3.22.66 \(\int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\) [2166]

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

[Out]

11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^3-7510/151263*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2/441*(1346+2
027*x)/(2+3*x)^3/(1-2*x)^(1/2)-3755/3087*(1-2*x)^(1/2)/(2+3*x)^2-3755/7203*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, 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 = {100, 149, 44, 65, 212} \begin {gather*} \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}} \end {gather*}

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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 100

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 149

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)/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 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 \text {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*}

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Mathematica [A]
time = 0.22, size = 68, normalized size = 0.57 \begin {gather*} \frac {-\frac {21 \left (-45383-150295 x-83306 x^2+150200 x^3+135180 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^3}-7510 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{151263} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*(-45383 - 150295*x - 83306*x^2 + 150200*x^3 + 135180*x^4))/((1 - 2*x)^(3/2)*(2 + 3*x)^3) - 7510*Sqrt[21]
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/151263

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Maple [A]
time = 0.14, size = 75, normalized size = 0.62

method	result	size
risch	\(\frac {135180 x^{4}+150200 x^{3}-83306 x^{2}-150295 x -45383}{7203 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}-\frac {7510 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\)	\(63\)
derivativedivides	\(\frac {2662}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {6534}{16807 \sqrt {1-2 x}}+\frac {-\frac {18708 \left (1-2 x \right )^{\frac {5}{2}}}{16807}+\frac {5260 \left (1-2 x \right )^{\frac {3}{2}}}{1029}-\frac {6040 \sqrt {1-2 x}}{1029}}{\left (-4-6 x \right )^{3}}-\frac {7510 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\)	\(75\)
default	\(\frac {2662}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {6534}{16807 \sqrt {1-2 x}}+\frac {-\frac {18708 \left (1-2 x \right )^{\frac {5}{2}}}{16807}+\frac {5260 \left (1-2 x \right )^{\frac {3}{2}}}{1029}-\frac {6040 \sqrt {1-2 x}}{1029}}{\left (-4-6 x \right )^{3}}-\frac {7510 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{151263}\)	\(75\)
trager	\(-\frac {\left (135180 x^{4}+150200 x^{3}-83306 x^{2}-150295 x -45383\right ) \sqrt {1-2 x}}{7203 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}+\frac {3755 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{151263}\)	\(89\)

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,method=_RETURNVERBOSE)

[Out]

2662/7203/(1-2*x)^(3/2)+6534/16807/(1-2*x)^(1/2)+54/16807*(-3118/9*(1-2*x)^(5/2)+128870/81*(1-2*x)^(3/2)-14798
0/81*(1-2*x)^(1/2))/(-4-6*x)^3-7510/151263*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.52, size = 110, normalized size = 0.92 \begin {gather*} \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 {gather*}

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 = 1.08, size = 114, normalized size = 0.95 \begin {gather*} \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 {gather*}

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(-1)] Timed out
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((3+5*x)**3/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [A]
time = 1.92, size = 95, normalized size = 0.79 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.07, size = 92, normalized size = 0.77 \begin {gather*} -\frac {\frac {14072\,{\left (2\,x-1\right )}^2}{3969}-\frac {968\,x}{441}+\frac {60080\,{\left (2\,x-1\right )}^3}{27783}+\frac {7510\,{\left (2\,x-1\right )}^4}{21609}-\frac {14278}{3969}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}-\frac {7510\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{151263} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((14072*(2*x - 1)^2)/3969 - (968*x)/441 + (60080*(2*x - 1)^3)/27783 + (7510*(2*x - 1)^4)/21609 - 14278/3969)
/((343*(1 - 2*x)^(3/2))/27 - (49*(1 - 2*x)^(5/2))/3 + 7*(1 - 2*x)^(7/2) - (1 - 2*x)^(9/2)) - (7510*21^(1/2)*at
anh((21^(1/2)*(1 - 2*x)^(1/2))/7))/151263

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