3.2166 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=120 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{2 (2027 x+1346)}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{3755 \sqrt{1-2 x}}{7203 (3 x+2)}-\frac{3755 \sqrt{1-2 x}}{3087 (3 x+2)^2}-\frac{7510 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203 \sqrt{21}} \]

[Out]

(-3755*Sqrt[1 - 2*x])/(3087*(2 + 3*x)^2) - (3755*Sqrt[1 - 2*x])/(7203*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2
*x)^(3/2)*(2 + 3*x)^3) + (2*(1346 + 2027*x))/(441*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (7510*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(7203*Sqrt[21])

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Rubi [A]  time = 0.0324268, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 144, 51, 63, 206} \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{2 (2027 x+1346)}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{3755 \sqrt{1-2 x}}{7203 (3 x+2)}-\frac{3755 \sqrt{1-2 x}}{3087 (3 x+2)^2}-\frac{7510 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3755*Sqrt[1 - 2*x])/(3087*(2 + 3*x)^2) - (3755*Sqrt[1 - 2*x])/(7203*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2
*x)^(3/2)*(2 + 3*x)^3) + (2*(1346 + 2027*x))/(441*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (7510*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(7203*Sqrt[21])

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 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -1]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1}{21} \int \frac{(-68-150 x) (3+5 x)}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2 (1346+2027 x)}{441 \sqrt{1-2 x} (2+3 x)^3}+\frac{7510}{441} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{3755 \sqrt{1-2 x}}{3087 (2+3 x)^2}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2 (1346+2027 x)}{441 \sqrt{1-2 x} (2+3 x)^3}+\frac{3755 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{1029}\\ &=-\frac{3755 \sqrt{1-2 x}}{3087 (2+3 x)^2}-\frac{3755 \sqrt{1-2 x}}{7203 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2 (1346+2027 x)}{441 \sqrt{1-2 x} (2+3 x)^3}+\frac{3755 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{7203}\\ &=-\frac{3755 \sqrt{1-2 x}}{3087 (2+3 x)^2}-\frac{3755 \sqrt{1-2 x}}{7203 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2 (1346+2027 x)}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{3755 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{7203}\\ &=-\frac{3755 \sqrt{1-2 x}}{3087 (2+3 x)^2}-\frac{3755 \sqrt{1-2 x}}{7203 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2 (1346+2027 x)}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{7510 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0290255, size = 59, normalized size = 0.49 \[ \frac{\frac{343 \left (1225 x^2-136 x+2091\right )}{(3 x+2)^3}-24032 (1-2 x)^2 \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{50421 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((343*(2091 - 136*x + 1225*x^2))/(2 + 3*x)^3 - 24032*(1 - 2*x)^2*Hypergeometric2F1[1/2, 4, 3/2, 3/7 - (6*x)/7]
)/(50421*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.013, size = 75, normalized size = 0.6 \begin{align*}{\frac{54}{16807\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{3118}{9} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{128870}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{147980}{81}\sqrt{1-2\,x}} \right ) }-{\frac{7510\,\sqrt{21}}{151263}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{2662}{7203} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{6534}{16807}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

54/16807*(-3118/9*(1-2*x)^(5/2)+128870/81*(1-2*x)^(3/2)-147980/81*(1-2*x)^(1/2))/(-6*x-4)^3-7510/151263*arctan
h(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2662/7203/(1-2*x)^(3/2)+6534/16807/(1-2*x)^(1/2)

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Maxima [A]  time = 1.52482, size = 149, normalized size = 1.24 \begin{align*} \frac{3755}{151263} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{4} + 210280 \,{\left (2 \, x - 1\right )}^{3} + 344764 \,{\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3755/151263*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/7203*(33795*(2*x -
1)^4 + 210280*(2*x - 1)^3 + 344764*(2*x - 1)^2 - 213444*x - 349811)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2
) + 441*(-2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.58183, size = 338, normalized size = 2.82 \begin{align*} \frac{3755 \, \sqrt{21}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (135180 \, x^{4} + 150200 \, x^{3} - 83306 \, x^{2} - 150295 \, x - 45383\right )} \sqrt{-2 \, x + 1}}{151263 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/151263*(3755*sqrt(21)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5
)/(3*x + 2)) - 21*(135180*x^4 + 150200*x^3 - 83306*x^2 - 150295*x - 45383)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4
- 45*x^3 - 58*x^2 + 4*x + 8)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.76359, size = 128, normalized size = 1.07 \begin{align*} \frac{3755}{151263} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{4} + 210280 \,{\left (2 \, x - 1\right )}^{3} + 344764 \,{\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3755/151263*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/7203*(3379
5*(2*x - 1)^4 + 210280*(2*x - 1)^3 + 344764*(2*x - 1)^2 - 213444*x - 349811)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x
+ 1))^3