3.19.26 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=161 \[ \frac {11 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac {11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {209 \sqrt {1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}} \]

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Rubi [A]  time = 0.06, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 12, 149, 146, 63, 206} \begin {gather*} \frac {11 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac {11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {209 \sqrt {1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

(-209*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(756*(2 + 3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(15*(2 + 3*x)^5) + (11*(1 -
2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^4) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)^3) - (11*Sqrt[1 - 2*x]
*(3911 + 6475*x))/(15876*(2 + 3*x)) - (146971*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7938*Sqrt[21])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 97

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

Rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}-\frac {11}{3} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11}{36} \int \frac {\sqrt {1-2 x} (3+5 x)^2 (24+18 x)}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11}{324} \int \frac {(-162-72 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \int \frac {(-9522-3330 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{13608}\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}+\frac {146971 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{15876}\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac {146971 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15876}\\ &=-\frac {209 \sqrt {1-2 x} (3+5 x)^2}{756 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}-\frac {11 \sqrt {1-2 x} (3911+6475 x)}{15876 (2+3 x)}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 89, normalized size = 0.55 \begin {gather*} -\frac {21 \left (52920000 x^6+226697490 x^5+288394965 x^4+106869513 x^3-43687652 x^2-40879074 x-7933096\right )-1469710 (3 x+2)^5 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )}{1666980 \sqrt {1-2 x} (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

-1/1666980*(21*(-7933096 - 40879074*x - 43687652*x^2 + 106869513*x^3 + 288394965*x^4 + 226697490*x^5 + 5292000
0*x^6) - 1469710*(2 + 3*x)^5*Sqrt[-21 + 42*x]*ArcTan[Sqrt[3/7]*Sqrt[-1 + 2*x]])/(Sqrt[1 - 2*x]*(2 + 3*x)^5)

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IntegrateAlgebraic [A]  time = 0.39, size = 97, normalized size = 0.60 \begin {gather*} \frac {\left (13230000 (1-2 x)^5-192728745 (1-2 x)^4+1053588690 (1-2 x)^3-2765406336 (1-2 x)^2+3528773710 (1-2 x)-1764386855\right ) \sqrt {1-2 x}}{39690 (3 (1-2 x)-7)^5}-\frac {146971 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7938 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

((-1764386855 + 3528773710*(1 - 2*x) - 2765406336*(1 - 2*x)^2 + 1053588690*(1 - 2*x)^3 - 192728745*(1 - 2*x)^4
+ 13230000*(1 - 2*x)^5)*Sqrt[1 - 2*x])/(39690*(-7 + 3*(1 - 2*x))^5) - (146971*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/(7938*Sqrt[21])

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fricas [A]  time = 1.31, size = 119, normalized size = 0.74 \begin {gather*} \frac {734855 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (26460000 \, x^{5} + 126578745 \, x^{4} + 207486855 \, x^{3} + 157178184 \, x^{2} + 56745266 \, x + 7933096\right )} \sqrt {-2 \, x + 1}}{1666980 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/1666980*(734855*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x
+ 1) - 5)/(3*x + 2)) + 21*(26460000*x^5 + 126578745*x^4 + 207486855*x^3 + 157178184*x^2 + 56745266*x + 793309
6)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 0.98, size = 125, normalized size = 0.78 \begin {gather*} \frac {146971}{333396} \, \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 {1000}{729} \, \sqrt {-2 \, x + 1} + \frac {345408705 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 2999598210 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 9762357024 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 14111613390 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7644051695 \, \sqrt {-2 \, x + 1}}{11430720 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="giac")

[Out]

146971/333396*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1000/729*s
qrt(-2*x + 1) + 1/11430720*(345408705*(2*x - 1)^4*sqrt(-2*x + 1) + 2999598210*(2*x - 1)^3*sqrt(-2*x + 1) + 976
2357024*(2*x - 1)^2*sqrt(-2*x + 1) - 14111613390*(-2*x + 1)^(3/2) + 7644051695*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.01, size = 84, normalized size = 0.52 \begin {gather*} -\frac {146971 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{166698}+\frac {1000 \sqrt {-2 x +1}}{729}+\frac {-\frac {284287 \left (-2 x +1\right )^{\frac {9}{2}}}{294}+\frac {226727 \left (-2 x +1\right )^{\frac {7}{2}}}{27}-\frac {11068432 \left (-2 x +1\right )^{\frac {5}{2}}}{405}+\frac {9599737 \left (-2 x +1\right )^{\frac {3}{2}}}{243}-\frac {31200211 \sqrt {-2 x +1}}{1458}}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^3/(3*x+2)^6,x)

[Out]

1000/729*(-2*x+1)^(1/2)+8/3*(-284287/784*(-2*x+1)^(9/2)+226727/72*(-2*x+1)^(7/2)-1383554/135*(-2*x+1)^(5/2)+95
99737/648*(-2*x+1)^(3/2)-31200211/3888*(-2*x+1)^(1/2))/(-6*x-4)^5-146971/166698*arctanh(1/7*21^(1/2)*(-2*x+1)^
(1/2))*21^(1/2)

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maxima [A]  time = 1.19, size = 137, normalized size = 0.85 \begin {gather*} \frac {146971}{333396} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1000}{729} \, \sqrt {-2 \, x + 1} + \frac {345408705 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 2999598210 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 9762357024 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 14111613390 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7644051695 \, \sqrt {-2 \, x + 1}}{357210 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="maxima")

[Out]

146971/333396*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1000/729*sqrt(-2*x
+ 1) + 1/357210*(345408705*(-2*x + 1)^(9/2) - 2999598210*(-2*x + 1)^(7/2) + 9762357024*(-2*x + 1)^(5/2) - 1411
1613390*(-2*x + 1)^(3/2) + 7644051695*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3
+ 30870*(2*x - 1)^2 + 72030*x - 19208)

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mupad [B]  time = 1.18, size = 116, normalized size = 0.72 \begin {gather*} \frac {1000\,\sqrt {1-2\,x}}{729}-\frac {146971\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{166698}+\frac {\frac {31200211\,\sqrt {1-2\,x}}{354294}-\frac {9599737\,{\left (1-2\,x\right )}^{3/2}}{59049}+\frac {11068432\,{\left (1-2\,x\right )}^{5/2}}{98415}-\frac {226727\,{\left (1-2\,x\right )}^{7/2}}{6561}+\frac {284287\,{\left (1-2\,x\right )}^{9/2}}{71442}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^3)/(3*x + 2)^6,x)

[Out]

(1000*(1 - 2*x)^(1/2))/729 - (146971*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/166698 + ((31200211*(1 - 2*
x)^(1/2))/354294 - (9599737*(1 - 2*x)^(3/2))/59049 + (11068432*(1 - 2*x)^(5/2))/98415 - (226727*(1 - 2*x)^(7/2
))/6561 + (284287*(1 - 2*x)^(9/2))/71442)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2
*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)

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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((1-2*x)**(5/2)*(3+5*x)**3/(2+3*x)**6,x)

[Out]

Timed out

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