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

Optimal. Leaf size=180 \[ \frac {3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac {251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac {2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac {30371 (5 x+3)^{3/2} \sqrt {1-2 x}}{448 (3 x+2)^2}-\frac {334081 \sqrt {5 x+3} \sqrt {1-2 x}}{6272 (3 x+2)}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]

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Rubi [A]  time = 0.05, 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 {3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac {251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac {2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac {30371 (5 x+3)^{3/2} \sqrt {1-2 x}}{448 (3 x+2)^2}-\frac {334081 \sqrt {5 x+3} \sqrt {1-2 x}}{6272 (3 x+2)}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-334081*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6272*(2 + 3*x)) + (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/(35*(2 + 3*x)^5)
+ (251*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(280*(2 + 3*x)^4) + (2761*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(336*(2 + 3
*x)^3) + (30371*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(448*(2 + 3*x)^2) - (3674891*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt
[3 + 5*x])])/(6272*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 {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251}{70} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761}{112} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371}{224} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {334081}{896} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {3674891 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {3674891 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 164, normalized size = 0.91 \begin {gather*} \frac {1}{70} \left (\frac {6 (5 x+3)^{3/2} (1-2 x)^{7/2}}{(3 x+2)^5}+\frac {251 \left (2352 (5 x+3)^{3/2} (1-2 x)^{5/2}+55 (3 x+2) \left (392 (1-2 x)^{3/2} (5 x+3)^{3/2}+33 (3 x+2) \left (7 \sqrt {1-2 x} \sqrt {5 x+3} (37 x+20)-121 \sqrt {7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )\right )\right )}{9408 (3 x+2)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5 + (251*(2352*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2) + 55*(2 + 3*x)*(
392*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2) + 33*(2 + 3*x)*(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(20 + 37*x) - 121*Sqrt[7]*(2
+ 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))))/(9408*(2 + 3*x)^4))/70

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IntegrateAlgebraic [A]  time = 0.40, size = 138, normalized size = 0.77 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {3765 (1-2 x)^4}{(5 x+3)^4}-\frac {190610 (1-2 x)^3}{(5 x+3)^3}-\frac {1574272 (1-2 x)^2}{(5 x+3)^2}-\frac {6026510 (1-2 x)}{5 x+3}-9039765\right )}{94080 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^5}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-9039765 + (3765*(1 - 2*x)^4)/(3 + 5*x)^4 - (190610*(1 - 2*x)^3)/(3 + 5*x)^3 - (1574272
*(1 - 2*x)^2)/(3 + 5*x)^2 - (6026510*(1 - 2*x))/(3 + 5*x)))/(94080*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^5)
- (3674891*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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fricas [A]  time = 0.83, size = 131, normalized size = 0.73 \begin {gather*} -\frac {55123365 \, \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 (390269835 \, x^{4} + 1058136330 \, x^{3} + 1076423732 \, x^{2} + 487066088 \, x + 82697568\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1317120*(55123365*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*(390269835*x^4 + 1058136330*x^3 + 1076423732*x^2 + 48
7066088*x + 82697568)*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 = 3.45, size = 426, normalized size = 2.37 \begin {gather*} \frac {3674891}{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 {14641 \, \sqrt {10} {\left (753 \, {\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} - 1524880 \, {\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} - 503767040 \, {\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} - 77139328000 \, {\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 {4628359680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {18513438720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3674891/878080*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)))) - 14641/9408*sqrt(10)*(753*((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 - 1524880*((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)))^7 - 503767040
*((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
- 77139328000*((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 - 4628359680000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 18513438720000*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 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (13394977695 \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 )}{1317120 \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((-2*x+1)^(5/2)*(5*x+3)^(1/2)/(3*x+2)^6,x)

[Out]

1/1317120*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(13394977695*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
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4+39688822800*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+13229607600*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+1763947680*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+1157765952*(-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 \begin {gather*} \frac {3674891}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {151855}{4704} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{15 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {73 \, {\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 {2573 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{336 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {91113 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3136 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1123727 \, \sqrt {-10 \, x^{2} - x + 3}}{18816 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3674891/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 151855/4704*sqrt(-10*x^2 - x + 3) +
7/15*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 73/40*(-10*x^2 - x + 3)^(
3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 2573/336*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8)
+ 91113/3136*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1123727/18816*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 {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\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)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^6,x)

[Out]

int(((1 - 2*x)^(5/2)*(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 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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