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

Optimal. Leaf size=181 \[ \frac {11 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {3223 \sqrt {1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac {11 \sqrt {1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac {33935 \sqrt {1-2 x}}{2333772 (3 x+2)}+\frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}} \]

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Rubi [A]  time = 0.07, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {97, 12, 149, 145, 51, 63, 206} \begin {gather*} \frac {11 \sqrt {1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac {3223 \sqrt {1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac {11 \sqrt {1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac {33935 \sqrt {1-2 x}}{2333772 (3 x+2)}+\frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(33935*Sqrt[1 - 2*x])/(2333772*(2 + 3*x)) - (3223*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2646*(2 + 3*x)^4) - ((1 - 2*x)^(
5/2)*(3 + 5*x)^3)/(21*(2 + 3*x)^7) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(189*(2 + 3*x)^6) + (11*Sqrt[1 - 2*x]*(3
+ 5*x)^3)/(7*(2 + 3*x)^5) - (11*Sqrt[1 - 2*x]*(187704 + 301765*x))/(333396*(2 + 3*x)^3) + (33935*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/(1166886*Sqrt[21])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {1}{21} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}-\frac {55}{21} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {55}{378} \int \frac {(42-18 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \int \frac {(3+5 x)^2 (-2016+2250 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{1134}\\ &=-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \int \frac {(3+5 x) (-137988+156780 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{95256}\\ &=-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}-\frac {33935 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{333396}\\ &=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}-\frac {33935 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2333772}\\ &=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}+\frac {33935 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2333772}\\ &=\frac {33935 \sqrt {1-2 x}}{2333772 (2+3 x)}-\frac {3223 \sqrt {1-2 x} (3+5 x)^2}{2646 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{21 (2+3 x)^7}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{189 (2+3 x)^6}+\frac {11 \sqrt {1-2 x} (3+5 x)^3}{7 (2+3 x)^5}-\frac {11 \sqrt {1-2 x} (187704+301765 x)}{333396 (2+3 x)^3}+\frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 52, normalized size = 0.29 \begin {gather*} \frac {(1-2 x)^{7/2} \left (\frac {823543 \left (18375 x^2+24448 x+8133\right )}{(3 x+2)^7}-4343680 \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{1089547389} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(7/2)*((823543*(8133 + 24448*x + 18375*x^2))/(2 + 3*x)^7 - 4343680*Hypergeometric2F1[7/2, 6, 9/2, 3
/7 - (6*x)/7]))/1089547389

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IntegrateAlgebraic [A]  time = 0.60, size = 106, normalized size = 0.59 \begin {gather*} \frac {33935 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1166886 \sqrt {21}}-\frac {\left (24738615 (1-2 x)^6+133793100 (1-2 x)^5-2174834277 (1-2 x)^4+8180415936 (1-2 x)^3-13834953363 (1-2 x)^2+11406910900 (1-2 x)-3992418815\right ) \sqrt {1-2 x}}{1166886 (3 (1-2 x)-7)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1166886*((-3992418815 + 11406910900*(1 - 2*x) - 13834953363*(1 - 2*x)^2 + 8180415936*(1 - 2*x)^3 - 21748342
77*(1 - 2*x)^4 + 133793100*(1 - 2*x)^5 + 24738615*(1 - 2*x)^6)*Sqrt[1 - 2*x])/(-7 + 3*(1 - 2*x))^7 + (33935*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1166886*Sqrt[21])

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fricas [A]  time = 1.28, size = 145, normalized size = 0.80 \begin {gather*} \frac {33935 \, \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 (24738615 \, x^{6} - 141112395 \, x^{5} - 283697388 \, x^{4} - 164222766 \, x^{3} - 39606312 \, x^{2} - 12384752 \, x - 4005436\right )} \sqrt {-2 \, x + 1}}{49009212 \, {\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)^(5/2)*(3+5*x)^3/(2+3*x)^8,x, algorithm="fricas")

[Out]

1/49009212*(33935*sqrt(21)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128
)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(24738615*x^6 - 141112395*x^5 - 283697388*x^4 - 1642
22766*x^3 - 39606312*x^2 - 12384752*x - 4005436)*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 = 0.97, size = 148, normalized size = 0.82 \begin {gather*} -\frac {33935}{49009212} \, \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 {24738615 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - 133793100 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 2174834277 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 8180415936 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 13834953363 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 11406910900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3992418815 \, \sqrt {-2 \, x + 1}}{149361408 \, {\left (3 \, x + 2\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^8,x, algorithm="giac")

[Out]

-33935/49009212*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/149361
408*(24738615*(2*x - 1)^6*sqrt(-2*x + 1) - 133793100*(2*x - 1)^5*sqrt(-2*x + 1) - 2174834277*(2*x - 1)^4*sqrt(
-2*x + 1) - 8180415936*(2*x - 1)^3*sqrt(-2*x + 1) - 13834953363*(2*x - 1)^2*sqrt(-2*x + 1) + 11406910900*(-2*x
+ 1)^(3/2) - 3992418815*sqrt(-2*x + 1))/(3*x + 2)^7

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maple [A]  time = 0.01, size = 93, normalized size = 0.51 \begin {gather*} \frac {33935 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{24504606}+\frac {-\frac {101805 \left (-2 x +1\right )^{\frac {13}{2}}}{4802}-\frac {353950 \left (-2 x +1\right )^{\frac {11}{2}}}{3087}+\frac {4931597 \left (-2 x +1\right )^{\frac {9}{2}}}{2646}-\frac {3091616 \left (-2 x +1\right )^{\frac {7}{2}}}{441}+\frac {1920721 \left (-2 x +1\right )^{\frac {5}{2}}}{162}-\frac {2375450 \left (-2 x +1\right )^{\frac {3}{2}}}{243}+\frac {1662815 \sqrt {-2 x +1}}{486}}{\left (-6 x -4\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^3/(3*x+2)^8,x)

[Out]

69984*(-33935/112021056*(-2*x+1)^(13/2)-176975/108020304*(-2*x+1)^(11/2)+4931597/185177664*(-2*x+1)^(9/2)-9661
3/964467*(-2*x+1)^(7/2)+1920721/11337408*(-2*x+1)^(5/2)-1187725/8503056*(-2*x+1)^(3/2)+1662815/34012224*(-2*x+
1)^(1/2))/(-6*x-4)^7+33935/24504606*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.35, size = 164, normalized size = 0.91 \begin {gather*} -\frac {33935}{49009212} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {24738615 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + 133793100 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2174834277 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 8180415936 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 13834953363 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 11406910900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3992418815 \, \sqrt {-2 \, x + 1}}{1166886 \, {\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)^(5/2)*(3+5*x)^3/(2+3*x)^8,x, algorithm="maxima")

[Out]

-33935/49009212*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1166886*(247386
15*(-2*x + 1)^(13/2) + 133793100*(-2*x + 1)^(11/2) - 2174834277*(-2*x + 1)^(9/2) + 8180415936*(-2*x + 1)^(7/2)
- 13834953363*(-2*x + 1)^(5/2) + 11406910900*(-2*x + 1)^(3/2) - 3992418815*sqrt(-2*x + 1))/(2187*(2*x - 1)^7
+ 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 49
41258*x - 1647086)

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mupad [B]  time = 1.19, size = 143, normalized size = 0.79 \begin {gather*} \frac {\frac {2375450\,{\left (1-2\,x\right )}^{3/2}}{531441}-\frac {1662815\,\sqrt {1-2\,x}}{1062882}-\frac {1920721\,{\left (1-2\,x\right )}^{5/2}}{354294}+\frac {3091616\,{\left (1-2\,x\right )}^{7/2}}{964467}-\frac {4931597\,{\left (1-2\,x\right )}^{9/2}}{5786802}+\frac {353950\,{\left (1-2\,x\right )}^{11/2}}{6751269}+\frac {33935\,{\left (1-2\,x\right )}^{13/2}}{3500658}}{\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 {33935\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{24504606} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^3)/(3*x + 2)^8,x)

[Out]

((2375450*(1 - 2*x)^(3/2))/531441 - (1662815*(1 - 2*x)^(1/2))/1062882 - (1920721*(1 - 2*x)^(5/2))/354294 + (30
91616*(1 - 2*x)^(7/2))/964467 - (4931597*(1 - 2*x)^(9/2))/5786802 + (353950*(1 - 2*x)^(11/2))/6751269 + (33935
*(1 - 2*x)^(13/2))/3500658)/((1647086*x)/729 + (117649*(2*x - 1)^2)/81 + (84035*(2*x - 1)^3)/81 + (12005*(2*x
- 1)^4)/27 + (343*(2*x - 1)^5)/3 + (49*(2*x - 1)^6)/3 + (2*x - 1)^7 - 1647086/2187) + (33935*21^(1/2)*atanh((2
1^(1/2)*(1 - 2*x)^(1/2))/7))/24504606

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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)**(5/2)*(3+5*x)**3/(2+3*x)**8,x)

[Out]

Timed out

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