3.1952 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac {347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac {8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (3 x+2)}+\frac {8051 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]

[Out]

-1/315*(1-2*x)^(7/2)/(2+3*x)^5+347/8820*(1-2*x)^(7/2)/(2+3*x)^4-8051/26460*(1-2*x)^(5/2)/(2+3*x)^3+8051/31752*
(1-2*x)^(3/2)/(2+3*x)^2+8051/333396*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-8051/31752*(1-2*x)^(1/2)/(2+3
*x)

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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, 47, 63, 206} \[ \frac {347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac {(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac {8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (3 x+2)}+\frac {8051 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - 2*x)^(7/2)/(315*(2 + 3*x)^5) + (347*(1 - 2*x)^(7/2))/(8820*(2 + 3*x)^4) - (8051*(1 - 2*x)^(5/2))/(26460*
(2 + 3*x)^3) + (8051*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)^2) - (8051*Sqrt[1 - 2*x])/(31752*(2 + 3*x)) + (8051*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(15876*Sqrt[21])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {(1-2 x)^{5/2} (1403+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}+\frac {8051 \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4} \, dx}{2940}\\ &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}-\frac {8051 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {8051 \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {8051 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{31752}\\ &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {8051 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31752}\\ &=-\frac {(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac {347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac {8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac {8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac {8051 \sqrt {1-2 x}}{31752 (2+3 x)}+\frac {8051 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 84, normalized size = 0.66 \[ -\frac {80510 (3 x+2)^5 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )-21 \left (14646690 x^5+17489565 x^4+4147953 x^3-2438512 x^2-1912794 x-503276\right )}{3333960 \sqrt {1-2 x} (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3333960*(-21*(-503276 - 1912794*x - 2438512*x^2 + 4147953*x^3 + 17489565*x^4 + 14646690*x^5) + 80510*(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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fricas [A]  time = 0.93, size = 115, normalized size = 0.90 \[ \frac {40255 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (7323345 \, x^{4} + 12406455 \, x^{3} + 8277204 \, x^{2} + 2919346 \, x + 503276\right )} \sqrt {-2 \, x + 1}}{3333960 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3333960*(40255*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*(7323345*x^4 + 12406455*x^3 + 8277204*x^2 + 2919346*x + 503276)*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 = 0.87, size = 116, normalized size = 0.91 \[ -\frac {8051}{666792} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7323345 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 54106290 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 151487616 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{2540160 \, {\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8051/666792*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/2540160*(
7323345*(2*x - 1)^4*sqrt(-2*x + 1) + 54106290*(2*x - 1)^3*sqrt(-2*x + 1) + 151487616*(2*x - 1)^2*sqrt(-2*x + 1
) - 193304510*(-2*x + 1)^(3/2) + 96652255*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.01, size = 75, normalized size = 0.59 \[ \frac {8051 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{333396}-\frac {3888 \left (-\frac {54247 \left (-2 x +1\right )^{\frac {9}{2}}}{2286144}+\frac {12269 \left (-2 x +1\right )^{\frac {7}{2}}}{69984}-\frac {16102 \left (-2 x +1\right )^{\frac {5}{2}}}{32805}+\frac {394499 \left (-2 x +1\right )^{\frac {3}{2}}}{629856}-\frac {394499 \sqrt {-2 x +1}}{1259712}\right )}{\left (-6 x -4\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3888*(-54247/2286144*(-2*x+1)^(9/2)+12269/69984*(-2*x+1)^(7/2)-16102/32805*(-2*x+1)^(5/2)+394499/629856*(-2*x
+1)^(3/2)-394499/1259712*(-2*x+1)^(1/2))/(-6*x-4)^5+8051/333396*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.26, size = 128, normalized size = 1.00 \[ -\frac {8051}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7323345 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 54106290 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 151487616 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 193304510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 96652255 \, \sqrt {-2 \, x + 1}}{79380 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8051/666792*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/79380*(7323345*(-2
*x + 1)^(9/2) - 54106290*(-2*x + 1)^(7/2) + 151487616*(-2*x + 1)^(5/2) - 193304510*(-2*x + 1)^(3/2) + 96652255
*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 = 0.07, size = 108, normalized size = 0.84 \[ \frac {8051\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396}-\frac {\frac {394499\,\sqrt {1-2\,x}}{78732}-\frac {394499\,{\left (1-2\,x\right )}^{3/2}}{39366}+\frac {257632\,{\left (1-2\,x\right )}^{5/2}}{32805}-\frac {12269\,{\left (1-2\,x\right )}^{7/2}}{4374}+\frac {54247\,{\left (1-2\,x\right )}^{9/2}}{142884}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8051*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/333396 - ((394499*(1 - 2*x)^(1/2))/78732 - (394499*(1 - 2*
x)^(3/2))/39366 + (257632*(1 - 2*x)^(5/2))/32805 - (12269*(1 - 2*x)^(7/2))/4374 + (54247*(1 - 2*x)^(9/2))/1428
84)/((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 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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