3.1968 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=121 \[ \frac{81}{520} (1-2 x)^{13/2}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{2 (1-2 x)^{5/2}}{15625}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{242 \sqrt{1-2 x}}{78125}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

[Out]

(242*Sqrt[1 - 2*x])/78125 + (22*(1 - 2*x)^(3/2))/46875 + (2*(1 - 2*x)^(5/2))/15625 - (136419*(1 - 2*x)^(7/2))/
35000 + (3819*(1 - 2*x)^(9/2))/1000 - (2889*(1 - 2*x)^(11/2))/2200 + (81*(1 - 2*x)^(13/2))/520 - (242*Sqrt[11/
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi [A]  time = 0.0416802, antiderivative size = 121, normalized size of antiderivative = 1., 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} \[ \frac{81}{520} (1-2 x)^{13/2}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{2 (1-2 x)^{5/2}}{15625}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{242 \sqrt{1-2 x}}{78125}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(242*Sqrt[1 - 2*x])/78125 + (22*(1 - 2*x)^(3/2))/46875 + (2*(1 - 2*x)^(5/2))/15625 - (136419*(1 - 2*x)^(7/2))/
35000 + (3819*(1 - 2*x)^(9/2))/1000 - (2889*(1 - 2*x)^(11/2))/2200 + (81*(1 - 2*x)^(13/2))/520 - (242*Sqrt[11/
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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)^{5/2} (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac{136419 (1-2 x)^{5/2}}{5000}-\frac{34371 (1-2 x)^{7/2}}{1000}+\frac{2889}{200} (1-2 x)^{9/2}-\frac{81}{40} (1-2 x)^{11/2}+\frac{(1-2 x)^{5/2}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{81}{520} (1-2 x)^{13/2}+\frac{1}{625} \int \frac{(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac{2 (1-2 x)^{5/2}}{15625}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{81}{520} (1-2 x)^{13/2}+\frac{11 \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac{22 (1-2 x)^{3/2}}{46875}+\frac{2 (1-2 x)^{5/2}}{15625}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{81}{520} (1-2 x)^{13/2}+\frac{121 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac{242 \sqrt{1-2 x}}{78125}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{2 (1-2 x)^{5/2}}{15625}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{81}{520} (1-2 x)^{13/2}+\frac{1331 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac{242 \sqrt{1-2 x}}{78125}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{2 (1-2 x)^{5/2}}{15625}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{81}{520} (1-2 x)^{13/2}-\frac{1331 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{78125}\\ &=\frac{242 \sqrt{1-2 x}}{78125}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{2 (1-2 x)^{5/2}}{15625}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{81}{520} (1-2 x)^{13/2}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125}\\ \end{align*}

Mathematica [A]  time = 0.0846605, size = 71, normalized size = 0.59 \[ \frac{5 \sqrt{1-2 x} \left (2338875000 x^6+2842087500 x^5-1540428750 x^4-2556079875 x^3+399578370 x^2+960784285 x-289133384\right )-726726 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1173046875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(-289133384 + 960784285*x + 399578370*x^2 - 2556079875*x^3 - 1540428750*x^4 + 2842087500*x^5
+ 2338875000*x^6) - 726726*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1173046875

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Maple [A]  time = 0.006, size = 83, normalized size = 0.7 \begin{align*}{\frac{22}{46875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{15625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{136419}{35000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{3819}{1000} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2889}{2200} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}+{\frac{81}{520} \left ( 1-2\,x \right ) ^{{\frac{13}{2}}}}-{\frac{242\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{78125}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22/46875*(1-2*x)^(3/2)+2/15625*(1-2*x)^(5/2)-136419/35000*(1-2*x)^(7/2)+3819/1000*(1-2*x)^(9/2)-2889/2200*(1-2
*x)^(11/2)+81/520*(1-2*x)^(13/2)-242/390625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/78125*(1-2*x)^(1
/2)

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Maxima [A]  time = 3.92524, size = 135, normalized size = 1.12 \begin{align*} \frac{81}{520} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{2889}{2200} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{3819}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{136419}{35000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{15625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{78125} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x, algorithm="maxima")

[Out]

81/520*(-2*x + 1)^(13/2) - 2889/2200*(-2*x + 1)^(11/2) + 3819/1000*(-2*x + 1)^(9/2) - 136419/35000*(-2*x + 1)^
(7/2) + 2/15625*(-2*x + 1)^(5/2) + 22/46875*(-2*x + 1)^(3/2) + 121/390625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*
x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/78125*sqrt(-2*x + 1)

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Fricas [A]  time = 1.33329, size = 306, normalized size = 2.53 \begin{align*} \frac{121}{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 (2338875000 \, x^{6} + 2842087500 \, x^{5} - 1540428750 \, x^{4} - 2556079875 \, x^{3} + 399578370 \, x^{2} + 960784285 \, x - 289133384\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

121/390625*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1/234609375*(23388750
00*x^6 + 2842087500*x^5 - 1540428750*x^4 - 2556079875*x^3 + 399578370*x^2 + 960784285*x - 289133384)*sqrt(-2*x
+ 1)

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Sympy [A]  time = 99.7535, size = 150, normalized size = 1.24 \begin{align*} \frac{81 \left (1 - 2 x\right )^{\frac{13}{2}}}{520} - \frac{2889 \left (1 - 2 x\right )^{\frac{11}{2}}}{2200} + \frac{3819 \left (1 - 2 x\right )^{\frac{9}{2}}}{1000} - \frac{136419 \left (1 - 2 x\right )^{\frac{7}{2}}}{35000} + \frac{2 \left (1 - 2 x\right )^{\frac{5}{2}}}{15625} + \frac{22 \left (1 - 2 x\right )^{\frac{3}{2}}}{46875} + \frac{242 \sqrt{1 - 2 x}}{78125} + \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 )}{78125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81*(1 - 2*x)**(13/2)/520 - 2889*(1 - 2*x)**(11/2)/2200 + 3819*(1 - 2*x)**(9/2)/1000 - 136419*(1 - 2*x)**(7/2)/
35000 + 2*(1 - 2*x)**(5/2)/15625 + 22*(1 - 2*x)**(3/2)/46875 + 242*sqrt(1 - 2*x)/78125 + 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))/78125

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Giac [A]  time = 2.39428, size = 186, normalized size = 1.54 \begin{align*} \frac{81}{520} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{2889}{2200} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{3819}{1000} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{136419}{35000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{15625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{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{242}{78125} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

81/520*(2*x - 1)^6*sqrt(-2*x + 1) + 2889/2200*(2*x - 1)^5*sqrt(-2*x + 1) + 3819/1000*(2*x - 1)^4*sqrt(-2*x + 1
) + 136419/35000*(2*x - 1)^3*sqrt(-2*x + 1) + 2/15625*(2*x - 1)^2*sqrt(-2*x + 1) + 22/46875*(-2*x + 1)^(3/2) +
121/390625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/78125*s
qrt(-2*x + 1)