3.2603 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=108 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{407 \sqrt{5 x+3}}{98 \sqrt{1-2 x}}+\frac{25}{6} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{147 \sqrt{7}} \]

[Out]

(-407*Sqrt[3 + 5*x])/(98*Sqrt[1 - 2*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)) + (25*Sqrt[5/2]*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/6 + (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(147*Sqrt[7])

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Rubi [A]  time = 0.0400216, 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} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{407 \sqrt{5 x+3}}{98 \sqrt{1-2 x}}+\frac{25}{6} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{147 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-407*Sqrt[3 + 5*x])/(98*Sqrt[1 - 2*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)) + (25*Sqrt[5/2]*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/6 + (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(147*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 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{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1}{21} \int \frac{\sqrt{3+5 x} \left (174+\frac{525 x}{2}\right )}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=-\frac{407 \sqrt{3+5 x}}{98 \sqrt{1-2 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1}{147} \int \frac{-\frac{6123}{2}-\frac{18375 x}{4}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{407 \sqrt{3+5 x}}{98 \sqrt{1-2 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{125}{12} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{407 \sqrt{3+5 x}}{98 \sqrt{1-2 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{2}{147} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{1}{6} \left (25 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{407 \sqrt{3+5 x}}{98 \sqrt{1-2 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac{25}{6} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{147 \sqrt{7}}\\ \end{align*}

Mathematica [C]  time = 0.063978, size = 97, normalized size = 0.9 \[ -\frac{-18865 \sqrt{22} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+56 \sqrt{5 x+3} (41 x+18)+24 \sqrt{7-14 x} (2 x-1) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{12348 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(56*Sqrt[3 + 5*x]*(18 + 41*x) + 24*Sqrt[7 - 14*x]*(-1 + 2*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])] -
18865*Sqrt[22]*Hypergeometric2F1[-3/2, -3/2, -1/2, (5*(1 - 2*x))/11])/(12348*(1 - 2*x)^(3/2))

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Maple [B]  time = 0.013, size = 191, normalized size = 1.8 \begin{align*}{\frac{1}{8232\, \left ( 2\,x-1 \right ) ^{2}} \left ( 34300\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-32\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-34300\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+32\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+8575\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -8\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +89936\,x\sqrt{-10\,{x}^{2}-x+3}-21252\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8232*(34300*10^(1/2)*arcsin(20/11*x+1/11)*x^2-32*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^2-34300*10^(1/2)*arcsin(20/11*x+1/11)*x+32*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+8575
*10^(1/2)*arcsin(20/11*x+1/11)-8*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+89936*x*(-10*x^2-x
+3)^(1/2)-21252*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [B]  time = 4.14112, size = 220, normalized size = 2.04 \begin{align*} -\frac{12233125 \, x^{2}}{3557763 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{625 \, x^{3}}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{25}{24} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{1029} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2446625}{7115526} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{12021894385 \, x}{697321548 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{16029625 \, x^{2}}{117612 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{6953014391}{697321548 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{12465295 \, x}{205821 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2681981}{274428 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12233125/3557763*x^2/sqrt(-10*x^2 - x + 3) + 625/6*x^3/(-10*x^2 - x + 3)^(3/2) + 25/24*sqrt(10)*arcsin(20/11*
x + 1/11) - 1/1029*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2446625/7115526*sqrt(-10*x^2 -
x + 3) - 12021894385/697321548*x/sqrt(-10*x^2 - x + 3) + 16029625/117612*x^2/(-10*x^2 - x + 3)^(3/2) - 6953014
391/697321548/sqrt(-10*x^2 - x + 3) + 12465295/205821*x/(-10*x^2 - x + 3)^(3/2) + 2681981/274428/(-10*x^2 - x
+ 3)^(3/2)

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Fricas [A]  time = 1.56876, size = 420, normalized size = 3.89 \begin{align*} -\frac{8575 \, \sqrt{5} \sqrt{2}{\left (4 \, x^{2} - 4 \, x + 1\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 ) - 8 \, \sqrt{7}{\left (4 \, x^{2} - 4 \, x + 1\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 ) - 308 \,{\left (292 \, x - 69\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{8232 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8232*(8575*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 8*sqrt(7)*(4*x^2 - 4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 308*(292*x - 69)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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

[Out]

Timed out

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Giac [B]  time = 2.60284, size = 243, normalized size = 2.25 \begin{align*} -\frac{1}{10290} \, \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{25}{24} \, \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{11 \,{\left (292 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1221 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{7350 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10290*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)))) + 25/24*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 11/7350*(292*sqrt
(5)*(5*x + 3) - 1221*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2