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

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

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Rubi [A]  time = 0.04, 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} \begin {gather*} -\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}} \end {gather*}

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 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 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 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*}

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Mathematica [C]  time = 0.02, size = 42, normalized size = 0.35 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.30, size = 88, normalized size = 0.73 \begin {gather*} \frac {17685 (1-2 x)^4-151305 (1-2 x)^3+468587 (1-2 x)^2-596967 (1-2 x)+241472}{9604 (3 (1-2 x)-7)^4 \sqrt {1-2 x}}-\frac {655 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(241472 - 596967*(1 - 2*x) + 468587*(1 - 2*x)^2 - 151305*(1 - 2*x)^3 + 17685*(1 - 2*x)^4)/(9604*(-7 + 3*(1 - 2
*x))^4*Sqrt[1 - 2*x]) - (655*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9604*Sqrt[21])

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fricas [A]  time = 1.38, size = 114, normalized size = 0.94 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.27, size = 109, normalized size = 0.90 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.02, size = 75, normalized size = 0.62 \begin {gather*} -\frac {655 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{201684}+\frac {176}{16807 \sqrt {-2 x +1}}+\frac {\frac {66771 \left (-2 x +1\right )^{\frac {7}{2}}}{67228}-\frac {75273 \left (-2 x +1\right )^{\frac {5}{2}}}{9604}+\frac {28925 \left (-2 x +1\right )^{\frac {3}{2}}}{1372}-\frac {3735 \sqrt {-2 x +1}}{196}}{\left (-6 x -4\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.50, size = 119, normalized size = 0.98 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.26, size = 98, normalized size = 0.81 \begin {gather*} \frac {\frac {4061\,x}{2646}+\frac {9563\,{\left (2\,x-1\right )}^2}{15876}+\frac {7205\,{\left (2\,x-1\right )}^3}{37044}+\frac {655\,{\left (2\,x-1\right )}^4}{28812}-\frac {7255}{15876}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {655\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{201684} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((4061*x)/2646 + (9563*(2*x - 1)^2)/15876 + (7205*(2*x - 1)^3)/37044 + (655*(2*x - 1)^4)/28812 - 7255/15876)/(
(2401*(1 - 2*x)^(1/2))/81 - (1372*(1 - 2*x)^(3/2))/27 + (98*(1 - 2*x)^(5/2))/3 - (28*(1 - 2*x)^(7/2))/3 + (1 -
2*x)^(9/2)) - (655*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/201684

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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