3.2336 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{28291441 \sqrt{5 x+3} \sqrt{1-2 x}}{1185408 (3 x+2)}+\frac{270463 \sqrt{5 x+3} \sqrt{1-2 x}}{84672 (3 x+2)^2}+\frac{7723 \sqrt{5 x+3} \sqrt{1-2 x}}{15120 (3 x+2)^3}+\frac{41 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{11988317 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

-((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(360*(2 + 3*x)^4) + (7723
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15120*(2 + 3*x)^3) + (270463*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 + 3*x)^2) +
(28291441*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1185408*(2 + 3*x)) - (11988317*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.0669546, 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 = {97, 149, 151, 12, 93, 204} \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{28291441 \sqrt{5 x+3} \sqrt{1-2 x}}{1185408 (3 x+2)}+\frac{270463 \sqrt{5 x+3} \sqrt{1-2 x}}{84672 (3 x+2)^2}+\frac{7723 \sqrt{5 x+3} \sqrt{1-2 x}}{15120 (3 x+2)^3}+\frac{41 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{11988317 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(360*(2 + 3*x)^4) + (7723
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15120*(2 + 3*x)^3) + (270463*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 + 3*x)^2) +
(28291441*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1185408*(2 + 3*x)) - (11988317*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(43904*Sqrt[7])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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)^{3/2} \sqrt{3+5 x}}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{\left (-\frac{13}{2}-20 x\right ) \sqrt{1-2 x}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}-\frac{1}{180} \int \frac{-\frac{1361}{4}+455 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{7723 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}-\frac{\int \frac{-\frac{244825}{8}+38615 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{3780}\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{7723 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{270463 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}-\frac{\int \frac{-\frac{29121535}{16}+\frac{6761575 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{52920}\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{7723 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{270463 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{28291441 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}-\frac{\int -\frac{1618422795}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{370440}\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{7723 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{270463 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{28291441 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}+\frac{11988317 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{87808}\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{7723 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{270463 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{28291441 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}+\frac{11988317 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{43904}\\ &=-\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x} \sqrt{3+5 x}}{360 (2+3 x)^4}+\frac{7723 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{270463 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{28291441 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}-\frac{11988317 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{43904 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0816251, size = 140, normalized size = 0.78 \[ \frac{9007 \left (7 \sqrt{1-2 x} \sqrt{5 x+3} \left (3103 x^2+4366 x+1488\right )-3993 \sqrt{7} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{921984 (3 x+2)^3}+\frac{153 (5 x+3)^{3/2} (1-2 x)^{5/2}}{392 (3 x+2)^4}+\frac{3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{35 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(35*(2 + 3*x)^5) + (153*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(392*(2 + 3*x)^4)
+ (9007*(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1488 + 4366*x + 3103*x^2) - 3993*Sqrt[7]*(2 + 3*x)^3*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/(921984*(2 + 3*x)^3)

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Maple [B]  time = 0.01, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{9219840\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 43697415465\,\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+49162327144\,{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)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x)

[Out]

1/9219840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(43697415465*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*
x^5+145658051550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+194210735400*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+17823607830*x^4*(-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^2+48324782100*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+49162327144*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 = 1.73689, size = 267, normalized size = 1.48 \begin{align*} \frac{11988317}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{495385}{32928} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{5 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{239 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{280 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{8395 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2352 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{297231 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{3665849 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11988317/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 495385/32928*sqrt(-10*x^2 - x + 3)
+ 1/5*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 239/280*(-10*x^2 - x +
3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 8395/2352*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x
+ 8) + 297231/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 3665849/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2
)

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Fricas [A]  time = 1.5446, size = 447, normalized size = 2.48 \begin{align*} -\frac{179824755 \, \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 (1273114845 \, x^{4} + 3451770150 \, x^{3} + 3511594796 \, x^{2} + 1588955864 \, x + 269759904\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{9219840 \,{\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)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/9219840*(179824755*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*(1273114845*x^4 + 3451770150*x^3 + 3511594796*x^2 +
1588955864*x + 269759904)*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)**(3/2)*(3+5*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.98616, size = 594, normalized size = 3.3 \begin{align*} \frac{11988317}{6146560} \, \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 (27021 \, \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} - 52500560 \, \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} - 18029240320 \, \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} - 2768103296000 \, \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} - 166086197760000 \, \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 )}}{65856 \,{\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)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

11988317/6146560*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)))) - 1331/65856*(27021*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 - 52500560*sqrt(1
0)*((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 - 18029240320*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 - 2768103296000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 166086197760000*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