3.2434 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=180 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{245529161 \sqrt{5 x+3} \sqrt{1-2 x}}{169344 (3 x+2)}+\frac{2347559 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^2}+\frac{67187 \sqrt{5 x+3} \sqrt{1-2 x}}{2160 (3 x+2)^3}+\frac{2023 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (2023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(360*(2 + 3*x)^4) + (6
7187*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2160*(2 + 3*x)^3) + (2347559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^
2) + (245529161*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(169344*(2 + 3*x)) - (104040277*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(6272*Sqrt[7])

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Rubi [A]  time = 0.0640718, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{245529161 \sqrt{5 x+3} \sqrt{1-2 x}}{169344 (3 x+2)}+\frac{2347559 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^2}+\frac{67187 \sqrt{5 x+3} \sqrt{1-2 x}}{2160 (3 x+2)^3}+\frac{2023 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (2023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(360*(2 + 3*x)^4) + (6
7187*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2160*(2 + 3*x)^3) + (2347559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^
2) + (245529161*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(169344*(2 + 3*x)) - (104040277*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(6272*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)^6 \sqrt{3+5 x}} \, dx &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{\left (\frac{421}{2}-190 x\right ) \sqrt{1-2 x}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}-\frac{1}{180} \int \frac{-\frac{79903}{4}+28825 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{67187 \sqrt{1-2 x} \sqrt{3+5 x}}{2160 (2+3 x)^3}-\frac{\int \frac{-\frac{14846615}{8}+2351545 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{3780}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{67187 \sqrt{1-2 x} \sqrt{3+5 x}}{2160 (2+3 x)^3}+\frac{2347559 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^2}-\frac{\int \frac{-\frac{1768979345}{16}+\frac{410822825 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{52920}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{67187 \sqrt{1-2 x} \sqrt{3+5 x}}{2160 (2+3 x)^3}+\frac{2347559 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^2}+\frac{245529161 \sqrt{1-2 x} \sqrt{3+5 x}}{169344 (2+3 x)}-\frac{\int -\frac{98318061765}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{370440}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{67187 \sqrt{1-2 x} \sqrt{3+5 x}}{2160 (2+3 x)^3}+\frac{2347559 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^2}+\frac{245529161 \sqrt{1-2 x} \sqrt{3+5 x}}{169344 (2+3 x)}+\frac{104040277 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{12544}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{67187 \sqrt{1-2 x} \sqrt{3+5 x}}{2160 (2+3 x)^3}+\frac{2347559 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^2}+\frac{245529161 \sqrt{1-2 x} \sqrt{3+5 x}}{169344 (2+3 x)}+\frac{104040277 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{6272}\\ &=\frac{7 (1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{2023 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{67187 \sqrt{1-2 x} \sqrt{3+5 x}}{2160 (2+3 x)^3}+\frac{2347559 \sqrt{1-2 x} \sqrt{3+5 x}}{12096 (2+3 x)^2}+\frac{245529161 \sqrt{1-2 x} \sqrt{3+5 x}}{169344 (2+3 x)}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{6272 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.100312, size = 135, normalized size = 0.75 \[ \frac{1}{35} \left (\frac{78167 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (15707 x^2+21638 x+7488\right )}{(3 x+2)^3}-19965 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{18816}+\frac{963 \sqrt{5 x+3} (1-2 x)^{7/2}}{56 (3 x+2)^4}+\frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{(3 x+2)^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

((3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(2 + 3*x)^5 + (963*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/(56*(2 + 3*x)^4) + (78167
*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(7488 + 21638*x + 15707*x^2))/(2 + 3*x)^3 - 19965*Sqrt[7]*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])]))/18816)/35

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Maple [B]  time = 0.013, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{1317120\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 379226809665\,\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]\,\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+426661359656\,{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)^6/(3+5*x)^(1/2),x)

[Out]

1/1317120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(379226809665*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)
*x^5+1264089365550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1685452487400*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+154683371430*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+419390813940*x^3*(-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+426661359656*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)^5

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Maxima [A]  time = 2.43333, size = 248, normalized size = 1.38 \begin{align*} \frac{104040277}{87808} \, \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}}{45 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{637 \, \sqrt{-10 \, x^{2} - x + 3}}{120 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{67187 \, \sqrt{-10 \, x^{2} - x + 3}}{2160 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{2347559 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{245529161 \, \sqrt{-10 \, x^{2} - x + 3}}{169344 \,{\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)^6/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

104040277/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/45*sqrt(-10*x^2 - x + 3)/(243*x
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 637/120*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 +
96*x + 16) + 67187/2160*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 2347559/12096*sqrt(-10*x^2 - x +
3)/(9*x^2 + 12*x + 4) + 245529161/169344*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.82145, size = 455, normalized size = 2.53 \begin{align*} -\frac{1560604155 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (11048812245 \, x^{4} + 29956486710 \, x^{3} + 30475811404 \, x^{2} + 13788819736 \, x + 2341358496\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1317120 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1317120*(1560604155*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(11048812245*x^4 + 29956486710*x^3 + 30475811404*x^
2 + 13788819736*x + 2341358496)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)

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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)**6/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 3.56952, size = 594, normalized size = 3.3 \begin{align*} \frac{104040277}{878080} \, \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{1331 \,{\left (706299 \, \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} + 493892560 \, \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} + 156884295680 \, \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} + 24022907776000 \, \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} + 1441374466560000 \, \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 )}}{9408 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

104040277/878080*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1331/9408*(706299*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 493892560*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))
)^7 + 156884295680*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 + 24022907776000*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 + 1441374466560000*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)^5