3.2104 \(\int \frac{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=106 \[ \frac{81}{160} (1-2 x)^{9/2}-\frac{43011 (1-2 x)^{7/2}}{5600}+\frac{507627 (1-2 x)^{5/2}}{10000}-\frac{1997451 (1-2 x)^{3/2}}{10000}+\frac{70752609 \sqrt{1-2 x}}{100000}+\frac{117649}{352 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

[Out]

117649/(352*Sqrt[1 - 2*x]) + (70752609*Sqrt[1 - 2*x])/100000 - (1997451*(1 - 2*x)^(3/2))/10000 + (507627*(1 -
2*x)^(5/2))/10000 - (43011*(1 - 2*x)^(7/2))/5600 + (81*(1 - 2*x)^(9/2))/160 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(34375*Sqrt[55])

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Rubi [A]  time = 0.0809923, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {87, 43, 63, 206} \[ \frac{81}{160} (1-2 x)^{9/2}-\frac{43011 (1-2 x)^{7/2}}{5600}+\frac{507627 (1-2 x)^{5/2}}{10000}-\frac{1997451 (1-2 x)^{3/2}}{10000}+\frac{70752609 \sqrt{1-2 x}}{100000}+\frac{117649}{352 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

117649/(352*Sqrt[1 - 2*x]) + (70752609*Sqrt[1 - 2*x])/100000 - (1997451*(1 - 2*x)^(3/2))/10000 + (507627*(1 -
2*x)^(5/2))/10000 - (43011*(1 - 2*x)^(7/2))/5600 + (81*(1 - 2*x)^(9/2))/160 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(34375*Sqrt[55])

Rule 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac{117649}{352 (1-2 x)^{3/2}}-\frac{31289679}{100000 \sqrt{1-2 x}}-\frac{4693491 x}{10000 \sqrt{1-2 x}}-\frac{479439 x^2}{1000 \sqrt{1-2 x}}-\frac{28431 x^3}{100 \sqrt{1-2 x}}-\frac{729 x^4}{10 \sqrt{1-2 x}}+\frac{1}{34375 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{117649}{352 \sqrt{1-2 x}}+\frac{31289679 \sqrt{1-2 x}}{100000}+\frac{\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{34375}-\frac{729}{10} \int \frac{x^4}{\sqrt{1-2 x}} \, dx-\frac{28431}{100} \int \frac{x^3}{\sqrt{1-2 x}} \, dx-\frac{4693491 \int \frac{x}{\sqrt{1-2 x}} \, dx}{10000}-\frac{479439 \int \frac{x^2}{\sqrt{1-2 x}} \, dx}{1000}\\ &=\frac{117649}{352 \sqrt{1-2 x}}+\frac{31289679 \sqrt{1-2 x}}{100000}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{34375}-\frac{729}{10} \int \left (\frac{1}{16 \sqrt{1-2 x}}-\frac{1}{4} \sqrt{1-2 x}+\frac{3}{8} (1-2 x)^{3/2}-\frac{1}{4} (1-2 x)^{5/2}+\frac{1}{16} (1-2 x)^{7/2}\right ) \, dx-\frac{28431}{100} \int \left (\frac{1}{8 \sqrt{1-2 x}}-\frac{3}{8} \sqrt{1-2 x}+\frac{3}{8} (1-2 x)^{3/2}-\frac{1}{8} (1-2 x)^{5/2}\right ) \, dx-\frac{4693491 \int \left (\frac{1}{2 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}\right ) \, dx}{10000}-\frac{479439 \int \left (\frac{1}{4 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}+\frac{1}{4} (1-2 x)^{3/2}\right ) \, dx}{1000}\\ &=\frac{117649}{352 \sqrt{1-2 x}}+\frac{70752609 \sqrt{1-2 x}}{100000}-\frac{1997451 (1-2 x)^{3/2}}{10000}+\frac{507627 (1-2 x)^{5/2}}{10000}-\frac{43011 (1-2 x)^{7/2}}{5600}+\frac{81}{160} (1-2 x)^{9/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0301286, size = 60, normalized size = 0.57 \[ \frac{14 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-99 \left (196875 x^5+1001250 x^4+2440575 x^3+4301000 x^2+10503235 x-10762494\right )}{1203125 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-99*(-10762494 + 10503235*x + 4301000*x^2 + 2440575*x^3 + 1001250*x^4 + 196875*x^5) + 14*Hypergeometric2F1[-1
/2, 1, 1/2, (5*(1 - 2*x))/11])/(1203125*Sqrt[1 - 2*x])

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Maple [A]  time = 0.007, size = 74, normalized size = 0.7 \begin{align*} -{\frac{1997451}{10000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{507627}{10000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{43011}{5600} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{81}{160} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2\,\sqrt{55}}{1890625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{117649}{352}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{70752609}{100000}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1997451/10000*(1-2*x)^(3/2)+507627/10000*(1-2*x)^(5/2)-43011/5600*(1-2*x)^(7/2)+81/160*(1-2*x)^(9/2)-2/189062
5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+117649/352/(1-2*x)^(1/2)[phone]/100000*(1-2*x)^(1/2)

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Maxima [A]  time = 1.53501, size = 123, normalized size = 1.16 \begin{align*} \frac{81}{160} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{43011}{5600} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{507627}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1997451}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{1890625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{70752609}{100000} \, \sqrt{-2 \, x + 1} + \frac{117649}{352 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x),x, algorithm="maxima")

[Out]

81/160*(-2*x + 1)^(9/2) - 43011/5600*(-2*x + 1)^(7/2) + 507627/10000*(-2*x + 1)^(5/2) - 1997451/10000*(-2*x +
1)^(3/2) + 1/1890625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70752609/100
000*sqrt(-2*x + 1) + 117649/352/sqrt(-2*x + 1)

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Fricas [A]  time = 1.55646, size = 271, normalized size = 2.56 \begin{align*} \frac{7 \, \sqrt{55}{\left (2 \, x - 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (3898125 \, x^{5} + 19824750 \, x^{4} + 48323385 \, x^{3} + 85159800 \, x^{2} + 207964053 \, x - 213097384\right )} \sqrt{-2 \, x + 1}}{13234375 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x),x, algorithm="fricas")

[Out]

1/13234375*(7*sqrt(55)*(2*x - 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(3898125*x^5 + 198247
50*x^4 + 48323385*x^3 + 85159800*x^2 + 207964053*x - 213097384)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [A]  time = 110.402, size = 138, normalized size = 1.3 \begin{align*} \frac{81 \left (1 - 2 x\right )^{\frac{9}{2}}}{160} - \frac{43011 \left (1 - 2 x\right )^{\frac{7}{2}}}{5600} + \frac{507627 \left (1 - 2 x\right )^{\frac{5}{2}}}{10000} - \frac{1997451 \left (1 - 2 x\right )^{\frac{3}{2}}}{10000} + \frac{70752609 \sqrt{1 - 2 x}}{100000} + \frac{2 \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 )}{34375} + \frac{117649}{352 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81*(1 - 2*x)**(9/2)/160 - 43011*(1 - 2*x)**(7/2)/5600 + 507627*(1 - 2*x)**(5/2)/10000 - 1997451*(1 - 2*x)**(3/
2)/10000 + 70752609*sqrt(1 - 2*x)/100000 + 2*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))/34375 + 117649/(352*sqrt(1 - 2*x)
)

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Giac [A]  time = 2.32525, size = 155, normalized size = 1.46 \begin{align*} \frac{81}{160} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{43011}{5600} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{507627}{10000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1997451}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{1890625} \, \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{70752609}{100000} \, \sqrt{-2 \, x + 1} + \frac{117649}{352 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(3/2)/(3+5*x),x, algorithm="giac")

[Out]

81/160*(2*x - 1)^4*sqrt(-2*x + 1) + 43011/5600*(2*x - 1)^3*sqrt(-2*x + 1) + 507627/10000*(2*x - 1)^2*sqrt(-2*x
+ 1) - 1997451/10000*(-2*x + 1)^(3/2) + 1/1890625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) + 70752609/100000*sqrt(-2*x + 1) + 117649/352/sqrt(-2*x + 1)