3.2348 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=238 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}+\frac{14677525921 \sqrt{1-2 x} \sqrt{5 x+3}}{464679936 (3 x+2)}+\frac{140331343 \sqrt{1-2 x} \sqrt{5 x+3}}{33191424 (3 x+2)^2}+\frac{4014523 \sqrt{1-2 x} \sqrt{5 x+3}}{5927040 (3 x+2)^3}+\frac{341917 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^4}-\frac{9901 \sqrt{1-2 x} \sqrt{5 x+3}}{52920 (3 x+2)^5}-\frac{6219452877 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{17210368 \sqrt{7}} \]

[Out]

(-9901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(52920*(2 + 3*x)^5) + (341917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2963520*(2 + 3
*x)^4) + (4014523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5927040*(2 + 3*x)^3) + (140331343*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(33191424*(2 + 3*x)^2) + (14677525921*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(464679936*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2))/(21*(2 + 3*x)^7) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(252*(2 + 3*x)^6) - (6219452877*ArcTan[S
qrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(17210368*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0976867, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}+\frac{14677525921 \sqrt{1-2 x} \sqrt{5 x+3}}{464679936 (3 x+2)}+\frac{140331343 \sqrt{1-2 x} \sqrt{5 x+3}}{33191424 (3 x+2)^2}+\frac{4014523 \sqrt{1-2 x} \sqrt{5 x+3}}{5927040 (3 x+2)^3}+\frac{341917 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^4}-\frac{9901 \sqrt{1-2 x} \sqrt{5 x+3}}{52920 (3 x+2)^5}-\frac{6219452877 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{17210368 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(52920*(2 + 3*x)^5) + (341917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2963520*(2 + 3
*x)^4) + (4014523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5927040*(2 + 3*x)^3) + (140331343*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(33191424*(2 + 3*x)^2) + (14677525921*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(464679936*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2))/(21*(2 + 3*x)^7) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(252*(2 + 3*x)^6) - (6219452877*ArcTan[S
qrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(17210368*Sqrt[7])

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 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{1}{21} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^7} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{1}{378} \int \frac{\sqrt{3+5 x} \left (-\frac{4941}{4}+1860 x\right )}{\sqrt{1-2 x} (2+3 x)^6} \, dx\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{\int \frac{-\frac{190077}{8}+28470 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx}{39690}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{\int \frac{-\frac{43274943}{16}+\frac{15386265 x}{4}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{1111320}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}+\frac{4014523 \sqrt{1-2 x} \sqrt{3+5 x}}{5927040 (2+3 x)^3}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{\int \frac{-\frac{7990392375}{32}+\frac{1264574745 x}{4}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{23337720}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}+\frac{4014523 \sqrt{1-2 x} \sqrt{3+5 x}}{5927040 (2+3 x)^3}+\frac{140331343 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{\int \frac{-\frac{951748581105}{64}+\frac{221021865225 x}{16}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{326728080}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}+\frac{4014523 \sqrt{1-2 x} \sqrt{3+5 x}}{5927040 (2+3 x)^3}+\frac{140331343 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)^2}+\frac{14677525921 \sqrt{1-2 x} \sqrt{3+5 x}}{464679936 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{\int -\frac{52896446718885}{128 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2287096560}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}+\frac{4014523 \sqrt{1-2 x} \sqrt{3+5 x}}{5927040 (2+3 x)^3}+\frac{140331343 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)^2}+\frac{14677525921 \sqrt{1-2 x} \sqrt{3+5 x}}{464679936 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}+\frac{6219452877 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{34420736}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}+\frac{4014523 \sqrt{1-2 x} \sqrt{3+5 x}}{5927040 (2+3 x)^3}+\frac{140331343 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)^2}+\frac{14677525921 \sqrt{1-2 x} \sqrt{3+5 x}}{464679936 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}+\frac{6219452877 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{17210368}\\ &=-\frac{9901 \sqrt{1-2 x} \sqrt{3+5 x}}{52920 (2+3 x)^5}+\frac{341917 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^4}+\frac{4014523 \sqrt{1-2 x} \sqrt{3+5 x}}{5927040 (2+3 x)^3}+\frac{140331343 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)^2}+\frac{14677525921 \sqrt{1-2 x} \sqrt{3+5 x}}{464679936 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac{6219452877 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{17210368 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.151599, size = 176, normalized size = 0.74 \[ \frac{1}{49} \left (\frac{141599 \left (7 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )-43923 \sqrt{7} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{2458624 (3 x+2)^4}+\frac{11841 (1-2 x)^{5/2} (5 x+3)^{5/2}}{280 (3 x+2)^5}+\frac{333 (1-2 x)^{5/2} (5 x+3)^{5/2}}{28 (3 x+2)^6}+\frac{3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7 + (333*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(28*(2 + 3*x)^6) + (1
1841*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(280*(2 + 3*x)^5) + (141599*(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 1459
40*x + 213240*x^2 + 100159*x^3) - 43923*Sqrt[7]*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/(2
458624*(2 + 3*x)^4))/49

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Maple [B]  time = 0.013, size = 394, normalized size = 1.7 \begin{align*}{\frac{1}{1204725760\, \left ( 2+3\,x \right ) ^{7}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 68009717209995\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+86046428675424\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1204725760*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(68009717209995*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^7+317378680313310*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+634757360626620*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+27740523990690*(-10*x^2-x+3)^(1/2)*x^6+7052859562518
00*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+112199818408020*x^5*(-10*x^2-x+3)^(1/2)+4701
90637501200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+189128663195472*x^4*(-10*x^2-x+3)^(
1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+170069285459584*x^3*(-10*x
^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+86046428675424*x^2*(
-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(
-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^7

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Maxima [A]  time = 3.67252, size = 437, normalized size = 1.84 \begin{align*} \frac{1167483755}{90354432} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{49 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac{333 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1372 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{11841 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{13720 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{424797 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{153664 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{15717489 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2151296 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{700490253 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{60236288 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{9509080845}{60236288} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{6219452877}{240945152} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{8378271231}{120472576} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{2771517227 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{361417728 \,{\left (3 \, x + 2\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)/(2+3*x)^8,x, algorithm="maxima")

[Out]

1167483755/90354432*(-10*x^2 - x + 3)^(3/2) + 3/49*(-10*x^2 - x + 3)^(5/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 +
22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 333/1372*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 486
0*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 11841/13720*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3
+ 720*x^2 + 240*x + 32) + 424797/153664*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 157
17489/2151296*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 700490253/60236288*(-10*x^2 - x + 3)^(5/2
)/(9*x^2 + 12*x + 4) + 9509080845/60236288*sqrt(-10*x^2 - x + 3)*x + 6219452877/240945152*sqrt(7)*arcsin(37/11
*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 8378271231/120472576*sqrt(-10*x^2 - x + 3) + 2771517227/361417728*(-10
*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.60062, size = 614, normalized size = 2.58 \begin{align*} -\frac{31097264385 \, \sqrt{7}{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (1981465999335 \, x^{6} + 8014272743430 \, x^{5} + 13509190228248 \, x^{4} + 12147806104256 \, x^{3} + 6146173476816 \, x^{2} + 1658923773088 \, x + 186609267072\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1204725760 \,{\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)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/1204725760*(31097264385*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344
*x + 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1981465999335*x
^6 + 8014272743430*x^5 + 13509190228248*x^4 + 12147806104256*x^3 + 6146173476816*x^2 + 1658923773088*x + 18660
9267072)*sqrt(5*x + 3)*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)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**8,x)

[Out]

Timed out

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Giac [B]  time = 5.68019, size = 759, normalized size = 3.19 \begin{align*} \frac{6219452877}{2409451520} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (424797 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{13} + 792954400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} - 748492373440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 270037116518400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 49241484970496000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 4873941796864000000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 204705555468288000000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{8605184 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\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)/(2+3*x)^8,x, algorithm="giac")

[Out]

6219452877/2409451520*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 14641/8605184*(424797*sqrt(10)*((sqrt(2
)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^13 + 79295
4400*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^11 - 748492373440*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 - 270037116518400*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5
*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 49241484970496000*sqrt(10)*((sqrt(2)*sqrt(
-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 4873941796864
000000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22)))^3 - 204705555468288000000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(
5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(
5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^7