3.19.23 \(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx\) [1823]

Optimal. Leaf size=107 \[ -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {92996 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \]

[Out]

-92996/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-53/189*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^2-1/9*(3+5*x)
^3*(1-2*x)^(1/2)/(2+3*x)^3+2/3969*(18016+26075*x)*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.00, 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 = {99, 154, 151, 65, 212} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{189 (3 x+2)^2}+\frac {2 \sqrt {1-2 x} (26075 x+18016)}{3969 (3 x+2)}-\frac {92996 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(189*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)^3) + (2*Sqrt[1 -
2*x]*(18016 + 26075*x))/(3969*(2 + 3*x)) - (92996*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3969*Sqrt[21])

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 151

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

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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 {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{378} \int \frac {(544-2980 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}+\frac {46498 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3969}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {46498 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3969}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {92996 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 63, normalized size = 0.59 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (112187+484618 x+695043 x^2+330750 x^3\right )}{(2+3 x)^3}-92996 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{83349} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*Sqrt[1 - 2*x]*(112187 + 484618*x + 695043*x^2 + 330750*x^3))/(2 + 3*x)^3 - 92996*Sqrt[21]*ArcTanh[Sqrt[3/
7]*Sqrt[1 - 2*x]])/83349

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Maple [A]
time = 0.12, size = 66, normalized size = 0.62

method	result	size
risch	\(-\frac {661500 x^{4}+1059336 x^{3}+274193 x^{2}-260244 x -112187}{3969 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {92996 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\)	\(56\)
derivativedivides	\(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\)	\(66\)
default	\(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\)	\(66\)
trager	\(\frac {\left (330750 x^{3}+695043 x^{2}+484618 x +112187\right ) \sqrt {1-2 x}}{3969 \left (2+3 x \right )^{3}}-\frac {46498 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{83349}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/81*(1-2*x)^(1/2)+2/3*(-3727/147*(1-2*x)^(5/2)+22046/189*(1-2*x)^(3/2)-3623/27*(1-2*x)^(1/2))/(-4-6*x)^3-92
996/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.61, size = 101, normalized size = 0.94 \begin {gather*} \frac {46498}{83349} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (33543 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}\right )}}{3969 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

46498/83349*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 250/81*sqrt(-2*x + 1)
+ 2/3969*(33543*(-2*x + 1)^(5/2) - 154322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*
x - 1)^2 + 882*x - 98)

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Fricas [A]
time = 0.80, size = 89, normalized size = 0.83 \begin {gather*} \frac {46498 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (330750 \, x^{3} + 695043 \, x^{2} + 484618 \, x + 112187\right )} \sqrt {-2 \, x + 1}}{83349 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/83349*(46498*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(
330750*x^3 + 695043*x^2 + 484618*x + 112187)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*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)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [A]
time = 1.69, size = 93, normalized size = 0.87 \begin {gather*} \frac {46498}{83349} \, \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 {250}{81} \, \sqrt {-2 \, x + 1} + \frac {33543 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}}{15876 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

46498/83349*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 250/81*sqrt(
-2*x + 1) + 1/15876*(33543*(2*x - 1)^2*sqrt(-2*x + 1) - 154322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(3*x
+ 2)^3

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Mupad [B]
time = 0.06, size = 80, normalized size = 0.75 \begin {gather*} \frac {250\,\sqrt {1-2\,x}}{81}-\frac {92996\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{83349}+\frac {\frac {7246\,\sqrt {1-2\,x}}{2187}-\frac {44092\,{\left (1-2\,x\right )}^{3/2}}{15309}+\frac {7454\,{\left (1-2\,x\right )}^{5/2}}{11907}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(250*(1 - 2*x)^(1/2))/81 - (92996*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/83349 + ((7246*(1 - 2*x)^(1/2)
)/2187 - (44092*(1 - 2*x)^(3/2))/15309 + (7454*(1 - 2*x)^(5/2))/11907)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3
- 98/27)

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