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]

-11988317/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/15*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3
*x)^5+41/360*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+7723/15120*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+270463/846
72*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+28291441/1185408*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.07, antiderivative size = 180, normalized size of antiderivative = 1.00, 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 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 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 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*}

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Mathematica [A]  time = 0.08, 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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fricas [A]  time = 1.96, size = 131, normalized size = 0.73 \[ -\frac {179824755 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1273114845 \, x^{4} + 3451770150 \, x^{3} + 3511594796 \, x^{2} + 1588955864 \, x + 269759904\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9219840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

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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giac [B]  time = 3.06, size = 426, normalized size = 2.37 \[ \frac {11988317}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (27021 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 52500560 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 18029240320 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 2768103296000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {166086197760000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {664344791040000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{65856 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]

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*sqrt(10)*(27021*((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
(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 - 1802
9240320*((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(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^3 - 166086197760000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 664344791040000
*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

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maple [B]  time = 0.01, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (43697415465 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{9219840 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9219840*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(43697415465*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1
/2))[phone]*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^3*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+129473823600*7^(1/2)*x^2*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+43157941200*7^(1/2)*x*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+5754392160*7^(1/2)*arctan(
1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+3776638656*(-10*x^2-x+3)^(1/2))/
(-10*x^2-x+3)^(1/2)/(3*x+2)^5

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maxima [A]  time = 1.35, size = 198, normalized size = 1.10 \[ \frac {11988317}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {495385}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {239 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {8395 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2352 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {297231 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {3665849 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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