3.2456 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{5 x+3}}+\frac{2992825 \sqrt{1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{25024175 \sqrt{1-2 x}}{1344 (5 x+3)^{3/2}}-\frac{519421265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

[Out]

(-25024175*Sqrt[1 - 2*x])/(1344*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)^(3/2)) + (847
*Sqrt[1 - 2*x])/(72*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (36817*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (
2992825*Sqrt[1 - 2*x])/(1344*(2 + 3*x)*(3 + 5*x)^(3/2)) + (227000875*Sqrt[1 - 2*x])/(1344*Sqrt[3 + 5*x]) - (51
9421265*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0724418, antiderivative size = 195, 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} \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{5 x+3}}+\frac{2992825 \sqrt{1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{25024175 \sqrt{1-2 x}}{1344 (5 x+3)^{3/2}}-\frac{519421265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-25024175*Sqrt[1 - 2*x])/(1344*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)^(3/2)) + (847
*Sqrt[1 - 2*x])/(72*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (36817*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (
2992825*Sqrt[1 - 2*x])/(1344*(2 + 3*x)*(3 + 5*x)^(3/2)) + (227000875*Sqrt[1 - 2*x])/(1344*Sqrt[3 + 5*x]) - (51
9421265*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*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)^5 (3+5 x)^{5/2}} \, dx &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{1}{12} \int \frac{\left (\frac{495}{2}-264 x\right ) \sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}-\frac{1}{108} \int \frac{-\frac{126423}{4}+49236 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}-\frac{\int \frac{-\frac{31923045}{8}+\frac{11597355 x}{2}}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx}{1512}\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{2992825 \sqrt{1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}-\frac{\int \frac{-\frac{5879900565}{16}+\frac{942739875 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{10584}\\ &=-\frac{25024175 \sqrt{1-2 x}}{1344 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{2992825 \sqrt{1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac{\int \frac{-\frac{663674731335}{32}+\frac{156075779475 x}{8}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{174636}\\ &=-\frac{25024175 \sqrt{1-2 x}}{1344 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{2992825 \sqrt{1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{3+5 x}}-\frac{\int -\frac{35635934727855}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{960498}\\ &=-\frac{25024175 \sqrt{1-2 x}}{1344 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{2992825 \sqrt{1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{3+5 x}}+\frac{519421265}{896} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{25024175 \sqrt{1-2 x}}{1344 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{2992825 \sqrt{1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{3+5 x}}+\frac{519421265}{448} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{25024175 \sqrt{1-2 x}}{1344 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{2992825 \sqrt{1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{3+5 x}}-\frac{519421265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{448 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.205854, size = 152, normalized size = 0.78 \[ \frac{65016 (3 x+2) (1-2 x)^{7/2}+7056 (1-2 x)^{7/2}+(3 x+2)^2 \left (716706 (1-2 x)^{7/2}+9444023 (3 x+2) \left (3 (1-2 x)^{5/2}-55 (3 x+2) \left (21 \sqrt{7} (5 x+3)^{3/2} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\sqrt{1-2 x} (107 x+62)\right )\right )\right )}{65856 (3 x+2)^4 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7056*(1 - 2*x)^(7/2) + 65016*(1 - 2*x)^(7/2)*(2 + 3*x) + (2 + 3*x)^2*(716706*(1 - 2*x)^(7/2) + 9444023*(2 + 3
*x)*(3*(1 - 2*x)^(5/2) - 55*(2 + 3*x)*(-(Sqrt[1 - 2*x]*(62 + 107*x)) + 21*Sqrt[7]*(3 + 5*x)^(3/2)*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))))/(65856*(2 + 3*x)^4*(3 + 5*x)^(3/2))

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Maple [B]  time = 0.015, size = 346, normalized size = 1.8 \begin{align*}{\frac{1}{18816\, \left ( 2+3\,x \right ) ^{4}} \left ( 3155484184875\,\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+3511408936896\,{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}}} \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)^5/(3+5*x)^(5/2),x)

[Out]

1/18816*(3155484184875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+12201205514850*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+19648148191155*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^4+1287094961250*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+4176132792300*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+5417063350650*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+3511408936896*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))*(1-2
*x)^(1/2)/(2+3*x)^4/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [B]  time = 4.34671, size = 439, normalized size = 2.25 \begin{align*} \frac{519421265}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{227000875 \, x}{672 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{79003515}{448 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{24449315 \, x}{288 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{324 \,{\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{37387}{648 \,{\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{571291}{864 \,{\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{60813781}{5184 \,{\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{237706249}{5184 \,{\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)^5/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

519421265/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 227000875/672*x/sqrt(-10*x^2 - x +
3) + 79003515/448/sqrt(-10*x^2 - x + 3) + 24449315/288*x/(-10*x^2 - x + 3)^(3/2) + 2401/324/(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)) + 37387/648/(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)) + 571291/864/(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)) + 60813781/5184/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^
2 - x + 3)^(3/2)) - 237706249/5184/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.83436, size = 531, normalized size = 2.72 \begin{align*} -\frac{1558263795 \, \sqrt{7}{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\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 (91935354375 \, x^{5} + 298295199450 \, x^{4} + 386933096475 \, x^{3} + 250814924064 \, x^{2} + 81243850516 \, x + 10520317456\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{18816 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18816*(1558263795*sqrt(7)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*arctan(1/
14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(91935354375*x^5 + 298295199450*x^4
+ 386933096475*x^3 + 250814924064*x^2 + 81243850516*x + 10520317456)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2025*x^6
+ 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)

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

[Out]

Timed out

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Giac [B]  time = 4.50425, size = 674, normalized size = 3.46 \begin{align*} -\frac{55}{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{103884253}{12544} \, \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{9295}{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{55 \,{\left (6089929 \, \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} + 4375094808 \, \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} + 1081495934400 \, \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} + 90973105216000 \, \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 )}}{224 \,{\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 )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-55/48*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 + 103884253/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 9295/2*sqrt(10)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 55/224*(608
9929*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 + 4375094808*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 1081495934400*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 + 90973105216000*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)^4