3.2606 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{15755 \sqrt{5 x+3}}{86436 \sqrt{1-2 x}}-\frac{2365 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)}-\frac{187 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^2}+\frac{32 \sqrt{5 x+3}}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{2585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(15755*Sqrt[3 + 5*x])/(86436*Sqrt[1 - 2*x]) + (32*Sqrt[3 + 5*x])/(441*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (187*Sqrt[3
+ 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (2365*Sqrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)
^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (2585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7]
)

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Rubi [A]  time = 0.0615776, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{15755 \sqrt{5 x+3}}{86436 \sqrt{1-2 x}}-\frac{2365 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)}-\frac{187 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^2}+\frac{32 \sqrt{5 x+3}}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{2585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^4),x]

[Out]

(15755*Sqrt[3 + 5*x])/(86436*Sqrt[1 - 2*x]) + (32*Sqrt[3 + 5*x])/(441*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (187*Sqrt[3
+ 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (2365*Sqrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)
^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (2585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*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{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1}{21} \int \frac{\left (-123-\frac{465 x}{2}\right ) \sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{\int \frac{-\frac{24783}{2}-\frac{43065 x}{2}}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{1323}\\ &=\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{187 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^2}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{\int \frac{-\frac{264495}{4}-117810 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{18522}\\ &=\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{187 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^2}-\frac{2365 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{\int \frac{-\frac{2149455}{8}-\frac{744975 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{129654}\\ &=\frac{15755 \sqrt{3+5 x}}{86436 \sqrt{1-2 x}}+\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{187 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^2}-\frac{2365 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{\int \frac{5374215}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{4991679}\\ &=\frac{15755 \sqrt{3+5 x}}{86436 \sqrt{1-2 x}}+\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{187 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^2}-\frac{2365 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2585 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{38416}\\ &=\frac{15755 \sqrt{3+5 x}}{86436 \sqrt{1-2 x}}+\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{187 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^2}-\frac{2365 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}+\frac{2585 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{19208}\\ &=\frac{15755 \sqrt{3+5 x}}{86436 \sqrt{1-2 x}}+\frac{32 \sqrt{3+5 x}}{441 \sqrt{1-2 x} (2+3 x)^3}-\frac{187 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^2}-\frac{2365 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{2585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{19208 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0699582, size = 100, normalized size = 0.58 \[ -\frac{7 \sqrt{5 x+3} \left (567180 x^4+552780 x^3-169221 x^2-304730 x-75888\right )-7755 \sqrt{7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{403368 (1-2 x)^{3/2} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^4),x]

[Out]

-(7*Sqrt[3 + 5*x]*(-75888 - 304730*x - 169221*x^2 + 552780*x^3 + 567180*x^4) - 7755*Sqrt[7 - 14*x]*(-1 + 2*x)*
(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(403368*(1 - 2*x)^(3/2)*(2 + 3*x)^3)

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Maple [B]  time = 0.014, size = 305, normalized size = 1.8 \begin{align*}{\frac{1}{806736\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) ^{2}} \left ( 837540\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+837540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-348975\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-7940520\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-449790\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-7738920\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+31020\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2369094\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+62040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4266220\,x\sqrt{-10\,{x}^{2}-x+3}+1062432\,\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)^4,x)

[Out]

1/806736*(837540*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+837540*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-348975*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-
7940520*x^4*(-10*x^2-x+3)^(1/2)-449790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-7738920*
x^3*(-10*x^2-x+3)^(1/2)+31020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2369094*x^2*(-10*x^
2-x+3)^(1/2)+62040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4266220*x*(-10*x^2-x+3)^(1/2)+10
62432*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.32041, size = 324, normalized size = 1.87 \begin{align*} \frac{2585}{268912} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{78775 \, x}{86436 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{11755}{172872 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{17875 \, x}{12348 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{1701 \,{\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{239}{15876 \,{\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{4997}{31752 \,{\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{901885}{666792 \,{\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)^4,x, algorithm="maxima")

[Out]

2585/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 78775/86436*x/sqrt(-10*x^2 - x + 3) +
11755/172872/sqrt(-10*x^2 - x + 3) + 17875/12348*x/(-10*x^2 - x + 3)^(3/2) - 1/1701/(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)) + 239/15876
/(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)) - 4997/31752/(3*(-
10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 901885/666792/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.55978, size = 396, normalized size = 2.29 \begin{align*} -\frac{7755 \, \sqrt{7}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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 (567180 \, x^{4} + 552780 \, x^{3} - 169221 \, x^{2} - 304730 \, x - 75888\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{806736 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

-1/806736*(7755*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5
*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(567180*x^4 + 552780*x^3 - 169221*x^2 - 304730*x - 75888)*sqrt(5
*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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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)**4,x)

[Out]

Timed out

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Giac [B]  time = 3.89756, size = 482, normalized size = 2.79 \begin{align*} \frac{517}{537824} \, \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{88 \,{\left (151 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1023 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1260525 \,{\left (2 \, x - 1\right )}^{2}} - \frac{11 \,{\left (3629 \, \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} + 2900800 \, \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} + 755384000 \, \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 )}}{67228 \,{\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 )}^{3}} \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)^4,x, algorithm="giac")

[Out]

517/537824*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)))) - 88/1260525*(151*sqrt(5)*(5*x + 3) - 1023*sqrt(5))*
sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 11/67228*(3629*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 + 2900800*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 + 755384000*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)^3