3.2081 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=121 \[ \frac{655}{28812 \sqrt{1-2 x}}-\frac{655}{24696 \sqrt{1-2 x} (3 x+2)}-\frac{131}{3528 \sqrt{1-2 x} (3 x+2)^2}-\frac{131}{1764 \sqrt{1-2 x} (3 x+2)^3}+\frac{1}{84 \sqrt{1-2 x} (3 x+2)^4}-\frac{655 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604 \sqrt{21}} \]

[Out]

655/(28812*Sqrt[1 - 2*x]) + 1/(84*Sqrt[1 - 2*x]*(2 + 3*x)^4) - 131/(1764*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 131/(352
8*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 655/(24696*Sqrt[1 - 2*x]*(2 + 3*x)) - (655*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9
604*Sqrt[21])

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Rubi [A]  time = 0.038164, antiderivative size = 128, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{655 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{655 \sqrt{1-2 x}}{8232 (3 x+2)^2}-\frac{131 \sqrt{1-2 x}}{588 (3 x+2)^3}+\frac{131}{294 \sqrt{1-2 x} (3 x+2)^3}+\frac{1}{84 \sqrt{1-2 x} (3 x+2)^4}-\frac{655 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(84*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 131/(294*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (131*Sqrt[1 - 2*x])/(588*(2 + 3*x)^3)
- (655*Sqrt[1 - 2*x])/(8232*(2 + 3*x)^2) - (655*Sqrt[1 - 2*x])/(19208*(2 + 3*x)) - (655*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/(9604*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)^5} \, dx &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{84} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{294 \sqrt{1-2 x} (2+3 x)^3}+\frac{131}{28} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{294 \sqrt{1-2 x} (2+3 x)^3}-\frac{131 \sqrt{1-2 x}}{588 (2+3 x)^3}+\frac{655}{588} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{294 \sqrt{1-2 x} (2+3 x)^3}-\frac{131 \sqrt{1-2 x}}{588 (2+3 x)^3}-\frac{655 \sqrt{1-2 x}}{8232 (2+3 x)^2}+\frac{655 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{2744}\\ &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{294 \sqrt{1-2 x} (2+3 x)^3}-\frac{131 \sqrt{1-2 x}}{588 (2+3 x)^3}-\frac{655 \sqrt{1-2 x}}{8232 (2+3 x)^2}-\frac{655 \sqrt{1-2 x}}{19208 (2+3 x)}+\frac{655 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{19208}\\ &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{294 \sqrt{1-2 x} (2+3 x)^3}-\frac{131 \sqrt{1-2 x}}{588 (2+3 x)^3}-\frac{655 \sqrt{1-2 x}}{8232 (2+3 x)^2}-\frac{655 \sqrt{1-2 x}}{19208 (2+3 x)}-\frac{655 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{19208}\\ &=\frac{1}{84 \sqrt{1-2 x} (2+3 x)^4}+\frac{131}{294 \sqrt{1-2 x} (2+3 x)^3}-\frac{131 \sqrt{1-2 x}}{588 (2+3 x)^3}-\frac{655 \sqrt{1-2 x}}{8232 (2+3 x)^2}-\frac{655 \sqrt{1-2 x}}{19208 (2+3 x)}-\frac{655 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0164584, size = 42, normalized size = 0.35 \[ \frac{2096 \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+\frac{2401}{(3 x+2)^4}}{201684 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2401/(2 + 3*x)^4 + 2096*Hypergeometric2F1[-1/2, 4, 1/2, 3/7 - (6*x)/7])/(201684*Sqrt[1 - 2*x])

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Maple [A]  time = 0.012, size = 75, normalized size = 0.6 \begin{align*}{\frac{1296}{16807\, \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{2473}{192} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{175637}{1728} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{1417325}{5184} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{142345}{576}\sqrt{1-2\,x}} \right ) }-{\frac{655\,\sqrt{21}}{201684}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{176}{16807}{\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)^5,x)

[Out]

1296/16807*(2473/192*(1-2*x)^(7/2)-175637/1728*(1-2*x)^(5/2)+1417325/5184*(1-2*x)^(3/2)-142345/576*(1-2*x)^(1/
2))/(-6*x-4)^4-655/201684*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+176/16807/(1-2*x)^(1/2)

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Maxima [A]  time = 2.91285, size = 161, normalized size = 1.33 \begin{align*} \frac{655}{403368} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17685 \,{\left (2 \, x - 1\right )}^{4} + 151305 \,{\left (2 \, x - 1\right )}^{3} + 468587 \,{\left (2 \, x - 1\right )}^{2} + 1193934 \, x - 355495}{9604 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2401 \, \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)^5,x, algorithm="maxima")

[Out]

655/403368*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/9604*(17685*(2*x - 1
)^4 + 151305*(2*x - 1)^3 + 468587*(2*x - 1)^2 + 1193934*x - 355495)/(81*(-2*x + 1)^(9/2) - 756*(-2*x + 1)^(7/2
) + 2646*(-2*x + 1)^(5/2) - 4116*(-2*x + 1)^(3/2) + 2401*sqrt(-2*x + 1))

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Fricas [A]  time = 1.70412, size = 339, normalized size = 2.8 \begin{align*} \frac{655 \, \sqrt{21}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (35370 \, x^{4} + 80565 \, x^{3} + 60391 \, x^{2} + 10742 \, x - 2566\right )} \sqrt{-2 \, x + 1}}{403368 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\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)^5,x, algorithm="fricas")

[Out]

1/403368*(655*sqrt(21)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) -
5)/(3*x + 2)) - 21*(35370*x^4 + 80565*x^3 + 60391*x^2 + 10742*x - 2566)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 +
216*x^3 - 24*x^2 - 64*x - 16)

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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)**5,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.50106, size = 147, normalized size = 1.21 \begin{align*} \frac{655}{403368} \, \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{176}{16807 \, \sqrt{-2 \, x + 1}} - \frac{66771 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 526911 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1417325 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1281105 \, \sqrt{-2 \, x + 1}}{1075648 \,{\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)^(3/2)/(2+3*x)^5,x, algorithm="giac")

[Out]

655/403368*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 176/16807/sqr
t(-2*x + 1) - 1/1075648*(66771*(2*x - 1)^3*sqrt(-2*x + 1) + 526911*(2*x - 1)^2*sqrt(-2*x + 1) - 1417325*(-2*x
+ 1)^(3/2) + 1281105*sqrt(-2*x + 1))/(3*x + 2)^4