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

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

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Rubi [A]  time = 0.07, antiderivative size = 195, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.22, size = 152, normalized size = 0.78 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 4.11, size = 228, normalized size = 1.17 \begin {gather*} \frac {5 \sqrt {11-2 (5 x+3)} \left (735482835 \sqrt {5} (5 x+3)^5+899565453 \sqrt {5} (5 x+3)^4+398378709 \sqrt {5} (5 x+3)^3+72615771 \sqrt {5} (5 x+3)^2+3715712 \sqrt {5} (5 x+3)-108416 \sqrt {5}\right )}{1344 (5 x+3)^{3/2} (3 (5 x+3)+1)^4}-\frac {519421265 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{448 \sqrt {7}}-\frac {519421265 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{448 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[11 - 2*(3 + 5*x)]*(-108416*Sqrt[5] + 3715712*Sqrt[5]*(3 + 5*x) + 72615771*Sqrt[5]*(3 + 5*x)^2 + 398378
709*Sqrt[5]*(3 + 5*x)^3 + 899565453*Sqrt[5]*(3 + 5*x)^4 + 735482835*Sqrt[5]*(3 + 5*x)^5))/(1344*(3 + 5*x)^(3/2
)*(1 + 3*(3 + 5*x))^4) - (519421265*ArcTan[(Sqrt[2/(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(
3 + 5*x)])])/(448*Sqrt[7]) - (519421265*ArcTan[(Sqrt[68 + 2*Sqrt[1155]]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2
*(3 + 5*x)])])/(448*Sqrt[7])

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fricas [A]  time = 1.34, size = 146, normalized size = 0.75 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 5.63, size = 495, normalized size = 2.54 \begin {gather*} \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 {55}{48} \, \sqrt {10} {\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 )}^{3} - \frac {4056 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {16224 \, \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 {gather*}

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]

103884253/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 55/48*sqrt(10)*(((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 4056*(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) + 16224*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 55/224*(6089929*sqr
t(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)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^5 + 1081495934400*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 90973105216000*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))))/(((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)))^2 + 280)^4

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maple [B]  time = 0.02, size = 346, normalized size = 1.77 \begin {gather*} \frac {\left (3155484184875 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{18816 \left (3 x +2\right )^{4} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18816*(3155484184875*7^(1/2)*x^6*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x
^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^5+16866647317080*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+8140370065080*7^(1/2)*x^2*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+2094306540480*7^(1/2)*x*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+224389986480*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+147284444384*(-10*x^2-x+3)^(1/2))*(-2*
x+1)^(1/2)/(3*x+2)^4/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [B]  time = 1.61, size = 325, normalized size = 1.67 \begin {gather*} \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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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