3.2080 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=101 \[ \frac{10}{147 \sqrt{1-2 x}}-\frac{5}{63 \sqrt{1-2 x} (3 x+2)}-\frac{1}{9 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{63 \sqrt{1-2 x} (3 x+2)^3}-\frac{10 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

[Out]

10/(147*Sqrt[1 - 2*x]) + 1/(63*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 1/(9*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 5/(63*Sqrt[1 - 2
*x]*(2 + 3*x)) - (10*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21])

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Rubi [A]  time = 0.0280667, antiderivative size = 108, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{5 \sqrt{1-2 x}}{49 (3 x+2)}-\frac{5 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{4}{9 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{63 \sqrt{1-2 x} (3 x+2)^3}-\frac{10 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

1/(63*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 4/(9*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[1 - 2*x])/(21*(2 + 3*x)^2) - (5*S
qrt[1 - 2*x])/(49*(2 + 3*x)) - (10*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*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 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)^{3/2} (2+3 x)^4} \, dx &=\frac{1}{63 \sqrt{1-2 x} (2+3 x)^3}+\frac{14}{9} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac{1}{63 \sqrt{1-2 x} (2+3 x)^3}+\frac{4}{9 \sqrt{1-2 x} (2+3 x)^2}+\frac{10}{3} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{1}{63 \sqrt{1-2 x} (2+3 x)^3}+\frac{4}{9 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{21 (2+3 x)^2}+\frac{5}{7} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{1}{63 \sqrt{1-2 x} (2+3 x)^3}+\frac{4}{9 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{21 (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{49 (2+3 x)}+\frac{5}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1}{63 \sqrt{1-2 x} (2+3 x)^3}+\frac{4}{9 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{21 (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{49 (2+3 x)}-\frac{5}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{1}{63 \sqrt{1-2 x} (2+3 x)^3}+\frac{4}{9 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{21 (2+3 x)^2}-\frac{5 \sqrt{1-2 x}}{49 (2+3 x)}-\frac{10 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0155165, size = 42, normalized size = 0.42 \[ \frac{16 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+\frac{7}{(3 x+2)^3}}{441 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(7/(2 + 3*x)^3 + 16*Hypergeometric2F1[-1/2, 3, 1/2, 3/7 - (6*x)/7])/(441*Sqrt[1 - 2*x])

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Maple [A]  time = 0.012, size = 66, normalized size = 0.7 \begin{align*}{\frac{216}{2401\, \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{113}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{1351}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{7007}{108}\sqrt{1-2\,x}} \right ) }-{\frac{10\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{88}{2401}{\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)^(3/2)/(2+3*x)^4,x)

[Out]

216/2401*(113/12*(1-2*x)^(5/2)-1351/27*(1-2*x)^(3/2)+7007/108*(1-2*x)^(1/2))/(-6*x-4)^3-10/1029*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)+88/2401/(1-2*x)^(1/2)

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Maxima [A]  time = 1.7402, size = 136, normalized size = 1.35 \begin{align*} \frac{5}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 280 \,{\left (2 \, x - 1\right )}^{2} + 1078 \, x - 231\right )}}{49 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

5/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/49*(45*(2*x - 1)^3 + 280
*(2*x - 1)^2 + 1078*x - 231)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 343*sqrt(-2*
x + 1))

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Fricas [A]  time = 1.68988, size = 267, normalized size = 2.64 \begin{align*} \frac{5 \, \sqrt{21}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (90 \, x^{3} + 145 \, x^{2} + 57 \, x + 1\right )} \sqrt{-2 \, x + 1}}{1029 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/1029*(5*sqrt(21)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) -
21*(90*x^3 + 145*x^2 + 57*x + 1)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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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)**(3/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.3396, size = 126, normalized size = 1.25 \begin{align*} \frac{5}{1029} \, \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{88}{2401 \, \sqrt{-2 \, x + 1}} - \frac{1017 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 5404 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 7007 \, \sqrt{-2 \, x + 1}}{9604 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

5/1029*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 88/2401/sqrt(-2*x
+ 1) - 1/9604*(1017*(2*x - 1)^2*sqrt(-2*x + 1) - 5404*(-2*x + 1)^(3/2) + 7007*sqrt(-2*x + 1))/(3*x + 2)^3