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

Optimal. Leaf size=80 \[ \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {\sqrt {1-2 x} (24825 x+15676)}{66550 (5 x+3)^2}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]

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Rubi [A]  time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {98, 145, 63, 206} \begin {gather*} \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {\sqrt {1-2 x} (24825 x+15676)}{66550 (5 x+3)^2}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (Sqrt[1 - 2*x]*(15676 + 24825*x))/(66550*(3 + 5*x)^2) - (7143
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(33275*Sqrt[55])

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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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)^3}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{11} \int \frac {(-16-3 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}+\frac {7143 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{66550}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}-\frac {7143 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{66550}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 59, normalized size = 0.74 \begin {gather*} \frac {14286 (5 x+3)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+11 \left (163350 x^2+195005 x+58186\right )}{332750 \sqrt {1-2 x} (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(58186 + 195005*x + 163350*x^2) + 14286*(3 + 5*x)^2*Hypergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11])/(
332750*Sqrt[1 - 2*x]*(3 + 5*x)^2)

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IntegrateAlgebraic [A]  time = 0.19, size = 70, normalized size = 0.88 \begin {gather*} \frac {215400 (1-2 x)^2-945527 (1-2 x)+1037575}{33275 (5 (1-2 x)-11)^2 \sqrt {1-2 x}}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1037575 - 945527*(1 - 2*x) + 215400*(1 - 2*x)^2)/(33275*(-11 + 5*(1 - 2*x))^2*Sqrt[1 - 2*x]) - (7143*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/(33275*Sqrt[55])

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fricas [A]  time = 1.16, size = 84, normalized size = 1.05 \begin {gather*} \frac {7143 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (430800 \, x^{2} + 514727 \, x + 153724\right )} \sqrt {-2 \, x + 1}}{3660250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3660250*(7143*sqrt(55)*(50*x^3 + 35*x^2 - 12*x - 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*
(430800*x^2 + 514727*x + 153724)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [A]  time = 1.27, size = 77, normalized size = 0.96 \begin {gather*} \frac {7143}{3660250} \, \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 {343}{1331 \, \sqrt {-2 \, x + 1}} + \frac {1025 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2277 \, \sqrt {-2 \, x + 1}}{133100 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7143/3660250*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 343/1331/s
qrt(-2*x + 1) + 1/133100*(1025*(-2*x + 1)^(3/2) - 2277*sqrt(-2*x + 1))/(5*x + 3)^2

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maple [A]  time = 0.02, size = 57, normalized size = 0.71 \begin {gather*} -\frac {7143 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1830125}+\frac {343}{1331 \sqrt {-2 x +1}}+\frac {\frac {41 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}-\frac {207 \sqrt {-2 x +1}}{3025}}{\left (-10 x -6\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/1331/(-2*x+1)^(1/2)+50/1331*(41/50*(-2*x+1)^(3/2)-2277/1250*(-2*x+1)^(1/2))/(-10*x-6)^2-7143/1830125*arcta
nh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)

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maxima [A]  time = 1.42, size = 83, normalized size = 1.04 \begin {gather*} \frac {7143}{3660250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (107700 \, {\left (2 \, x - 1\right )}^{2} + 945527 \, x + 46024\right )}}{33275 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7143/3660250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/33275*(107700*(2*x
- 1)^2 + 945527*x + 46024)/(25*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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mupad [B]  time = 1.20, size = 62, normalized size = 0.78 \begin {gather*} \frac {\frac {171914\,x}{75625}+\frac {8616\,{\left (2\,x-1\right )}^2}{33275}+\frac {8368}{75625}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}-\frac {7143\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1830125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((171914*x)/75625 + (8616*(2*x - 1)^2)/33275 + 8368/75625)/((121*(1 - 2*x)^(1/2))/25 - (22*(1 - 2*x)^(3/2))/5
+ (1 - 2*x)^(5/2)) - (7143*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1830125

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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