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

Optimal. Leaf size=134 \[ \frac {3 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^4}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{315 (3 x+2)^3}-\frac {2 \sqrt {1-2 x} (15074 x+9529)}{9261 (3 x+2)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \]

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Rubi [A]  time = 0.04, antiderivative size = 134, 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 = {97, 149, 145, 63, 206} \begin {gather*} \frac {3 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^4}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{315 (3 x+2)^3}-\frac {2 \sqrt {1-2 x} (15074 x+9529)}{9261 (3 x+2)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-67*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(315*(2 + 3*x)^3) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(15*(2 + 3*x)^5) + (3*Sqrt[1
- 2*x]*(3 + 5*x)^3)/(5*(2 + 3*x)^4) - (2*Sqrt[1 - 2*x]*(9529 + 15074*x))/(9261*(2 + 3*x)^2) - (13892*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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 {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {1}{180} \int \frac {(3+5 x)^2 (-684+180 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {\int \frac {(3+5 x) (-48960+6840 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{11340}\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}+\frac {6946 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{9261}\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {6946 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{9261}\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 52, normalized size = 0.39 \begin {gather*} \frac {(1-2 x)^{5/2} \left (\frac {86436 \left (30625 x^2+40790 x+13583\right )}{(3 x+2)^5}-8001792 \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{63530460} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(5/2)*((86436*(13583 + 40790*x + 30625*x^2))/(2 + 3*x)^5 - 8001792*Hypergeometric2F1[5/2, 4, 7/2, 3
/7 - (6*x)/7]))/63530460

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IntegrateAlgebraic [A]  time = 0.40, size = 88, normalized size = 0.66 \begin {gather*} -\frac {4 \sqrt {1-2 x} \left (2452185 (1-2 x)^4-20184570 (1-2 x)^3+61826142 (1-2 x)^2-83386730 (1-2 x)[phone]\right )}{46305 (3 (1-2 x)-7)^5}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(41693365 - 83386730*(1 - 2*x) + 61826142*(1 - 2*x)^2 - 20184570*(1 - 2*x)^3 + 2452185*(1 - 2*x)^4)*Sqrt[1
- 2*x])/(46305*(-7 + 3*(1 - 2*x))^5) - (13892*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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fricas [A]  time = 1.65, size = 114, normalized size = 0.85 \begin {gather*} \frac {34730 \, \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 (4904370 \, x^{4} + 10375830 \, x^{3} + 7992771 \, x^{2} + 2619854 \, x + 300049\right )} \sqrt {-2 \, x + 1}}{972405 \, {\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((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/972405*(34730*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*(4904370*x^4 + 10375830*x^3 + 7992771*x^2 + 2619854*x + 300049)*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 = 1.12, size = 116, normalized size = 0.87 \begin {gather*} \frac {6946}{194481} \, \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 {2452185 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 20184570 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 61826142 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}}{370440 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6946/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/370440*(24
52185*(2*x - 1)^4*sqrt(-2*x + 1) + 20184570*(2*x - 1)^3*sqrt(-2*x + 1) + 61826142*(2*x - 1)^2*sqrt(-2*x + 1) -
83386730*(-2*x + 1)^(3/2) + 41693365*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.01, size = 75, normalized size = 0.56 \begin {gather*} -\frac {13892 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{194481}+\frac {-\frac {217972 \left (-2 x +1\right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (-2 x +1\right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (-2 x +1\right )^{\frac {5}{2}}}{315}+\frac {194488 \left (-2 x +1\right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {-2 x +1}}{27}}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1944*(-54493/500094*(-2*x+1)^(9/2)+4577/5103*(-2*x+1)^(7/2)-210293/76545*(-2*x+1)^(5/2)+24311/6561*(-2*x+1)^(3
/2)-24311/13122*(-2*x+1)^(1/2))/(-6*x-4)^5-13892/194481*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.26, size = 128, normalized size = 0.96 \begin {gather*} \frac {6946}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (2452185 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 20184570 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 61826142 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}\right )}}{46305 \, {\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((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^6,x, algorithm="maxima")

[Out]

6946/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/46305*(2452185*(-2*
x + 1)^(9/2) - 20184570*(-2*x + 1)^(7/2) + 61826142*(-2*x + 1)^(5/2) - 83386730*(-2*x + 1)^(3/2) + 41693365*sq
rt(-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 = 0.08, size = 107, normalized size = 0.80 \begin {gather*} \frac {\frac {97244\,\sqrt {1-2\,x}}{6561}-\frac {194488\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {1682344\,{\left (1-2\,x\right )}^{5/2}}{76545}-\frac {36616\,{\left (1-2\,x\right )}^{7/2}}{5103}+\frac {217972\,{\left (1-2\,x\right )}^{9/2}}{250047}}{\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}}-\frac {13892\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((97244*(1 - 2*x)^(1/2))/6561 - (194488*(1 - 2*x)^(3/2))/6561 + (1682344*(1 - 2*x)^(5/2))/76545 - (36616*(1 -
2*x)^(7/2))/5103 + (217972*(1 - 2*x)^(9/2))/250047)/((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 - 19208/243) - (13892*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/19448
1

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

[Out]

Timed out

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