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

Optimal. Leaf size=121 \[ -\frac {243 (1-2 x)^{13/2}}{1040}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {2 (1-2 x)^{3/2}}{46875}+\frac {22 \sqrt {1-2 x}}{78125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]

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Rubi [A]  time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \begin {gather*} -\frac {243 (1-2 x)^{13/2}}{1040}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {2 (1-2 x)^{3/2}}{46875}+\frac {22 \sqrt {1-2 x}}{78125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(22*Sqrt[1 - 2*x])/78125 + (2*(1 - 2*x)^(3/2))/46875 - (4774713*(1 - 2*x)^(5/2))/250000 + (806121*(1 - 2*x)^(7
/2))/35000 - (5673*(1 - 2*x)^(9/2))/500 + (5751*(1 - 2*x)^(11/2))/2200 - (243*(1 - 2*x)^(13/2))/1040 - (22*Sqr
t[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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)^{3/2} (2+3 x)^5}{3+5 x} \, dx &=\int \left (\frac {4774713 (1-2 x)^{3/2}}{50000}-\frac {806121 (1-2 x)^{5/2}}{5000}+\frac {51057}{500} (1-2 x)^{7/2}-\frac {5751}{200} (1-2 x)^{9/2}+\frac {243}{80} (1-2 x)^{11/2}+\frac {(1-2 x)^{3/2}}{3125 (3+5 x)}\right ) \, dx\\ &=-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}+\frac {\int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}+\frac {11 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {22 \sqrt {1-2 x}}{78125}+\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {22 \sqrt {1-2 x}}{78125}+\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}-\frac {121 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {22 \sqrt {1-2 x}}{78125}+\frac {2 (1-2 x)^{3/2}}{46875}-\frac {4774713 (1-2 x)^{5/2}}{250000}+\frac {806121 (1-2 x)^{7/2}}{35000}-\frac {5673}{500} (1-2 x)^{9/2}+\frac {5751 (1-2 x)^{11/2}}{2200}-\frac {243 (1-2 x)^{13/2}}{1040}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 71, normalized size = 0.59 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (3508312500 x^6+9100350000 x^5+6683000625 x^4-1659418875 x^3-4276774170 x^2-1321809935 x+1180568944\right )-66066 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1173046875} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x]*(1180568944 - 1321809935*x - 4276774170*x^2 - 1659418875*x^3 + 6683000625*x^4 + 9100350000*x
^5 + 3508312500*x^6) - 66066*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1173046875

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IntegrateAlgebraic [A]  time = 0.06, size = 112, normalized size = 0.93 \begin {gather*} \frac {-877078125 (1-2 x)^{13/2}[phone] (1-2 x)^{11/2}-42590047500 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-71692315695 (1-2 x)^{5/2}+160160 (1-2 x)^{3/2}+1057056 \sqrt {1-2 x}}{3753750000}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1057056*Sqrt[1 - 2*x] + 160160*(1 - 2*x)^(3/2) - 71692315695*(1 - 2*x)^(5/2) + 86456477250*(1 - 2*x)^(7/2) -
42590047500*(1 - 2*x)^(9/2) + 9812643750*(1 - 2*x)^(11/2) - 877078125*(1 - 2*x)^(13/2))/3753750000 - (22*Sqrt[
11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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fricas [A]  time = 1.53, size = 76, normalized size = 0.63 \begin {gather*} \frac {11}{390625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{234609375} \, {\left (3508312500 \, x^{6} + 9100350000 \, x^{5} + 6683000625 \, x^{4} - 1659418875 \, x^{3} - 4276774170 \, x^{2} - 1321809935 \, x + 1180568944\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/390625*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 1/234609375*(350831250
0*x^6 + 9100350000*x^5 + 6683000625*x^4 - 1659418875*x^3 - 4276774170*x^2 - 1321809935*x + 1180568944)*sqrt(-2
*x + 1)

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giac [A]  time = 1.12, size = 138, normalized size = 1.14 \begin {gather*} -\frac {243}{1040} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {5751}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {5673}{500} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {806121}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4774713}{250000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{390625} \, \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 {22}{78125} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/1040*(2*x - 1)^6*sqrt(-2*x + 1) - 5751/2200*(2*x - 1)^5*sqrt(-2*x + 1) - 5673/500*(2*x - 1)^4*sqrt(-2*x +
1) - 806121/35000*(2*x - 1)^3*sqrt(-2*x + 1) - 4774713/250000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/46875*(-2*x + 1)
^(3/2) + 11/390625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/7
8125*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 83, normalized size = 0.69 \begin {gather*} -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{390625}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{46875}-\frac {4774713 \left (-2 x +1\right )^{\frac {5}{2}}}{250000}+\frac {806121 \left (-2 x +1\right )^{\frac {7}{2}}}{35000}-\frac {5673 \left (-2 x +1\right )^{\frac {9}{2}}}{500}+\frac {5751 \left (-2 x +1\right )^{\frac {11}{2}}}{2200}-\frac {243 \left (-2 x +1\right )^{\frac {13}{2}}}{1040}+\frac {22 \sqrt {-2 x +1}}{78125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/46875*(-2*x+1)^(3/2)-4774713/250000*(-2*x+1)^(5/2)+806121/35000*(-2*x+1)^(7/2)-5673/500*(-2*x+1)^(9/2)+5751/
2200*(-2*x+1)^(11/2)-243/1040*(-2*x+1)^(13/2)-22/390625*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+22/7812
5*(-2*x+1)^(1/2)

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maxima [A]  time = 1.16, size = 100, normalized size = 0.83 \begin {gather*} -\frac {243}{1040} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {5751}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {5673}{500} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {806121}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4774713}{250000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{78125} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/1040*(-2*x + 1)^(13/2) + 5751/2200*(-2*x + 1)^(11/2) - 5673/500*(-2*x + 1)^(9/2) + 806121/35000*(-2*x + 1
)^(7/2) - 4774713/250000*(-2*x + 1)^(5/2) + 2/46875*(-2*x + 1)^(3/2) + 11/390625*sqrt(55)*log(-(sqrt(55) - 5*s
qrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/78125*sqrt(-2*x + 1)

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mupad [B]  time = 0.07, size = 84, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{78125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{46875}-\frac {4774713\,{\left (1-2\,x\right )}^{5/2}}{250000}+\frac {806121\,{\left (1-2\,x\right )}^{7/2}}{35000}-\frac {5673\,{\left (1-2\,x\right )}^{9/2}}{500}+\frac {5751\,{\left (1-2\,x\right )}^{11/2}}{2200}-\frac {243\,{\left (1-2\,x\right )}^{13/2}}{1040}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/390625 + (22*(1 - 2*x)^(1/2))/78125 + (2*(1 - 2*x)^(3/2)
)/46875 - (4774713*(1 - 2*x)^(5/2))/250000 + (806121*(1 - 2*x)^(7/2))/35000 - (5673*(1 - 2*x)^(9/2))/500 + (57
51*(1 - 2*x)^(11/2))/2200 - (243*(1 - 2*x)^(13/2))/1040

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sympy [A]  time = 107.20, size = 150, normalized size = 1.24 \begin {gather*} - \frac {243 \left (1 - 2 x\right )^{\frac {13}{2}}}{1040} + \frac {5751 \left (1 - 2 x\right )^{\frac {11}{2}}}{2200} - \frac {5673 \left (1 - 2 x\right )^{\frac {9}{2}}}{500} + \frac {806121 \left (1 - 2 x\right )^{\frac {7}{2}}}{35000} - \frac {4774713 \left (1 - 2 x\right )^{\frac {5}{2}}}{250000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{46875} + \frac {22 \sqrt {1 - 2 x}}{78125} + \frac {242 \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 )}{78125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*(1 - 2*x)**(13/2)/1040 + 5751*(1 - 2*x)**(11/2)/2200 - 5673*(1 - 2*x)**(9/2)/500 + 806121*(1 - 2*x)**(7/2
)/35000 - 4774713*(1 - 2*x)**(5/2)/250000 + 2*(1 - 2*x)**(3/2)/46875 + 22*sqrt(1 - 2*x)/78125 + 242*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))/78125

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