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

Optimal. Leaf size=127 \[ -\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac {\sqrt {1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac {11237 \sqrt {1-2 x}}{111132 (3 x+2)}+\frac {11237 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}} \]

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Rubi [A]  time = 0.04, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 149, 145, 51, 63, 206} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac {\sqrt {1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac {11237 \sqrt {1-2 x}}{111132 (3 x+2)}+\frac {11237 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11237*Sqrt[1 - 2*x])/(111132*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(630*(2 + 3*x)^4) - (Sqrt[1 - 2*x]*(
3 + 5*x)^3)/(15*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(37224 + 59665*x))/(79380*(2 + 3*x)^3) + (11237*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(55566*Sqrt[21])

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 97

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 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

Rule 149

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] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^3}{(2+3 x)^6} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}+\frac {\int \frac {(346-3310 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{1260}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}-\frac {11237 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{15876}\\ &=\frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}-\frac {11237 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{111132}\\ &=\frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac {11237 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{111132}\\ &=\frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac {11237 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 52, normalized size = 0.41 \begin {gather*} \frac {(1-2 x)^{3/2} \left (\frac {4802 \left (78750 x^2+104667 x+34784\right )}{(3 x+2)^5}-1797920 \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{27227340} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(3/2)*((4802*(34784 + 104667*x + 78750*x^2))/(2 + 3*x)^5 - 1797920*Hypergeometric2F1[3/2, 4, 5/2, 3
/7 - (6*x)/7]))/27227340

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IntegrateAlgebraic [A]  time = 0.38, size = 88, normalized size = 0.69 \begin {gather*} \frac {11237 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}}-\frac {\left (4550985 (1-2 x)^4-18685170 (1-2 x)^3-11651808 (1-2 x)^2+128927330 (1-2 x)-134900185\right ) \sqrt {1-2 x}}{277830 (3 (1-2 x)-7)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/277830*((-134900185 + 128927330*(1 - 2*x) - 11651808*(1 - 2*x)^2 - 18685170*(1 - 2*x)^3 + 4550985*(1 - 2*x)
^4)*Sqrt[1 - 2*x])/(-7 + 3*(1 - 2*x))^5 + (11237*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(55566*Sqrt[21])

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fricas [A]  time = 1.53, size = 115, normalized size = 0.91 \begin {gather*} \frac {56185 \, \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 (4550985 \, x^{4} + 240615 \, x^{3} - 10100352 \, x^{2} - 8471518 \, x - 1984928\right )} \sqrt {-2 \, x + 1}}{11668860 \, {\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)^3*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/11668860*(56185*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*(4550985*x^4 + 240615*x^3 - 10100352*x^2 - 8471518*x - 1984928)*sqrt(-2*x + 1))/(24
3*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 1.28, size = 116, normalized size = 0.91 \begin {gather*} -\frac {11237}{2333772} \, \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 {4550985 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 18685170 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 11651808 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 128927330 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 134900185 \, \sqrt {-2 \, x + 1}}{8890560 \, {\left (3 \, x + 2\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-11237/2333772*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/8890560
*(4550985*(2*x - 1)^4*sqrt(-2*x + 1) + 18685170*(2*x - 1)^3*sqrt(-2*x + 1) - 11651808*(2*x - 1)^2*sqrt(-2*x +
1) + 128927330*(-2*x + 1)^(3/2) - 134900185*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 {11237 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1166886}+\frac {-\frac {11237 \left (-2 x +1\right )^{\frac {9}{2}}}{686}+\frac {4237 \left (-2 x +1\right )^{\frac {7}{2}}}{63}+\frac {39632 \left (-2 x +1\right )^{\frac {5}{2}}}{945}-\frac {263117 \left (-2 x +1\right )^{\frac {3}{2}}}{567}+\frac {78659 \sqrt {-2 x +1}}{162}}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1944*(-11237/1333584*(-2*x+1)^(9/2)+4237/122472*(-2*x+1)^(7/2)+4954/229635*(-2*x+1)^(5/2)-263117/1102248*(-2*x
+1)^(3/2)+78659/314928*(-2*x+1)^(1/2))/(-6*x-4)^5+11237/1166886*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.33, size = 128, normalized size = 1.01 \begin {gather*} -\frac {11237}{2333772} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4550985 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 18685170 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 11651808 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 128927330 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 134900185 \, \sqrt {-2 \, x + 1}}{277830 \, {\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)^3*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

-11237/2333772*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/277830*(4550985*
(-2*x + 1)^(9/2) - 18685170*(-2*x + 1)^(7/2) - 11651808*(-2*x + 1)^(5/2) + 128927330*(-2*x + 1)^(3/2) - 134900
185*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 - 19
208)

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mupad [B]  time = 1.18, size = 108, normalized size = 0.85 \begin {gather*} \frac {11237\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1166886}-\frac {\frac {78659\,\sqrt {1-2\,x}}{39366}-\frac {263117\,{\left (1-2\,x\right )}^{3/2}}{137781}+\frac {39632\,{\left (1-2\,x\right )}^{5/2}}{229635}+\frac {4237\,{\left (1-2\,x\right )}^{7/2}}{15309}-\frac {11237\,{\left (1-2\,x\right )}^{9/2}}{166698}}{\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)^3)/(3*x + 2)^6,x)

[Out]

(11237*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1166886 - ((78659*(1 - 2*x)^(1/2))/39366 - (263117*(1 - 2
*x)^(3/2))/137781 + (39632*(1 - 2*x)^(5/2))/229635 + (4237*(1 - 2*x)^(7/2))/15309 - (11237*(1 - 2*x)^(9/2))/16
6698)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/2
43)

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

[Out]

Timed out

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