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

Optimal. Leaf size=134 \[ \frac {243}{800} (1-2 x)^{15/2}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {2 (1-2 x)^{3/2}}{234375}+\frac {22 \sqrt {1-2 x}}{390625}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625} \]

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Rubi [A]  time = 0.04, antiderivative size = 134, 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}{800} (1-2 x)^{15/2}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {2 (1-2 x)^{3/2}}{234375}+\frac {22 \sqrt {1-2 x}}{390625}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(22*Sqrt[1 - 2*x])/390625 + (2*(1 - 2*x)^(3/2))/234375 - (167115051*(1 - 2*x)^(5/2))/2500000 + (70752609*(1 -
2*x)^(7/2))/700000 - (665817*(1 - 2*x)^(9/2))/10000 + (507627*(1 - 2*x)^(11/2))/22000 - (43011*(1 - 2*x)^(13/2
))/10400 + (243*(1 - 2*x)^(15/2))/800 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/390625

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)^6}{3+5 x} \, dx &=\int \left (\frac {167115051 (1-2 x)^{3/2}}{500000}-\frac {70752609 (1-2 x)^{5/2}}{100000}+\frac {5992353 (1-2 x)^{7/2}}{10000}-\frac {507627 (1-2 x)^{9/2}}{2000}+\frac {43011}{800} (1-2 x)^{11/2}-\frac {729}{160} (1-2 x)^{13/2}+\frac {(1-2 x)^{3/2}}{15625 (3+5 x)}\right ) \, dx\\ &=-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}+\frac {\int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{15625}\\ &=\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}+\frac {11 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{78125}\\ &=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{390625}\\ &=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}-\frac {121 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{390625}\\ &=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 76, normalized size = 0.57 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (45608062500 x^7+150857437500 x^6+174123928125 x^5+49094797500 x^4-61883481375 x^3-56176961670 x^2-9645684935 x+15379193944\right )-66066 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5865234375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x]*(15379193944 - 9645684935*x - 56176961670*x^2 - 61883481375*x^3 + 49094797500*x^4 + 17412392
8125*x^5 + 150857437500*x^6 + 45608062500*x^7) - 66066*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5865234375

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IntegrateAlgebraic [A]  time = 0.07, size = 123, normalized size = 0.92 \begin {gather*} \frac {11402015625 (1-2 x)^{15/2}-155242828125 (1-2 x)^{13/2}[phone] (1-2 x)^{11/2}-2499310563750 (1-2 x)^{9/2}[phone] (1-2 x)^{7/2}-2509232490765 (1-2 x)^{5/2}+320320 (1-2 x)^{3/2}+2114112 \sqrt {1-2 x}}{37537500000}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2114112*Sqrt[1 - 2*x] + 320320*(1 - 2*x)^(3/2) - 2509232490765*(1 - 2*x)^(5/2) + 3794108657625*(1 - 2*x)^(7/2
) - 2499310563750*(1 - 2*x)^(9/2) + 866138568750*(1 - 2*x)^(11/2) - 155242828125*(1 - 2*x)^(13/2) + 1140201562
5*(1 - 2*x)^(15/2))/37537500000 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/390625

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fricas [A]  time = 1.46, size = 81, normalized size = 0.60 \begin {gather*} \frac {11}{1953125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{1173046875} \, {\left (45608062500 \, x^{7} + 150857437500 \, x^{6} + 174123928125 \, x^{5} + 49094797500 \, x^{4} - 61883481375 \, x^{3} - 56176961670 \, x^{2} - 9645684935 \, x + 15379193944\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)^6/(3+5*x),x, algorithm="fricas")

[Out]

11/1953125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 1/1173046875*(4560806
2500*x^7 + 150857437500*x^6 + 174123928125*x^5 + 49094797500*x^4 - 61883481375*x^3 - 56176961670*x^2 - 9645684
935*x + 15379193944)*sqrt(-2*x + 1)

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giac [A]  time = 0.97, size = 154, normalized size = 1.15 \begin {gather*} -\frac {243}{800} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {43011}{10400} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {507627}{22000} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {665817}{10000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {70752609}{700000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {167115051}{2500000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{234375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{1953125} \, \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}{390625} \, \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)^6/(3+5*x),x, algorithm="giac")

[Out]

-243/800*(2*x - 1)^7*sqrt(-2*x + 1) - 43011/10400*(2*x - 1)^6*sqrt(-2*x + 1) - 507627/22000*(2*x - 1)^5*sqrt(-
2*x + 1) - 665817/10000*(2*x - 1)^4*sqrt(-2*x + 1) - 70752609/700000*(2*x - 1)^3*sqrt(-2*x + 1) - 167115051/25
00000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/234375*(-2*x + 1)^(3/2) + 11/1953125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 1
0*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/390625*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 92, normalized size = 0.69 \begin {gather*} -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1953125}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{234375}-\frac {167115051 \left (-2 x +1\right )^{\frac {5}{2}}}{2500000}+\frac {70752609 \left (-2 x +1\right )^{\frac {7}{2}}}{700000}-\frac {665817 \left (-2 x +1\right )^{\frac {9}{2}}}{10000}+\frac {507627 \left (-2 x +1\right )^{\frac {11}{2}}}{22000}-\frac {43011 \left (-2 x +1\right )^{\frac {13}{2}}}{10400}+\frac {243 \left (-2 x +1\right )^{\frac {15}{2}}}{800}+\frac {22 \sqrt {-2 x +1}}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/234375*(-2*x+1)^(3/2)-167115051/2500000*(-2*x+1)^(5/2)[phone]/700000*(-2*x+1)^(7/2)-665817/10000*(-2*x+1)^
(9/2)+507627/22000*(-2*x+1)^(11/2)-43011/10400*(-2*x+1)^(13/2)+243/800*(-2*x+1)^(15/2)-22/1953125*arctanh(1/11
*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+22/390625*(-2*x+1)^(1/2)

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maxima [A]  time = 1.45, size = 109, normalized size = 0.81 \begin {gather*} \frac {243}{800} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {43011}{10400} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {507627}{22000} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {665817}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {70752609}{700000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {167115051}{2500000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{234375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{1953125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{390625} \, \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)^6/(3+5*x),x, algorithm="maxima")

[Out]

243/800*(-2*x + 1)^(15/2) - 43011/10400*(-2*x + 1)^(13/2) + 507627/22000*(-2*x + 1)^(11/2) - 665817/10000*(-2*
x + 1)^(9/2) + 70752609/700000*(-2*x + 1)^(7/2) - 167115051/2500000*(-2*x + 1)^(5/2) + 2/234375*(-2*x + 1)^(3/
2) + 11/1953125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/390625*sqrt(-2
*x + 1)

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mupad [B]  time = 1.18, size = 93, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{390625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{234375}-\frac {167115051\,{\left (1-2\,x\right )}^{5/2}}{2500000}+\frac {70752609\,{\left (1-2\,x\right )}^{7/2}}{700000}-\frac {665817\,{\left (1-2\,x\right )}^{9/2}}{10000}+\frac {507627\,{\left (1-2\,x\right )}^{11/2}}{22000}-\frac {43011\,{\left (1-2\,x\right )}^{13/2}}{10400}+\frac {243\,{\left (1-2\,x\right )}^{15/2}}{800}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{1953125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/1953125 + (22*(1 - 2*x)^(1/2))/390625 + (2*(1 - 2*x)^(3/
2))/234375 - (167115051*(1 - 2*x)^(5/2))/2500000 + (70752609*(1 - 2*x)^(7/2))/700000 - (665817*(1 - 2*x)^(9/2)
)/10000 + (507627*(1 - 2*x)^(11/2))/22000 - (43011*(1 - 2*x)^(13/2))/10400 + (243*(1 - 2*x)^(15/2))/800

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sympy [A]  time = 125.77, size = 162, normalized size = 1.21 \begin {gather*} \frac {243 \left (1 - 2 x\right )^{\frac {15}{2}}}{800} - \frac {43011 \left (1 - 2 x\right )^{\frac {13}{2}}}{10400} + \frac {507627 \left (1 - 2 x\right )^{\frac {11}{2}}}{22000} - \frac {665817 \left (1 - 2 x\right )^{\frac {9}{2}}}{10000} + \frac {70752609 \left (1 - 2 x\right )^{\frac {7}{2}}}{700000} - \frac {167115051 \left (1 - 2 x\right )^{\frac {5}{2}}}{2500000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{234375} + \frac {22 \sqrt {1 - 2 x}}{390625} + \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 )}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243*(1 - 2*x)**(15/2)/800 - 43011*(1 - 2*x)**(13/2)/10400 + 507627*(1 - 2*x)**(11/2)/22000 - 665817*(1 - 2*x)*
*(9/2)/10000 + 70752609*(1 - 2*x)**(7/2)/700000 - 167115051*(1 - 2*x)**(5/2)/2500000 + 2*(1 - 2*x)**(3/2)/2343
75 + 22*sqrt(1 - 2*x)/390625 + 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))/390625

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