3.1811 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=88 \[ \frac {23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac {(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac {2381 \sqrt {1-2 x}}{2646 (3 x+2)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {21}} \]

[Out]

-1/189*(1-2*x)^(3/2)/(2+3*x)^3+23/294*(1-2*x)^(3/2)/(2+3*x)^2+2381/27783*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)-2381/2646*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.02, antiderivative size = 88, 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 = {89, 78, 47, 63, 206} \[ \frac {23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac {(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac {2381 \sqrt {1-2 x}}{2646 (3 x+2)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - 2*x)^(3/2)/(189*(2 + 3*x)^3) + (23*(1 - 2*x)^(3/2))/(294*(2 + 3*x)^2) - (2381*Sqrt[1 - 2*x])/(2646*(2 +
3*x)) + (2381*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1323*Sqrt[21])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 78

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {1}{189} \int \frac {\sqrt {1-2 x} (843+1575 x)}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}+\frac {2381}{882} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}-\frac {2381 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2646}\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}+\frac {2381 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2646}\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 74, normalized size = 0.84 \[ -\frac {4762 (3 x+2)^3 \sqrt {42 x-21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )-21 \left (45342 x^3+34831 x^2-10503 x-9124\right )}{55566 \sqrt {1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/55566*(-21*(-9124 - 10503*x + 34831*x^2 + 45342*x^3) + 4762*(2 + 3*x)^3*Sqrt[-21 + 42*x]*ArcTan[Sqrt[3/7]*S
qrt[-1 + 2*x]])/(Sqrt[1 - 2*x]*(2 + 3*x)^3)

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fricas [A]  time = 0.79, size = 85, normalized size = 0.97 \[ \frac {2381 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (22671 \, x^{2} + 28751 \, x + 9124\right )} \sqrt {-2 \, x + 1}}{55566 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/55566*(2381*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*(2
2671*x^2 + 28751*x + 9124)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 1.21, size = 84, normalized size = 0.95 \[ -\frac {2381}{55566} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {22671 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 102844 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 116669 \, \sqrt {-2 \, x + 1}}{10584 \, {\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2381/55566*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/10584*(226
71*(2*x - 1)^2*sqrt(-2*x + 1) - 102844*(-2*x + 1)^(3/2) + 116669*sqrt(-2*x + 1))/(3*x + 2)^3

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maple [A]  time = 0.01, size = 57, normalized size = 0.65 \[ \frac {2381 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{27783}-\frac {108 \left (-\frac {2519 \left (-2 x +1\right )^{\frac {5}{2}}}{15876}+\frac {3673 \left (-2 x +1\right )^{\frac {3}{2}}}{5103}-\frac {2381 \sqrt {-2 x +1}}{2916}\right )}{\left (-6 x -4\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-108*(-2519/15876*(-2*x+1)^(5/2)+3673/5103*(-2*x+1)^(3/2)-2381/2916*(-2*x+1)^(1/2))/(-6*x-4)^3+2381/27783*arct
anh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.30, size = 92, normalized size = 1.05 \[ -\frac {2381}{55566} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {22671 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 102844 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 116669 \, \sqrt {-2 \, x + 1}}{1323 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2381/55566*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/1323*(22671*(-2*x +
1)^(5/2) - 102844*(-2*x + 1)^(3/2) + 116669*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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mupad [B]  time = 1.20, size = 72, normalized size = 0.82 \[ \frac {2381\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{27783}-\frac {\frac {2381\,\sqrt {1-2\,x}}{729}-\frac {14692\,{\left (1-2\,x\right )}^{3/2}}{5103}+\frac {2519\,{\left (1-2\,x\right )}^{5/2}}{3969}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2381*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/27783 - ((2381*(1 - 2*x)^(1/2))/729 - (14692*(1 - 2*x)^(3/
2))/5103 + (2519*(1 - 2*x)^(5/2))/3969)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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