3.17.74 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac {7 (1-2 x)^{3/2}}{180 (3 x+2)^4}-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}+\frac {31 \sqrt {1-2 x}}{3528 (3 x+2)}+\frac {31 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {31 \sqrt {1-2 x}}{108 (3 x+2)^3}+\frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}} \]

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Rubi [A]  time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {89, 78, 47, 51, 63, 206} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{180 (3 x+2)^4}-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}+\frac {31 \sqrt {1-2 x}}{3528 (3 x+2)}+\frac {31 \sqrt {1-2 x}}{1512 (3 x+2)^2}-\frac {31 \sqrt {1-2 x}}{108 (3 x+2)^3}+\frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-(1 - 2*x)^(3/2)/(315*(2 + 3*x)^5) + (7*(1 - 2*x)^(3/2))/(180*(2 + 3*x)^4) - (31*Sqrt[1 - 2*x])/(108*(2 + 3*x)
^3) + (31*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2) + (31*Sqrt[1 - 2*x])/(3528*(2 + 3*x)) + (31*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/(1764*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 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 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)^6} \, dx &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {\sqrt {1-2 x} (1407+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}+\frac {31}{12} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}-\frac {31}{108} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}-\frac {31}{504} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}-\frac {31 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3528}\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}+\frac {31 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3528}\\ &=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}+\frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 47, normalized size = 0.37 \begin {gather*} \frac {(1-2 x)^{3/2} \left (\frac {343 (147 x+94)}{(3 x+2)^5}-2480 \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{432180} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

((1 - 2*x)^(3/2)*((343*(94 + 147*x))/(2 + 3*x)^5 - 2480*Hypergeometric2F1[3/2, 4, 5/2, 3/7 - (6*x)/7]))/432180

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IntegrateAlgebraic [A]  time = 0.33, size = 88, normalized size = 0.69 \begin {gather*} \frac {31 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}}-\frac {\left (12555 (1-2 x)^4-136710 (1-2 x)^3+348096 (1-2 x)^2-68810 (1-2 x)-372155\right ) \sqrt {1-2 x}}{8820 (3 (1-2 x)-7)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-1/8820*((-372155 - 68810*(1 - 2*x) + 348096*(1 - 2*x)^2 - 136710*(1 - 2*x)^3 + 12555*(1 - 2*x)^4)*Sqrt[1 - 2*
x])/(-7 + 3*(1 - 2*x))^5 + (31*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1764*Sqrt[21])

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fricas [A]  time = 1.27, size = 115, normalized size = 0.90 \begin {gather*} \frac {155 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (12555 \, x^{4} + 43245 \, x^{3} + 3324 \, x^{2} - 33434 \, x - 13564\right )} \sqrt {-2 \, x + 1}}{370440 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/370440*(155*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt(21)*sqrt(-2*x + 1
) - 5)/(3*x + 2)) + 21*(12555*x^4 + 43245*x^3 + 3324*x^2 - 33434*x - 13564)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 1.36, size = 116, normalized size = 0.91 \begin {gather*} -\frac {31}{74088} \, \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 {12555 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 136710 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 348096 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 68810 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 372155 \, \sqrt {-2 \, x + 1}}{282240 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

-31/74088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/282240*(1255
5*(2*x - 1)^4*sqrt(-2*x + 1) + 136710*(2*x - 1)^3*sqrt(-2*x + 1) + 348096*(2*x - 1)^2*sqrt(-2*x + 1) - 68810*(
-2*x + 1)^(3/2) - 372155*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.01, size = 75, normalized size = 0.59 \begin {gather*} \frac {31 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{37044}-\frac {3888 \left (\frac {31 \left (-2 x +1\right )^{\frac {9}{2}}}{84672}-\frac {31 \left (-2 x +1\right )^{\frac {7}{2}}}{7776}+\frac {37 \left (-2 x +1\right )^{\frac {5}{2}}}{3645}-\frac {983 \left (-2 x +1\right )^{\frac {3}{2}}}{489888}-\frac {1519 \sqrt {-2 x +1}}{139968}\right )}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^2*(-2*x+1)^(1/2)/(3*x+2)^6,x)

[Out]

-3888*(31/84672*(-2*x+1)^(9/2)-31/7776*(-2*x+1)^(7/2)+37/3645*(-2*x+1)^(5/2)-983/489888*(-2*x+1)^(3/2)-1519/13
9968*(-2*x+1)^(1/2))/(-6*x-4)^5+31/37044*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.20, size = 128, normalized size = 1.00 \begin {gather*} -\frac {31}{74088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12555 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 136710 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 348096 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 68810 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 372155 \, \sqrt {-2 \, x + 1}}{8820 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

-31/74088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/8820*(12555*(-2*x + 1
)^(9/2) - 136710*(-2*x + 1)^(7/2) + 348096*(-2*x + 1)^(5/2) - 68810*(-2*x + 1)^(3/2) - 372155*sqrt(-2*x + 1))/
(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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mupad [B]  time = 0.07, size = 108, normalized size = 0.84 \begin {gather*} \frac {31\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{37044}-\frac {\frac {1519\,\sqrt {1-2\,x}}{8748}+\frac {983\,{\left (1-2\,x\right )}^{3/2}}{30618}-\frac {592\,{\left (1-2\,x\right )}^{5/2}}{3645}+\frac {31\,{\left (1-2\,x\right )}^{7/2}}{486}-\frac {31\,{\left (1-2\,x\right )}^{9/2}}{5292}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^2)/(3*x + 2)^6,x)

[Out]

(31*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/37044 - ((1519*(1 - 2*x)^(1/2))/8748 + (983*(1 - 2*x)^(3/2))
/30618 - (592*(1 - 2*x)^(5/2))/3645 + (31*(1 - 2*x)^(7/2))/486 - (31*(1 - 2*x)^(9/2))/5292)/((24010*x)/81 + (3
430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)

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sympy [F(-1)]  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)**2*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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