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

Optimal. Leaf size=180 \[ \frac {17 \sqrt {1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}-\frac {187 \sqrt {1-2 x} (5 x+3)^{5/2}}{1680 (3 x+2)^3}-\frac {2057 \sqrt {1-2 x} (5 x+3)^{3/2}}{9408 (3 x+2)^2}-\frac {22627 \sqrt {1-2 x} \sqrt {5 x+3}}{43904 (3 x+2)}-\frac {248897 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]

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Rubi [A]  time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {17 \sqrt {1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}-\frac {187 \sqrt {1-2 x} (5 x+3)^{5/2}}{1680 (3 x+2)^3}-\frac {2057 \sqrt {1-2 x} (5 x+3)^{3/2}}{9408 (3 x+2)^2}-\frac {22627 \sqrt {1-2 x} \sqrt {5 x+3}}{43904 (3 x+2)}-\frac {248897 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-22627*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (2057*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(9408*(2 + 3*x)^
2) - (187*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1680*(2 + 3*x)^3) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(35*(2 + 3*x
)^5) + (17*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (248897*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(43904*Sqrt[7])

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 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17}{10} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {187}{80} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {187 \sqrt {1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {2057}{672} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {2057 \sqrt {1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac {187 \sqrt {1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {22627 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{6272}\\ &=-\frac {22627 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {2057 \sqrt {1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac {187 \sqrt {1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {248897 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {22627 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {2057 \sqrt {1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac {187 \sqrt {1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac {248897 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {22627 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {2057 \sqrt {1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac {187 \sqrt {1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac {248897 \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 = 0.07, size = 84, normalized size = 0.47 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (27422145 x^4+74915550 x^3+74550556 x^2+32206264 x+5112864\right )}{(3 x+2)^5}-3733455 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4609920} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5112864 + 32206264*x + 74550556*x^2 + 74915550*x^3 + 27422145*x^4))/(2 + 3*x)
^5 - 3733455*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4609920

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IntegrateAlgebraic [A]  time = 0.35, size = 138, normalized size = 0.77 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {255 (1-2 x)^4}{(5 x+3)^4}+\frac {8330 (1-2 x)^3}{(5 x+3)^3}+\frac {106624 (1-2 x)^2}{(5 x+3)^2}-\frac {282730 (1-2 x)}{5 x+3}-612255\right )}{658560 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^5}-\frac {248897 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-612255 + (255*(1 - 2*x)^4)/(3 + 5*x)^4 + (8330*(1 - 2*x)^3)/(3 + 5*x)^3 + (106624*(1 -
2*x)^2)/(3 + 5*x)^2 - (282730*(1 - 2*x))/(3 + 5*x)))/(658560*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^5) - (24
8897*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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fricas [A]  time = 1.52, size = 131, normalized size = 0.73 \begin {gather*} -\frac {3733455 \, \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 (27422145 \, x^{4} + 74915550 \, x^{3} + 74550556 \, x^{2} + 32206264 \, x + 5112864\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9219840*(3733455*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*(27422145*x^4 + 74915550*x^3 + 74550556*x^2 + 32206264
*x + 5112864)*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 = 3.19, size = 426, normalized size = 2.37 \begin {gather*} \frac {248897}{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 {14641 \, \sqrt {10} {\left (51 \, {\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} + 66640 \, {\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} + 34119680 \, {\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} - 3618944000 \, {\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 {313474560000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1253898240000 \, \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

248897/6146560*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)))) - 14641/65856*sqrt(10)*(51*((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 + 66640*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 34119680*((
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 -
3618944000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))^3 - 313474560000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1253898240000*sqrt(5*x + 3)/(sq
rt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(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 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (907229565 \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9219840*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(907229565*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*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+2688087600*7^(1/2)*x^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^3+896029200*7^(1/2)*x*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^2+119470560*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+71580096*(-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 = 1.21, size = 198, normalized size = 1.10 \begin {gather*} \frac {248897}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {10285}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{105 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{40 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {45 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {6171 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {76109 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

248897/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 10285/32928*sqrt(-10*x^2 - x + 3) +
1/105*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) - 3/40*(-10*x^2 - x + 3)^(
3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 45/784*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) +
6171/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 76109/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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