3.2435 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=209 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{31603880465 \sqrt{5 x+3} \sqrt{1-2 x}}{4741632 (3 x+2)}+\frac{302171615 \sqrt{5 x+3} \sqrt{1-2 x}}{338688 (3 x+2)^2}+\frac{1729615 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^3}+\frac{21199 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}+\frac{497 \sqrt{5 x+3} \sqrt{1-2 x}}{108 (3 x+2)^5}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (497*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(108*(2 + 3*x)^5) + (21
199*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4) + (1729615*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^3)
+ (302171615*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(338688*(2 + 3*x)^2) + (31603880465*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4
741632*(2 + 3*x)) - (13391796605*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0814239, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{31603880465 \sqrt{5 x+3} \sqrt{1-2 x}}{4741632 (3 x+2)}+\frac{302171615 \sqrt{5 x+3} \sqrt{1-2 x}}{338688 (3 x+2)^2}+\frac{1729615 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^3}+\frac{21199 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}+\frac{497 \sqrt{5 x+3} \sqrt{1-2 x}}{108 (3 x+2)^5}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (497*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(108*(2 + 3*x)^5) + (21
199*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4) + (1729615*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^3)
+ (302171615*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(338688*(2 + 3*x)^2) + (31603880465*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4
741632*(2 + 3*x)) - (13391796605*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*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 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)^7 \sqrt{3+5 x}} \, dx &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{\left (\frac{487}{2}-256 x\right ) \sqrt{1-2 x}}{(2+3 x)^6 \sqrt{3+5 x}} \, dx\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}-\frac{1}{270} \int \frac{-\frac{121615}{4}+47140 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}-\frac{\int \frac{-\frac{30857925}{8}+\frac{11129475 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{7560}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}+\frac{1729615 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^3}-\frac{\int \frac{-\frac{5733084525}{16}+\frac{908047875 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{158760}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}+\frac{1729615 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^3}+\frac{302171615 \sqrt{1-2 x} \sqrt{3+5 x}}{338688 (2+3 x)^2}-\frac{\int \frac{-\frac{683095555275}{32}+\frac{158640097875 x}{8}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{2222640}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}+\frac{1729615 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^3}+\frac{302171615 \sqrt{1-2 x} \sqrt{3+5 x}}{338688 (2+3 x)^2}+\frac{31603880465 \sqrt{1-2 x} \sqrt{3+5 x}}{4741632 (2+3 x)}-\frac{\int -\frac{37965743375175}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{15558480}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}+\frac{1729615 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^3}+\frac{302171615 \sqrt{1-2 x} \sqrt{3+5 x}}{338688 (2+3 x)^2}+\frac{31603880465 \sqrt{1-2 x} \sqrt{3+5 x}}{4741632 (2+3 x)}+\frac{13391796605 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{351232}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}+\frac{1729615 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^3}+\frac{302171615 \sqrt{1-2 x} \sqrt{3+5 x}}{338688 (2+3 x)^2}+\frac{31603880465 \sqrt{1-2 x} \sqrt{3+5 x}}{4741632 (2+3 x)}+\frac{13391796605 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{175616}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{18 (2+3 x)^6}+\frac{497 \sqrt{1-2 x} \sqrt{3+5 x}}{108 (2+3 x)^5}+\frac{21199 \sqrt{1-2 x} \sqrt{3+5 x}}{864 (2+3 x)^4}+\frac{1729615 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^3}+\frac{302171615 \sqrt{1-2 x} \sqrt{3+5 x}}{338688 (2+3 x)^2}+\frac{31603880465 \sqrt{1-2 x} \sqrt{3+5 x}}{4741632 (2+3 x)}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{175616 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.197426, size = 193, normalized size = 0.92 \[ \frac{1}{42} \left (\frac{237 \sqrt{5 x+3} (1-2 x)^{7/2}}{14 (3 x+2)^5}+\frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{(3 x+2)^6}+\frac{8332464 \sqrt{5 x+3} (1-2 x)^{7/2}+2012291 (3 x+2) \left (56 \sqrt{5 x+3} (1-2 x)^{5/2}+55 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (95 x+68)-363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )}{87808 (3 x+2)^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]

[Out]

((3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(2 + 3*x)^6 + (237*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(14*(2 + 3*x)^5) + (83324
64*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x] + 2012291*(2 + 3*x)*(56*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x] + 55*(2 + 3*x)*(7*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x]*(68 + 95*x) - 363*Sqrt[7]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])))/
(87808*(2 + 3*x)^4))/42

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Maple [B]  time = 0.014, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 29287859175135\,\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+36876342922048\,{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 ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(29287859175135*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^6+117151436700540*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+195252394500900*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+11946266815770*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+40353920114760*x^4*(-10*x^2-x+3)^(1/2)[phone]
000400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+54544410839520*x^3*(-10*x^2-x+3)^(1/2)+2
3141024533440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+36876342922048*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))/(-10*x^2-x+3)^(1/2)/(2+3*x)^6

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Maxima [A]  time = 2.50107, size = 311, normalized size = 1.49 \begin{align*} \frac{13391796605}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{54 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{469 \, \sqrt{-10 \, x^{2} - x + 3}}{108 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{21199 \, \sqrt{-10 \, x^{2} - x + 3}}{864 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1729615 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{302171615 \, \sqrt{-10 \, x^{2} - x + 3}}{338688 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{31603880465 \, \sqrt{-10 \, x^{2} - x + 3}}{4741632 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

13391796605/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/54*sqrt(-10*x^2 - x + 3)/(7
29*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 469/108*sqrt(-10*x^2 - x + 3)/(243*x^5 + 81
0*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 21199/864*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x
+ 16) + 1729615/12096*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 302171615/338688*sqrt(-10*x^2 - x +
3)/(9*x^2 + 12*x + 4) + 31603880465/4741632*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.89489, size = 529, normalized size = 2.53 \begin{align*} -\frac{40175389815 \, \sqrt{7}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (853304772555 \, x^{5} + 2882422865340 \, x^{4} + 3896029345680 \, x^{3} + 2634024494432 \, x^{2} + 890768460368 \, x + 120549503808\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{7375872 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/7375872*(40175389815*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14
*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(853304772555*x^5 + 2882422865340*x^4
+ 3896029345680*x^3 + 2634024494432*x^2 + 890768460368*x + 120549503808)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x
^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**7/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 3.40483, size = 676, normalized size = 3.23 \begin{align*} \frac{2678359321}{4917248} \, \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{6655 \,{\left (20305527 \, \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 )}^{11} + 17887837240 \, \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 )}^{9} + 7599643632000 \, \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} + 1749282956467200 \, \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} + 210267345272320000 \, \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} + 10389680589926400000 \, \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 )}}{263424 \,{\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 )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

2678359321/4917248*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)))) + 6655/263424*(20305527*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)))^11 + 17887837
240*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)))^9 + 7599643632000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s
qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 1749282956467200*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 + 210267345272320000*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 + 103896805899
26400000*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)^6