3.18.55 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=174 \[ \frac {2 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^6}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{735 (3 x+2)^5}-\frac {\sqrt {1-2 x} (237807 x+146585)}{185220 (3 x+2)^4}-\frac {4369 \sqrt {1-2 x}}{1210104 (3 x+2)}-\frac {4369 \sqrt {1-2 x}}{518616 (3 x+2)^2}-\frac {4369 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}} \]

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Rubi [A]  time = 0.06, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {2 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^6}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {173 \sqrt {1-2 x} (5 x+3)^2}{735 (3 x+2)^5}-\frac {\sqrt {1-2 x} (237807 x+146585)}{185220 (3 x+2)^4}-\frac {4369 \sqrt {1-2 x}}{1210104 (3 x+2)}-\frac {4369 \sqrt {1-2 x}}{518616 (3 x+2)^2}-\frac {4369 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

(-4369*Sqrt[1 - 2*x])/(518616*(2 + 3*x)^2) - (4369*Sqrt[1 - 2*x])/(1210104*(2 + 3*x)) - (173*Sqrt[1 - 2*x]*(3
+ 5*x)^2)/(735*(2 + 3*x)^5) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(21*(2 + 3*x)^7) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(
7*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(146585 + 237807*x))/(185220*(2 + 3*x)^4) - (4369*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(605052*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 {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {1}{378} \int \frac {(3+5 x)^2 (-1674+2160 x)}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\int \frac {(3+5 x) (-118854+144450 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{39690}\\ &=-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}+\frac {4369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{37044}\\ &=-\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}+\frac {4369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{172872}\\ &=-\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}+\frac {4369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1210104}\\ &=-\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}-\frac {4369 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1210104}\\ &=-\frac {4369 \sqrt {1-2 x}}{518616 (2+3 x)^2}-\frac {4369 \sqrt {1-2 x}}{1210104 (2+3 x)}-\frac {173 \sqrt {1-2 x} (3+5 x)^2}{735 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^6}-\frac {\sqrt {1-2 x} (146585+237807 x)}{185220 (2+3 x)^4}-\frac {4369 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 52, normalized size = 0.30 \begin {gather*} \frac {(1-2 x)^{5/2} \left (\frac {84035 \left (8575 x^2+11393 x+3785\right )}{(3 x+2)^7}-279616 \, _2F_1\left (\frac {5}{2},6;\frac {7}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{86472015} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

((1 - 2*x)^(5/2)*((84035*(3785 + 11393*x + 8575*x^2))/(2 + 3*x)^7 - 279616*Hypergeometric2F1[5/2, 6, 7/2, 3/7
- (6*x)/7]))/86472015

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IntegrateAlgebraic [A]  time = 0.56, size = 106, normalized size = 0.61 \begin {gather*} \frac {\left (15925005 (1-2 x)^6-247722300 (1-2 x)^5+829056921 (1-2 x)^4+304480512 (1-2 x)^3-5224501569 (1-2 x)^2+7342978300 (1-2 x)-2570042405\right ) \sqrt {1-2 x}}{3025260 (3 (1-2 x)-7)^7}-\frac {4369 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

((-2570042405 + 7342978300*(1 - 2*x) - 5224501569*(1 - 2*x)^2 + 304480512*(1 - 2*x)^3 + 829056921*(1 - 2*x)^4
- 247722300*(1 - 2*x)^5 + 15925005*(1 - 2*x)^6)*Sqrt[1 - 2*x])/(3025260*(-7 + 3*(1 - 2*x))^7) - (4369*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(605052*Sqrt[21])

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fricas [A]  time = 0.90, size = 144, normalized size = 0.83 \begin {gather*} \frac {21845 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (15925005 \, x^{6} + 76086135 \, x^{5} - 42669876 \, x^{4} - 182748162 \, x^{3} - 98441652 \, x^{2} + 606784 \, x + 7033976\right )} \sqrt {-2 \, x + 1}}{127060920 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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)^8,x, algorithm="fricas")

[Out]

1/127060920*(21845*sqrt(21)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 12
8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(15925005*x^6 + 76086135*x^5 - 42669876*x^4 - 18274
8162*x^3 - 98441652*x^2 + 606784*x + 7033976)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 +
15120*x^3 + 6048*x^2 + 1344*x + 128)

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giac [A]  time = 0.94, size = 148, normalized size = 0.85 \begin {gather*} \frac {4369}{25412184} \, \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 {15925005 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 247722300 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 829056921 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 304480512 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 5224501569 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 7342978300 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2570042405 \, \sqrt {-2 \, x + 1}}{387233280 \, {\left (3 \, x + 2\right )}^{7}} \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)^8,x, algorithm="giac")

[Out]

4369/25412184*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/38723328
0*(15925005*(2*x - 1)^6*sqrt(-2*x + 1) + 247722300*(2*x - 1)^5*sqrt(-2*x + 1) + 829056921*(2*x - 1)^4*sqrt(-2*
x + 1) - 304480512*(2*x - 1)^3*sqrt(-2*x + 1) - 5224501569*(2*x - 1)^2*sqrt(-2*x + 1) + 7342978300*(-2*x + 1)^
(3/2) - 2570042405*sqrt(-2*x + 1))/(3*x + 2)^7

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maple [A]  time = 0.01, size = 93, normalized size = 0.53 \begin {gather*} -\frac {4369 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{12706092}+\frac {\frac {353889 \left (-2 x +1\right )^{\frac {13}{2}}}{67228}-\frac {196605 \left (-2 x +1\right )^{\frac {11}{2}}}{2401}+\frac {5639843 \left (-2 x +1\right )^{\frac {9}{2}}}{20580}+\frac {172608 \left (-2 x +1\right )^{\frac {7}{2}}}{1715}-\frac {725323 \left (-2 x +1\right )^{\frac {5}{2}}}{420}+\frac {21845 \left (-2 x +1\right )^{\frac {3}{2}}}{9}-\frac {30583 \sqrt {-2 x +1}}{36}}{\left (-6 x -4\right )^{7}} \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)^8,x)

[Out]

69984*(4369/58084992*(-2*x+1)^(13/2)-21845/18670176*(-2*x+1)^(11/2)+5639843/1440270720*(-2*x+1)^(9/2)+1798/125
0235*(-2*x+1)^(7/2)-725323/29393280*(-2*x+1)^(5/2)+21845/629856*(-2*x+1)^(3/2)-30583/2519424*(-2*x+1)^(1/2))/(
-6*x-4)^7-4369/12706092*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.09, size = 164, normalized size = 0.94 \begin {gather*} \frac {4369}{25412184} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {15925005 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 247722300 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 829056921 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 304480512 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 5224501569 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 7342978300 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2570042405 \, \sqrt {-2 \, x + 1}}{3025260 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\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)^8,x, algorithm="maxima")

[Out]

4369/25412184*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3025260*(15925005
*(-2*x + 1)^(13/2) - 247722300*(-2*x + 1)^(11/2) + 829056921*(-2*x + 1)^(9/2) + 304480512*(-2*x + 1)^(7/2) - 5
224501569*(-2*x + 1)^(5/2) + 7342978300*(-2*x + 1)^(3/2) - 2570042405*sqrt(-2*x + 1))/(2187*(2*x - 1)^7 + 3572
1*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 4941258*
x - 1647086)

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mupad [B]  time = 1.19, size = 144, normalized size = 0.83 \begin {gather*} -\frac {\frac {21845\,{\left (1-2\,x\right )}^{3/2}}{19683}-\frac {30583\,\sqrt {1-2\,x}}{78732}-\frac {725323\,{\left (1-2\,x\right )}^{5/2}}{918540}+\frac {57536\,{\left (1-2\,x\right )}^{7/2}}{1250235}+\frac {5639843\,{\left (1-2\,x\right )}^{9/2}}{45008460}-\frac {21845\,{\left (1-2\,x\right )}^{11/2}}{583443}+\frac {4369\,{\left (1-2\,x\right )}^{13/2}}{1815156}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}-\frac {4369\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12706092} \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)^8,x)

[Out]

- ((21845*(1 - 2*x)^(3/2))/19683 - (30583*(1 - 2*x)^(1/2))/78732 - (725323*(1 - 2*x)^(5/2))/918540 + (57536*(1
- 2*x)^(7/2))/1250235 + (5639843*(1 - 2*x)^(9/2))/45008460 - (21845*(1 - 2*x)^(11/2))/583443 + (4369*(1 - 2*x
)^(13/2))/1815156)/((1647086*x)/729 + (117649*(2*x - 1)^2)/81 + (84035*(2*x - 1)^3)/81 + (12005*(2*x - 1)^4)/2
7 + (343*(2*x - 1)^5)/3 + (49*(2*x - 1)^6)/3 + (2*x - 1)^7 - 1647086/2187) - (4369*21^(1/2)*atanh((21^(1/2)*(1
- 2*x)^(1/2))/7))/12706092

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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)**8,x)

[Out]

Timed out

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