3.2165 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{720 x+487}{294 \sqrt{1-2 x} (3 x+2)^2}+\frac{905 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

(905*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - (487 + 720*x)/(294*
Sqrt[1 - 2*x]*(2 + 3*x)^2) + (905*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi [A]  time = 0.0263209, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{720 x+487}{294 \sqrt{1-2 x} (3 x+2)^2}+\frac{905 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(905*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - (487 + 720*x)/(294*
Sqrt[1 - 2*x]*(2 + 3*x)^2) + (905*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*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)^3} \, dx &=\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{1}{21} \int \frac{(3+5 x) (31+15 x)}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{487+720 x}{294 \sqrt{1-2 x} (2+3 x)^2}-\frac{905}{294} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{905 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{487+720 x}{294 \sqrt{1-2 x} (2+3 x)^2}-\frac{905 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac{905 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{487+720 x}{294 \sqrt{1-2 x} (2+3 x)^2}+\frac{905 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2058}\\ &=\frac{905 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{487+720 x}{294 \sqrt{1-2 x} (2+3 x)^2}+\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0208737, size = 62, normalized size = 0.62 \[ -\frac{-7240 \left (6 x^2+x-2\right )^2 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )-343 \left (875 x^2+1303 x+128\right )}{21609 (1-2 x)^{3/2} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-343*(128 + 1303*x + 875*x^2) - 7240*(-2 + x + 6*x^2)^2*Hypergeometric2F1[1/2, 3, 3/2, 3/7 - (6*x)/7])/(2160
9*(1 - 2*x)^(3/2)*(2 + 3*x)^2)

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Maple [A]  time = 0.013, size = 66, normalized size = 0.7 \begin{align*} -{\frac{18}{2401\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{199}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1379}{54}\sqrt{1-2\,x}} \right ) }+{\frac{905\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1331}{1029} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{726}{2401}{\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)^3,x)

[Out]

-18/2401*(-199/18*(1-2*x)^(3/2)+1379/54*(1-2*x)^(1/2))/(-6*x-4)^2+905/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2)
)*21^(1/2)+1331/1029/(1-2*x)^(3/2)-726/2401/(1-2*x)^(1/2)

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Maxima [A]  time = 2.83944, size = 124, normalized size = 1.24 \begin{align*} -\frac{905}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2715 \,{\left (2 \, x - 1\right )}^{3} + 24850 \,{\left (2 \, x - 1\right )}^{2} + 142296 \, x - 5929}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\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)^3,x, algorithm="maxima")

[Out]

-905/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1029*(2715*(2*x - 1)
^3 + 24850*(2*x - 1)^2 + 142296*x - 5929)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.35391, size = 284, normalized size = 2.84 \begin{align*} \frac{905 \, \sqrt{21}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (10860 \, x^{3} + 33410 \, x^{2} + 29593 \, x + 8103\right )} \sqrt{-2 \, x + 1}}{43218 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

1/43218*(905*sqrt(21)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2))
+ 21*(10860*x^3 + 33410*x^2 + 29593*x + 8103)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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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)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.56091, size = 120, normalized size = 1.2 \begin{align*} -\frac{905}{43218} \, \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{121 \,{\left (36 \, x + 59\right )}}{7203 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{597 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1379 \, \sqrt{-2 \, x + 1}}{28812 \,{\left (3 \, x + 2\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

-905/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 121/7203*(36*
x + 59)/((2*x - 1)*sqrt(-2*x + 1)) + 1/28812*(597*(-2*x + 1)^(3/2) - 1379*sqrt(-2*x + 1))/(3*x + 2)^2