3.1894 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx\)

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.0593299, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 145, 51, 63, 206} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{7 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac{173 \sqrt{1-2 x} (5 x+3)^2}{735 (3 x+2)^5}-\frac{\sqrt{1-2 x} (237807 x+146585)}{185220 (3 x+2)^4}-\frac{4369 \sqrt{1-2 x}}{1210104 (3 x+2)}-\frac{4369 \sqrt{1-2 x}}{518616 (3 x+2)^2}-\frac{4369 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{605052 \sqrt{21}} \]

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 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 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 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 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 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*}

Mathematica [C]  time = 0.0316445, size = 52, normalized size = 0.3 \[ \frac{(1-2 x)^{5/2} \left (\frac{84035 \left (8575 x^2+11393 x+3785\right )}{(3 x+2)^7}-279616 \, _2F_1\left (\frac{5}{2},6;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{86472015} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 93, normalized size = 0.5 \begin{align*} 69984\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{7}} \left ({\frac{4369\, \left ( 1-2\,x \right ) ^{13/2}}{58084992}}-{\frac{21845\, \left ( 1-2\,x \right ) ^{11/2}}{18670176}}+{\frac{5639843\, \left ( 1-2\,x \right ) ^{9/2}}{1440270720}}+{\frac{1798\, \left ( 1-2\,x \right ) ^{7/2}}{1250235}}-{\frac{725323\, \left ( 1-2\,x \right ) ^{5/2}}{29393280}}+{\frac{21845\, \left ( 1-2\,x \right ) ^{3/2}}{629856}}-{\frac{30583\,\sqrt{1-2\,x}}{2519424}} \right ) }-{\frac{4369\,\sqrt{21}}{12706092}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

69984*(4369/58084992*(1-2*x)^(13/2)-21845/18670176*(1-2*x)^(11/2)+5639843/1440270720*(1-2*x)^(9/2)+1798/125023
5*(1-2*x)^(7/2)-725323/29393280*(1-2*x)^(5/2)+21845/629856*(1-2*x)^(3/2)-30583/2519424*(1-2*x)^(1/2))/(-6*x-4)
^7-4369/12706092*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 5.33708, size = 221, normalized size = 1.27 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.23842, size = 497, normalized size = 2.86 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.72416, size = 200, normalized size = 1.15 \begin{align*} \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{align*}

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