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

Optimal. Leaf size=128 \[ -\frac {9145 \sqrt {1-2 x}}{57624 (3 x+2)}-\frac {9145 \sqrt {1-2 x}}{24696 (3 x+2)^2}-\frac {1829 \sqrt {1-2 x}}{1764 (3 x+2)^3}-\frac {2179 \sqrt {1-2 x}}{588 (3 x+2)^4}+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^4}-\frac {9145 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \]

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Rubi [A]  time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \begin {gather*} -\frac {9145 \sqrt {1-2 x}}{57624 (3 x+2)}-\frac {9145 \sqrt {1-2 x}}{24696 (3 x+2)^2}-\frac {1829 \sqrt {1-2 x}}{1764 (3 x+2)^3}-\frac {2179 \sqrt {1-2 x}}{588 (3 x+2)^4}+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^4}-\frac {9145 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

121/(14*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (2179*Sqrt[1 - 2*x])/(588*(2 + 3*x)^4) - (1829*Sqrt[1 - 2*x])/(1764*(2 +
3*x)^3) - (9145*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2) - (9145*Sqrt[1 - 2*x])/(57624*(2 + 3*x)) - (9145*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/(28812*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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{14} \int \frac {-1336+175 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}+\frac {1829}{84} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}+\frac {9145 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1764}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}+\frac {9145 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{8232}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {9145 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{57624}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}-\frac {9145 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{57624}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}-\frac {9145 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 60, normalized size = 0.47 \begin {gather*} \frac {29264 (2 x-1) (3 x+2)^4 \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )+747397 (2 x-1)+1743126}{201684 \sqrt {1-2 x} (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1743126 + 747397*(-1 + 2*x) + 29264*(-1 + 2*x)*(2 + 3*x)^4*Hypergeometric2F1[1/2, 4, 3/2, 3/7 - (6*x)/7])/(20
1684*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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IntegrateAlgebraic [A]  time = 0.30, size = 88, normalized size = 0.69 \begin {gather*} \frac {246915 (1-2 x)^4-2112495 (1-2 x)^3+6542333 (1-2 x)^2-8609153 (1-2 x)+3984288}{28812 (3 (1-2 x)-7)^4 \sqrt {1-2 x}}-\frac {9145 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3984288 - 8609153*(1 - 2*x) + 6542333*(1 - 2*x)^2 - 2112495*(1 - 2*x)^3 + 246915*(1 - 2*x)^4)/(28812*(-7 + 3*
(1 - 2*x))^4*Sqrt[1 - 2*x]) - (9145*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(28812*Sqrt[21])

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fricas [A]  time = 1.48, size = 114, normalized size = 0.89 \begin {gather*} \frac {9145 \, \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 (493830 \, x^{4} + 1124835 \, x^{3} + 843169 \, x^{2} + 218578 \, x + 6486\right )} \sqrt {-2 \, x + 1}}{1210104 \, {\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)^2/(1-2*x)^(3/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/1210104*(9145*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*(493830*x^4 + 1124835*x^3 + 843169*x^2 + 218578*x + 6486)*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 = 1.33, size = 109, normalized size = 0.85 \begin {gather*} \frac {9145}{1210104} \, \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 {968}{16807 \, \sqrt {-2 \, x + 1}} - \frac {787509 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 6005769 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 15060395 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 12452615 \, \sqrt {-2 \, x + 1}}{3226944 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9145/1210104*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 968/16807/s
qrt(-2*x + 1) - 1/3226944*(787509*(2*x - 1)^3*sqrt(-2*x + 1) + 6005769*(2*x - 1)^2*sqrt(-2*x + 1) - 15060395*(
-2*x + 1)^(3/2) + 12452615*sqrt(-2*x + 1))/(3*x + 2)^4

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maple [A]  time = 0.01, size = 75, normalized size = 0.59 \begin {gather*} -\frac {9145 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{605052}+\frac {968}{16807 \sqrt {-2 x +1}}+\frac {\frac {262503 \left (-2 x +1\right )^{\frac {7}{2}}}{67228}-\frac {285989 \left (-2 x +1\right )^{\frac {5}{2}}}{9604}+\frac {307355 \left (-2 x +1\right )^{\frac {3}{2}}}{4116}-\frac {36305 \sqrt {-2 x +1}}{588}}{\left (-6 x -4\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

968/16807/(-2*x+1)^(1/2)+648/16807*(29167/288*(-2*x+1)^(7/2)-2001923/2592*(-2*x+1)^(5/2)[phone]/7776*(-2*x+1
)^(3/2)-12452615/7776*(-2*x+1)^(1/2))/(-6*x-4)^4-9145/605052*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.23, size = 119, normalized size = 0.93 \begin {gather*} \frac {9145}{1210104} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {246915 \, {\left (2 \, x - 1\right )}^{4} + 2112495 \, {\left (2 \, x - 1\right )}^{3} + 6542333 \, {\left (2 \, x - 1\right )}^{2} + 17218306 \, x - 4624865}{28812 \, {\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)^2/(1-2*x)^(3/2)/(2+3*x)^5,x, algorithm="maxima")

[Out]

9145/1210104*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/28812*(246915*(2*x
- 1)^4 + 2112495*(2*x - 1)^3 + 6542333*(2*x - 1)^2 + 17218306*x - 4624865)/(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 = 0.08, size = 98, normalized size = 0.77 \begin {gather*} \frac {\frac {175697\,x}{23814}+\frac {133517\,{\left (2\,x-1\right )}^2}{47628}+\frac {100595\,{\left (2\,x-1\right )}^3}{111132}+\frac {9145\,{\left (2\,x-1\right )}^4}{86436}-\frac {94385}{47628}}{\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 {9145\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{605052} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((175697*x)/23814 + (133517*(2*x - 1)^2)/47628 + (100595*(2*x - 1)^3)/111132 + (9145*(2*x - 1)^4)/86436 - 9438
5/47628)/((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)) - (9145*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/605052

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

[Out]

Timed out

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