3.1967 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^8} \, dx\)

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

[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])

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Rubi [A]  time = 0.0747664, antiderivative size = 181, normalized size of antiderivative = 1., 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} \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{7 (3 x+2)^5}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{189 (3 x+2)^6}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac{3223 \sqrt{1-2 x} (5 x+3)^2}{2646 (3 x+2)^4}-\frac{11 \sqrt{1-2 x} (301765 x+187704)}{333396 (3 x+2)^3}+\frac{33935 \sqrt{1-2 x}}{2333772 (3 x+2)}+\frac{33935 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1166886 \sqrt{21}} \]

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 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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)^{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*}

Mathematica [C]  time = 0.0348526, size = 52, normalized size = 0.29 \[ \frac{(1-2 x)^{7/2} \left (\frac{823543 \left (18375 x^2+24448 x+8133\right )}{(3 x+2)^7}-4343680 \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{1089547389} \]

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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Maple [A]  time = 0.011, size = 93, normalized size = 0.5 \begin{align*} 69984\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{7}} \left ( -{\frac{33935\, \left ( 1-2\,x \right ) ^{13/2}}{112021056}}-{\frac{176975\, \left ( 1-2\,x \right ) ^{11/2}}{108020304}}+{\frac{4931597\, \left ( 1-2\,x \right ) ^{9/2}}{185177664}}-{\frac{96613\, \left ( 1-2\,x \right ) ^{7/2}}{964467}}+{\frac{1920721\, \left ( 1-2\,x \right ) ^{5/2}}{11337408}}-{\frac{1187725\, \left ( 1-2\,x \right ) ^{3/2}}{8503056}}+{\frac{1662815\,\sqrt{1-2\,x}}{34012224}} \right ) }+{\frac{33935\,\sqrt{21}}{24504606}{\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)^(5/2)*(3+5*x)^3/(2+3*x)^8,x)

[Out]

69984*(-33935/112021056*(1-2*x)^(13/2)-176975/108020304*(1-2*x)^(11/2)+4931597/185177664*(1-2*x)^(9/2)-96613/9
64467*(1-2*x)^(7/2)+1920721/11337408*(1-2*x)^(5/2)-1187725/8503056*(1-2*x)^(3/2)+1662815/34012224*(1-2*x)^(1/2
))/(-6*x-4)^7+33935/24504606*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 5.28164, size = 221, normalized size = 1.22 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.39861, size = 501, normalized size = 2.77 \begin{align*} \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{align*}

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

[Out]

Timed out

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Giac [A]  time = 2.40031, size = 200, normalized size = 1.1 \begin{align*} -\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{align*}

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