3.1895 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx\)

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

[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

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Rubi [A]  time = 0.0384962, antiderivative size = 134, normalized size of antiderivative = 1., 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} \[ \frac{243}{800} (1-2 x)^{15/2}-\frac{43011 (1-2 x)^{13/2}}{10400}+\frac{507627 (1-2 x)^{11/2}}{22000}-\frac{665817 (1-2 x)^{9/2}}{10000}+\frac{70752609 (1-2 x)^{7/2}}{700000}-\frac{167115051 (1-2 x)^{5/2}}{2500000}+\frac{2 (1-2 x)^{3/2}}{234375}+\frac{22 \sqrt{1-2 x}}{390625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{390625} \]

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 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 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 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*}

Mathematica [A]  time = 0.0844573, size = 76, normalized size = 0.57 \[ \frac{-5 \sqrt{1-2 x} \left (45608062500 x^7+150857437500 x^6+174123928125 x^5+49094797500 x^4-61883481375 x^3-56176961670 x^2-9645684935 x+15379193944\right )-66066 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5865234375} \]

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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Maple [A]  time = 0.008, size = 92, normalized size = 0.7 \begin{align*}{\frac{2}{234375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{167115051}{2500000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{70752609}{700000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{665817}{10000} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{507627}{22000} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{43011}{10400} \left ( 1-2\,x \right ) ^{{\frac{13}{2}}}}+{\frac{243}{800} \left ( 1-2\,x \right ) ^{{\frac{15}{2}}}}-{\frac{22\,\sqrt{55}}{1953125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{390625}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.18316, size = 147, normalized size = 1.1 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.38292, size = 347, normalized size = 2.59 \begin{align*} \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{align*}

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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Sympy [A]  time = 89.832, size = 162, normalized size = 1.21 \begin{align*} \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{align*}

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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Giac [A]  time = 1.46722, size = 208, normalized size = 1.55 \begin{align*} -\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{align*}

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)