 4*x + 1)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 7*(12
*x + 71)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 15.4183, size = 90, normalized size = 1.61 \begin{align*} - \frac{6 \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 )}{49} - \frac{2}{49 \sqrt{1 - 2 x}} + \frac{11}{21 \left (1 - 2 x\right )^{\frac{3}{2}}} \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),x)

[Out]

-6*Piecewise((-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))/49 - 2/(49*sqrt(1 - 2*x)) + 11/(21*(1 - 2*x)**(3/2))

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Giac [A]  time = 2.14231, size = 82, normalized size = 1.46 \begin{align*} -\frac{1}{343} \, \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{12 \, x + 71}{147 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \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),x, algorithm="giac")

[Out]

-1/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/147*(12*x + 71)
/((2*x - 1)*sqrt(-2*x + 1))