3.18.58 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=108 \[ \frac {81}{440} (1-2 x)^{11/2}-\frac {321}{200} (1-2 x)^{9/2}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {2 (1-2 x)^{3/2}}{9375}+\frac {22 \sqrt {1-2 x}}{15625}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]

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Rubi [A]  time = 0.03, antiderivative size = 108, 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 {81}{440} (1-2 x)^{11/2}-\frac {321}{200} (1-2 x)^{9/2}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {2 (1-2 x)^{3/2}}{9375}+\frac {22 \sqrt {1-2 x}}{15625}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(22*Sqrt[1 - 2*x])/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5/2))/25000 + (34371*(1 - 2*x)^(7/2))
/7000 - (321*(1 - 2*x)^(9/2))/200 + (81*(1 - 2*x)^(11/2))/440 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/15625

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)^4}{3+5 x} \, dx &=\int \left (\frac {136419 (1-2 x)^{3/2}}{5000}-\frac {34371 (1-2 x)^{5/2}}{1000}+\frac {2889}{200} (1-2 x)^{7/2}-\frac {81}{40} (1-2 x)^{9/2}+\frac {(1-2 x)^{3/2}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}+\frac {1}{625} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}+\frac {11 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {121 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625}\\ &=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {22 \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 = 0.08, size = 66, normalized size = 0.61 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (21262500 x^5+39532500 x^4+9559125 x^3-21433590 x^2-12144995 x+7095688\right )-5082 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18046875} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(-5*Sqrt[1 - 2*x]*(7095688 - 12144995*x - 21433590*x^2 + 9559125*x^3 + 39532500*x^4 + 21262500*x^5) - 5082*Sqr
t[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18046875

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IntegrateAlgebraic [A]  time = 0.05, size = 101, normalized size = 0.94 \begin {gather*} \frac {5315625 (1-2 x)^{11/2}-46344375 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-157563945 (1-2 x)^{5/2}+6160 (1-2 x)^{3/2}+40656 \sqrt {1-2 x}}{28875000}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(40656*Sqrt[1 - 2*x] + 6160*(1 - 2*x)^(3/2) - 157563945*(1 - 2*x)^(5/2) + 141780375*(1 - 2*x)^(7/2) - 46344375
*(1 - 2*x)^(9/2) + 5315625*(1 - 2*x)^(11/2))/28875000 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1562
5

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fricas [A]  time = 1.26, size = 71, normalized size = 0.66 \begin {gather*} \frac {11}{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 (21262500 \, x^{5} + 39532500 \, x^{4} + 9559125 \, x^{3} - 21433590 \, x^{2} - 12144995 \, x + 7095688\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)^4/(3+5*x),x, algorithm="fricas")

[Out]

11/78125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 1/3609375*(21262500*x^5
+ 39532500*x^4 + 9559125*x^3 - 21433590*x^2 - 12144995*x + 7095688)*sqrt(-2*x + 1)

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giac [A]  time = 0.97, size = 122, normalized size = 1.13 \begin {gather*} -\frac {81}{440} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {321}{200} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {34371}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {136419}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{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 {22}{15625} \, \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)^4/(3+5*x),x, algorithm="giac")

[Out]

-81/440*(2*x - 1)^5*sqrt(-2*x + 1) - 321/200*(2*x - 1)^4*sqrt(-2*x + 1) - 34371/7000*(2*x - 1)^3*sqrt(-2*x + 1
) - 136419/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/9375*(-2*x + 1)^(3/2) + 11/78125*sqrt(55)*log(1/2*abs(-2*sqrt(
55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/15625*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 74, normalized size = 0.69 \begin {gather*} -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{78125}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (-2 x +1\right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (-2 x +1\right )^{\frac {7}{2}}}{7000}-\frac {321 \left (-2 x +1\right )^{\frac {9}{2}}}{200}+\frac {81 \left (-2 x +1\right )^{\frac {11}{2}}}{440}+\frac {22 \sqrt {-2 x +1}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(3*x+2)^4/(5*x+3),x)

[Out]

2/9375*(-2*x+1)^(3/2)-136419/25000*(-2*x+1)^(5/2)+34371/7000*(-2*x+1)^(7/2)-321/200*(-2*x+1)^(9/2)+81/440*(-2*
x+1)^(11/2)-22/78125*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+22/15625*(-2*x+1)^(1/2)

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maxima [A]  time = 1.23, size = 91, normalized size = 0.84 \begin {gather*} \frac {81}{440} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {321}{200} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {34371}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {136419}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{15625} \, \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)^4/(3+5*x),x, algorithm="maxima")

[Out]

81/440*(-2*x + 1)^(11/2) - 321/200*(-2*x + 1)^(9/2) + 34371/7000*(-2*x + 1)^(7/2) - 136419/25000*(-2*x + 1)^(5
/2) + 2/9375*(-2*x + 1)^(3/2) + 11/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 22/15625*sqrt(-2*x + 1)

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mupad [B]  time = 1.17, size = 75, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{15625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{9375}-\frac {136419\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {34371\,{\left (1-2\,x\right )}^{7/2}}{7000}-\frac {321\,{\left (1-2\,x\right )}^{9/2}}{200}+\frac {81\,{\left (1-2\,x\right )}^{11/2}}{440}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{78125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(3*x + 2)^4)/(5*x + 3),x)

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/78125 + (22*(1 - 2*x)^(1/2))/15625 + (2*(1 - 2*x)^(3/2))
/9375 - (136419*(1 - 2*x)^(5/2))/25000 + (34371*(1 - 2*x)^(7/2))/7000 - (321*(1 - 2*x)^(9/2))/200 + (81*(1 - 2
*x)^(11/2))/440

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sympy [A]  time = 78.50, size = 138, normalized size = 1.28 \begin {gather*} \frac {81 \left (1 - 2 x\right )^{\frac {11}{2}}}{440} - \frac {321 \left (1 - 2 x\right )^{\frac {9}{2}}}{200} + \frac {34371 \left (1 - 2 x\right )^{\frac {7}{2}}}{7000} - \frac {136419 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {22 \sqrt {1 - 2 x}}{15625} + \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 )}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x),x)

[Out]

81*(1 - 2*x)**(11/2)/440 - 321*(1 - 2*x)**(9/2)/200 + 34371*(1 - 2*x)**(7/2)/7000 - 136419*(1 - 2*x)**(5/2)/25
000 + 2*(1 - 2*x)**(3/2)/9375 + 22*sqrt(1 - 2*x)/15625 + 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))/15625

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