3.2457 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{5 x+3}}+\frac{270667969 \sqrt{1-2 x}}{18816 (3 x+2) (5 x+3)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (3 x+2)^4 (5 x+3)^{3/2}}-\frac{754386765 \sqrt{1-2 x}}{6272 (5 x+3)^{3/2}}-\frac{46975917593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(-754386765*Sqrt[1 - 2*x])/(6272*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*(3 + 5*x)^(3/2)) + (10
01*Sqrt[1 - 2*x])/(120*(2 + 3*x)^4*(3 + 5*x)^(3/2)) + (53009*Sqrt[1 - 2*x])/(720*(2 + 3*x)^3*(3 + 5*x)^(3/2))
+ (3329689*Sqrt[1 - 2*x])/(4032*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (270667969*Sqrt[1 - 2*x])/(18816*(2 + 3*x)*(3 +
5*x)^(3/2)) + (20529722435*Sqrt[1 - 2*x])/(18816*Sqrt[3 + 5*x]) - (46975917593*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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Rubi [A]  time = 0.0939754, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{5 x+3}}+\frac{270667969 \sqrt{1-2 x}}{18816 (3 x+2) (5 x+3)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (3 x+2)^4 (5 x+3)^{3/2}}-\frac{754386765 \sqrt{1-2 x}}{6272 (5 x+3)^{3/2}}-\frac{46975917593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-754386765*Sqrt[1 - 2*x])/(6272*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*(3 + 5*x)^(3/2)) + (10
01*Sqrt[1 - 2*x])/(120*(2 + 3*x)^4*(3 + 5*x)^(3/2)) + (53009*Sqrt[1 - 2*x])/(720*(2 + 3*x)^3*(3 + 5*x)^(3/2))
+ (3329689*Sqrt[1 - 2*x])/(4032*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (270667969*Sqrt[1 - 2*x])/(18816*(2 + 3*x)*(3 +
5*x)^(3/2)) + (20529722435*Sqrt[1 - 2*x])/(18816*Sqrt[3 + 5*x]) - (46975917593*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)^{5/2}} \, dx &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1}{15} \int \frac{\left (\frac{561}{2}-330 x\right ) \sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}-\frac{1}{180} \int \frac{-\frac{177903}{4}+72435 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}-\frac{\int \frac{-\frac{57169035}{8}[phone] x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx}{3780}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}-\frac{\int \frac{-\frac{14435442945}{16}+\frac{5244260175 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx}{52920}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{270667969 \sqrt{1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}-\frac{\int \frac{-\frac{2658860407665}{32}+\frac{426302051175 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{370440}\\ &=-\frac{754386765 \sqrt{1-2 x}}{6272 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{270667969 \sqrt{1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac{\int \frac{-\frac{300110253247035}{64}+\frac{70576653799575 x}{16}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{6112260}\\ &=-\frac{754386765 \sqrt{1-2 x}}{6272 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{270667969 \sqrt{1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{3+5 x}}-\frac{\int -\frac{16114383891514755}{128 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{33617430}\\ &=-\frac{754386765 \sqrt{1-2 x}}{6272 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{270667969 \sqrt{1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{3+5 x}}+\frac{46975917593 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{12544}\\ &=-\frac{754386765 \sqrt{1-2 x}}{6272 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{270667969 \sqrt{1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{3+5 x}}+\frac{46975917593 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{6272}\\ &=-\frac{754386765 \sqrt{1-2 x}}{6272 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{270667969 \sqrt{1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{3+5 x}}-\frac{46975917593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{6272 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.233065, size = 170, normalized size = 0.76 \[ \frac{3252816 (3 x+2) (1-2 x)^{7/2}+395136 (1-2 x)^{7/2}+(3 x+2)^2 \left (29407896 (1-2 x)^{7/2}+(3 x+2) \left (324091386 (1-2 x)^{7/2}[phone] (3 x+2) \left (3 (1-2 x)^{5/2}-55 (3 x+2) \left (21 \sqrt{7} (5 x+3)^{3/2} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\sqrt{1-2 x} (107 x+62)\right )\right )\right )\right )}{4609920 (3 x+2)^5 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(395136*(1 - 2*x)^(7/2) + 3252816*(1 - 2*x)^(7/2)*(2 + 3*x) + (2 + 3*x)^2*(29407896*(1 - 2*x)^(7/2) + (2 + 3*x
)*(324091386*(1 - 2*x)^(7/2) + 4270537963*(2 + 3*x)*(3*(1 - 2*x)^(5/2) - 55*(2 + 3*x)*(-(Sqrt[1 - 2*x]*(62 + 1
07*x)) + 21*Sqrt[7]*(3 + 5*x)^(3/2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])))))/(4609920*(2 + 3*x)^5*(3
+ 5*x)^(3/2))

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Maple [B]  time = 0.014, size = 394, normalized size = 1.8 \begin{align*}{\frac{1}{1317120\, \left ( 2+3\,x \right ) ^{5}} \left ( 4280680490662125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}+19405751557668300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+37689013564451865\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+1746052893096750\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+40650610289102550\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+6829311689562600\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+26297118668561400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+11125554365281230\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+10203169301199600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+9662658051124260\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2198472943352400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+4718679545989416\,{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 ) +1228469050319504\,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}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

1/1317120*(4280680490662125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^7+19405751557668300*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+37689013564451865*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+1746052893096750*(-10*x^2-x+3)^(1/2)*x^6+40650610289102550*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+6829311689562600*x^5*(-10*x^2-x+3)^(1/2)+26297118668561400*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+11125554365281230*x^4*(-10*x^2-x+3)^(1/2)[phone]
199600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+9662658051124260*x^3*(-10*x^2-x+3)^(1/2)
+2198472943352400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+4718679545989416*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))+1228469050319504*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)^(3/2)

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Maxima [B]  time = 4.33624, size = 576, normalized size = 2.57 \begin{align*} \frac{46975917593}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{20529722435 \, x}{9408 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{21434986553}{18816 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2211170555 \, x}{4032 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{405 \,{\left (243 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + 810 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 1080 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 720 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 240 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 32 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{43561}{1080 \,{\left (81 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 96 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 16 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{2438681}{6480 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{110694619}{25920 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1309509421}{17280 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{21497905297}{72576 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="maxima")

[Out]

46975917593/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 20529722435/9408*x/sqrt(-10*x^2
- x + 3) + 21434986553/18816/sqrt(-10*x^2 - x + 3) + 2211170555/4032*x/(-10*x^2 - x + 3)^(3/2) + 2401/405/(243
*(-10*x^2 - x + 3)^(3/2)*x^5 + 810*(-10*x^2 - x + 3)^(3/2)*x^4 + 1080*(-10*x^2 - x + 3)^(3/2)*x^3 + 720*(-10*x
^2 - x + 3)^(3/2)*x^2 + 240*(-10*x^2 - x + 3)^(3/2)*x + 32*(-10*x^2 - x + 3)^(3/2)) + 43561/1080/(81*(-10*x^2
- x + 3)^(3/2)*x^4 + 216*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 - x + 3)^
(3/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 2438681/6480/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3
/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 110694619/25920/(9*(-10*x^2 - x + 3)^(3/
2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 1309509421/17280/(3*(-10*x^2 - x + 3)^(3/
2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 21497905297/72576/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.9752, size = 628, normalized size = 2.8 \begin{align*} -\frac{704638763895 \, \sqrt{7}{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\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 (124718063792625 \, x^{6} + 487807977825900 \, x^{5} + 794682454662945 \, x^{4} + 690189860794590 \, x^{3} + 337048538999244 \, x^{2} + 87747789308536 \, x + 9514465420576\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1317120 \,{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\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)^(5/2),x, algorithm="fricas")

[Out]

-1/1317120*(704638763895*sqrt(7)*(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*
x + 288)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(124718063792625*
x^6 + 487807977825900*x^5 + 794682454662945*x^4 + 690189860794590*x^3 + 337048538999244*x^2 + 87747789308536*x
+ 9514465420576)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14
480*x^2 + 3120*x + 288)

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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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 4.7624, size = 756, normalized size = 3.38 \begin{align*} -\frac{275}{48} \, \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} + \frac{46975917593}{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 )} + 27775 \, \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{11 \,{\left (3277500437 \, \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} + 3147123544880 \, \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} + 1168996576419840 \, \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} + 196941720284288000 \, \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} + 12621260024737280000 \, \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)^(5/2),x, algorithm="giac")

[Out]

-275/48*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 + 46975917593/878080*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)))) + 27775*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))) + 11/3136*
(3277500437*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 + 3147123544880*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 + 1168996576419840*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 + 196941720284288000*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 + 1262
1260024737280000*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