3.2446 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt{5 x+3}}+\frac{102293609 \sqrt{1-2 x}}{18816 (3 x+2) \sqrt{5 x+3}}+\frac{587477 \sqrt{1-2 x}}{1344 (3 x+2)^2 \sqrt{5 x+3}}+\frac{12023 \sqrt{1-2 x}}{240 (3 x+2)^3 \sqrt{5 x+3}}+\frac{2513 \sqrt{1-2 x}}{360 (3 x+2)^4 \sqrt{5 x+3}}-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{5 x+3}}+\frac{3538809681 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(-4639661185*Sqrt[1 - 2*x])/(56448*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*Sqrt[3 + 5*x]) + (2513
*Sqrt[1 - 2*x])/(360*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (12023*Sqrt[1 - 2*x])/(240*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (587
477*Sqrt[1 - 2*x])/(1344*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (102293609*Sqrt[1 - 2*x])/(18816*(2 + 3*x)*Sqrt[3 + 5*x]
) + (3538809681*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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Rubi [A]  time = 0.0771676, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 149, 151, 152, 12, 93, 204} \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt{5 x+3}}+\frac{102293609 \sqrt{1-2 x}}{18816 (3 x+2) \sqrt{5 x+3}}+\frac{587477 \sqrt{1-2 x}}{1344 (3 x+2)^2 \sqrt{5 x+3}}+\frac{12023 \sqrt{1-2 x}}{240 (3 x+2)^3 \sqrt{5 x+3}}+\frac{2513 \sqrt{1-2 x}}{360 (3 x+2)^4 \sqrt{5 x+3}}-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{5 x+3}}+\frac{3538809681 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(3/2)),x]

[Out]

(-4639661185*Sqrt[1 - 2*x])/(56448*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*Sqrt[3 + 5*x]) + (2513
*Sqrt[1 - 2*x])/(360*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (12023*Sqrt[1 - 2*x])/(240*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (587
477*Sqrt[1 - 2*x])/(1344*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (102293609*Sqrt[1 - 2*x])/(18816*(2 + 3*x)*Sqrt[3 + 5*x]
) + (3538809681*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{1}{15} \int \frac{\left (\frac{491}{2}-260 x\right ) \sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}-\frac{1}{180} \int \frac{-\frac{124003}{4}+48180 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}-\frac{\int \frac{-\frac{31387125}{8}+\frac{11361735 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx}{3780}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}+\frac{587477 \sqrt{1-2 x}}{1344 (2+3 x)^2 \sqrt{3+5 x}}-\frac{\int \frac{-\frac{5806022145}{16}+\frac{925276275 x}{2}}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{52920}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}+\frac{587477 \sqrt{1-2 x}}{1344 (2+3 x)^2 \sqrt{3+5 x}}+\frac{102293609 \sqrt{1-2 x}}{18816 (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{-\frac{685091891715}{32}+\frac{161112434175 x}{8}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{370440}\\ &=-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}+\frac{587477 \sqrt{1-2 x}}{1344 (2+3 x)^2 \sqrt{3+5 x}}+\frac{102293609 \sqrt{1-2 x}}{18816 (2+3 x) \sqrt{3+5 x}}+\frac{\int -\frac{36785926633995}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2037420}\\ &=-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}+\frac{587477 \sqrt{1-2 x}}{1344 (2+3 x)^2 \sqrt{3+5 x}}+\frac{102293609 \sqrt{1-2 x}}{18816 (2+3 x) \sqrt{3+5 x}}-\frac{3538809681 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{12544}\\ &=-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}+\frac{587477 \sqrt{1-2 x}}{1344 (2+3 x)^2 \sqrt{3+5 x}}+\frac{102293609 \sqrt{1-2 x}}{18816 (2+3 x) \sqrt{3+5 x}}-\frac{3538809681 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{6272}\\ &=-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt{3+5 x}}+\frac{2513 \sqrt{1-2 x}}{360 (2+3 x)^4 \sqrt{3+5 x}}+\frac{12023 \sqrt{1-2 x}}{240 (2+3 x)^3 \sqrt{3+5 x}}+\frac{587477 \sqrt{1-2 x}}{1344 (2+3 x)^2 \sqrt{3+5 x}}+\frac{102293609 \sqrt{1-2 x}}{18816 (2+3 x) \sqrt{3+5 x}}+\frac{3538809681 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{6272 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.188896, size = 157, normalized size = 0.78 \[ \frac{131376 (3 x+2) (1-2 x)^{7/2}+18816 (1-2 x)^{7/2}+(3 x+2)^2 \left (973656 (1-2 x)^{7/2}+9748787 (3 x+2) \left (2 (1-2 x)^{5/2}+55 (3 x+2) \left (33 \sqrt{7} (3 x+2) \sqrt{5 x+3} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\sqrt{1-2 x} (101 x+65)\right )\right )\right )}{219520 (3 x+2)^5 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(3/2)),x]

[Out]

(18816*(1 - 2*x)^(7/2) + 131376*(1 - 2*x)^(7/2)*(2 + 3*x) + (2 + 3*x)^2*(973656*(1 - 2*x)^(7/2) + 9748787*(2 +
3*x)*(2*(1 - 2*x)^(5/2) + 55*(2 + 3*x)*(-(Sqrt[1 - 2*x]*(65 + 101*x)) + 33*Sqrt[7]*(2 + 3*x)*Sqrt[3 + 5*x]*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))))/(219520*(2 + 3*x)^5*Sqrt[3 + 5*x])

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Maple [B]  time = 0.015, size = 346, normalized size = 1.7 \begin{align*} -{\frac{1}{439040\, \left ( 2+3\,x \right ) ^{5}} \left ( 21498268812075\,\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{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+25455981805688\,{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 ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x)

[Out]

-1/439040*(21498268812075*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+84559857327495*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+138544399011150*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^4+8768959639650*x^5*(-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^3+29036530544490*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+38452412617500*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+25455981805688*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arct
an(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))*(1-2*x)^(1/2)/(2+3*x)^5/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [B]  time = 1.90346, size = 537, normalized size = 2.66 \begin{align*} -\frac{3538809681}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{4639661185 \, x}{28224 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{4844248403}{56448 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{135 \,{\left (243 \, \sqrt{-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt{-10 \, x^{2} - x + 3} x + 32 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{5341}{360 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{242879}{2160 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{315689}{320 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{33314567}{2688 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-3538809681/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4639661185/28224*x/sqrt(-10*x^2
- x + 3) - 4844248403/56448/sqrt(-10*x^2 - x + 3) + 343/135/(243*sqrt(-10*x^2 - x + 3)*x^5 + 810*sqrt(-10*x^2
- x + 3)*x^4 + 1080*sqrt(-10*x^2 - x + 3)*x^3 + 720*sqrt(-10*x^2 - x + 3)*x^2 + 240*sqrt(-10*x^2 - x + 3)*x +
32*sqrt(-10*x^2 - x + 3)) + 5341/360/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(
-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 242879/2160/(27*sqrt(-10*x^2 -
x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 315689/32
0/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 33314567/2688/(3*sqrt
(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.86699, size = 528, normalized size = 2.61 \begin{align*} \frac{17694048405 \, \sqrt{7}{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\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 (626354259975 \, x^{5} + 2074037896035 \, x^{4} + 2746600901250 \, x^{3} + 1818284414692 \, x^{2} + 601741553688 \, x + 79638637088\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{439040 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/439040*(17694048405*sqrt(7)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*arctan(1/14*
sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(626354259975*x^5 + 2074037896035*x^4
+ 2746600901250*x^3 + 1818284414692*x^2 + 601741553688*x + 79638637088)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(1215*x^
6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)

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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)/(2+3*x)**6/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 4.47627, size = 674, normalized size = 3.34 \begin{align*} -\frac{3538809681}{878080} \, \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{3025}{2} \, \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 )} - \frac{121 \,{\left (34728039 \, \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} + 30879615760 \, \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} + 10961021460480 \, \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} + 1791349451136000 \, \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} + 112299870108160000 \, \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 )}}{3136 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-3538809681/878080*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)))) - 3025/2*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))) - 121/3136*(34728039*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 + 30879615760*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 + 10961021460480*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 1791349451136000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 112299870108160000*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)^5