3.2452 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{22 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{8}{75} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{98}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-22*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (814*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]) - (8*Sqrt[2/5]*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/75 - (98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3

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Rubi [A]  time = 0.0398689, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 150, 157, 54, 216, 93, 204} \[ -\frac{22 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{8}{75} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{98}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-22*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (814*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]) - (8*Sqrt[2/5]*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/75 - (98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3

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 150

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] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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) (3+5 x)^{5/2}} \, dx &=-\frac{22 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{2}{15} \int \frac{\left (\frac{237}{2}-6 x\right ) \sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}-\frac{4}{75} \int \frac{-\frac{8559}{4}+6 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}-\frac{8}{75} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{343}{3} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}+\frac{686}{3} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{16 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{75 \sqrt{5}}\\ &=-\frac{22 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}-\frac{8}{75} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{98}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.122946, size = 121, normalized size = 1.12 \[ -\frac{2 \left (55 \left (1130 x^2+91 x-328\right ) \sqrt{5 x+3}-4 \sqrt{10-20 x} (5 x+3)^2 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+6125 \sqrt{7-14 x} (5 x+3)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{375 \sqrt{1-2 x} (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(55*Sqrt[3 + 5*x]*(-328 + 91*x + 1130*x^2) - 4*Sqrt[10 - 20*x]*(3 + 5*x)^2*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]
] + 6125*Sqrt[7 - 14*x]*(3 + 5*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/(375*Sqrt[1 - 2*x]*(3 + 5*
x)^2)

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Maple [B]  time = 0.012, size = 184, normalized size = 1.7 \begin{align*}{\frac{1}{375} \left ( 153125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-100\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+183750\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-120\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+55125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -36\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +62150\,x\sqrt{-10\,{x}^{2}-x+3}+36080\,\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)/(3+5*x)^(5/2),x)

[Out]

1/375*(153125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-100*10^(1/2)*arcsin(20/11*x+1/11)
*x^2+183750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-120*10^(1/2)*arcsin(20/11*x+1/11)*x+5
5125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-36*10^(1/2)*arcsin(20/11*x+1/11)+62150*x*(-10*
x^2-x+3)^(1/2)+36080*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [B]  time = 3.00346, size = 220, normalized size = 2.04 \begin{align*} \frac{626336 \, x^{2}}{17788815 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{16 \, x^{3}}{15 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{4}{375} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{49}{3} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{313168}{88944075} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{5905573412 \, x}{88944075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3286544 \, x^{2}}{735075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{3102773174}{88944075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{11007824 \, x}{735075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{2075846}{245025 \,{\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)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

626336/17788815*x^2/sqrt(-10*x^2 - x + 3) - 16/15*x^3/(-10*x^2 - x + 3)^(3/2) - 4/375*sqrt(10)*arcsin(20/11*x
+ 1/11) + 49/3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 313168/88944075*sqrt(-10*x^2 - x +
3) - 5905573412/88944075*x/sqrt(-10*x^2 - x + 3) + 3286544/735075*x^2/(-10*x^2 - x + 3)^(3/2) + 3102773174/889
44075/sqrt(-10*x^2 - x + 3) + 11007824/735075*x/(-10*x^2 - x + 3)^(3/2) - 2075846/245025/(-10*x^2 - x + 3)^(3/
2)

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Fricas [A]  time = 1.8017, size = 427, normalized size = 3.95 \begin{align*} \frac{4 \, \sqrt{5} \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 6125 \, \sqrt{7}{\left (25 \, x^{2} + 30 \, x + 9\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 ) + 110 \,{\left (565 \, x + 328\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/375*(4*sqrt(5)*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 6125*sqrt(7)*(25*x^2 + 30*x + 9)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*
x + 1)/(10*x^2 + x - 3)) + 110*(565*x + 328)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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Giac [B]  time = 2.88725, size = 352, normalized size = 3.26 \begin{align*} -\frac{11}{6000} \, \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{49}{30} \, \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{4}{375} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{407}{250} \, \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/6000*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 + 49/30*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)))) - 4/375*sqrt(10)*(pi + 2*arctan(-1/4*
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)))) +
407/250*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)))