3.1969 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=108 \[ -\frac {27}{220} (1-2 x)^{11/2}+\frac {18}{25} (1-2 x)^{9/2}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {2 (1-2 x)^{5/2}}{3125}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {242 \sqrt {1-2 x}}{15625}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]

[Out]

22/9375*(1-2*x)^(3/2)+2/3125*(1-2*x)^(5/2)-3897/3500*(1-2*x)^(7/2)+18/25*(1-2*x)^(9/2)-27/220*(1-2*x)^(11/2)-2
42/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/15625*(1-2*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ -\frac {27}{220} (1-2 x)^{11/2}+\frac {18}{25} (1-2 x)^{9/2}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {2 (1-2 x)^{5/2}}{3125}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {242 \sqrt {1-2 x}}{15625}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(242*Sqrt[1 - 2*x])/15625 + (22*(1 - 2*x)^(3/2))/9375 + (2*(1 - 2*x)^(5/2))/3125 - (3897*(1 - 2*x)^(7/2))/3500
+ (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(11/2))/220 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15
625

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -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} (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac {3897}{500} (1-2 x)^{5/2}-\frac {162}{25} (1-2 x)^{7/2}+\frac {27}{20} (1-2 x)^{9/2}+\frac {(1-2 x)^{5/2}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {1}{125} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {11}{625} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {121 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}+\frac {1331 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}-\frac {1331 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625}\\ &=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}-\frac {242 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 66, normalized size = 0.61 \[ \frac {5 \sqrt {1-2 x} \left (14175000 x^5+6142500 x^4-15572250 x^3-3564885 x^2+7726195 x-1796318\right )-55902 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18046875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(-1796318 + 7726195*x - 3564885*x^2 - 15572250*x^3 + 6142500*x^4 + 14175000*x^5) - 55902*Sqrt
[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18046875

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fricas [A]  time = 0.95, size = 71, normalized size = 0.66 \[ \frac {121}{78125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{3609375} \, {\left (14175000 \, x^{5} + 6142500 \, x^{4} - 15572250 \, x^{3} - 3564885 \, x^{2} + 7726195 \, x - 1796318\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/78125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1/3609375*(14175000*x^
5 + 6142500*x^4 - 15572250*x^3 - 3564885*x^2 + 7726195*x - 1796318)*sqrt(-2*x + 1)

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giac [A]  time = 1.01, size = 122, normalized size = 1.13 \[ \frac {27}{220} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {18}{25} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3897}{3500} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{3125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{15625} \, \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/220*(2*x - 1)^5*sqrt(-2*x + 1) + 18/25*(2*x - 1)^4*sqrt(-2*x + 1) + 3897/3500*(2*x - 1)^3*sqrt(-2*x + 1) +
2/3125*(2*x - 1)^2*sqrt(-2*x + 1) + 22/9375*(-2*x + 1)^(3/2) + 121/78125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10
*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/15625*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 74, normalized size = 0.69 \[ -\frac {242 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{78125}+\frac {22 \left (-2 x +1\right )^{\frac {3}{2}}}{9375}+\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{3125}-\frac {3897 \left (-2 x +1\right )^{\frac {7}{2}}}{3500}+\frac {18 \left (-2 x +1\right )^{\frac {9}{2}}}{25}-\frac {27 \left (-2 x +1\right )^{\frac {11}{2}}}{220}+\frac {242 \sqrt {-2 x +1}}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22/9375*(-2*x+1)^(3/2)+2/3125*(-2*x+1)^(5/2)-3897/3500*(-2*x+1)^(7/2)+18/25*(-2*x+1)^(9/2)-27/220*(-2*x+1)^(11
/2)-242/78125*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+242/15625*(-2*x+1)^(1/2)

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maxima [A]  time = 1.21, size = 91, normalized size = 0.84 \[ -\frac {27}{220} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {18}{25} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3897}{3500} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{15625} \, \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/220*(-2*x + 1)^(11/2) + 18/25*(-2*x + 1)^(9/2) - 3897/3500*(-2*x + 1)^(7/2) + 2/3125*(-2*x + 1)^(5/2) + 22
/9375*(-2*x + 1)^(3/2) + 121/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 242/15625*sqrt(-2*x + 1)

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mupad [B]  time = 0.06, size = 75, normalized size = 0.69 \[ \frac {242\,\sqrt {1-2\,x}}{15625}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{9375}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{3125}-\frac {3897\,{\left (1-2\,x\right )}^{7/2}}{3500}+\frac {18\,{\left (1-2\,x\right )}^{9/2}}{25}-\frac {27\,{\left (1-2\,x\right )}^{11/2}}{220}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{78125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*242i)/78125 + (242*(1 - 2*x)^(1/2))/15625 + (22*(1 - 2*x)^(3/
2))/9375 + (2*(1 - 2*x)^(5/2))/3125 - (3897*(1 - 2*x)^(7/2))/3500 + (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(1
1/2))/220

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sympy [A]  time = 76.95, size = 138, normalized size = 1.28 \[ - \frac {27 \left (1 - 2 x\right )^{\frac {11}{2}}}{220} + \frac {18 \left (1 - 2 x\right )^{\frac {9}{2}}}{25} - \frac {3897 \left (1 - 2 x\right )^{\frac {7}{2}}}{3500} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{3125} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {242 \sqrt {1 - 2 x}}{15625} + \frac {2662 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*(1 - 2*x)**(11/2)/220 + 18*(1 - 2*x)**(9/2)/25 - 3897*(1 - 2*x)**(7/2)/3500 + 2*(1 - 2*x)**(5/2)/3125 + 22
*(1 - 2*x)**(3/2)/9375 + 242*sqrt(1 - 2*x)/15625 + 2662*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/
55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 > -11/5))/15625

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