3.340 \(\int \frac {(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^5} \, dx\)

Optimal. Leaf size=188 \[ -\frac {14477995 \left (2 x^2-x+3\right )^{5/2}}{23887872 (2 x+5)^2}+\frac {224815 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^3}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{2304 (2 x+5)^4}+\frac {(67865260 x+762984903) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)}+\frac {(2339916063-389975609 x) \sqrt {2 x^2-x+3}}{31850496}-\frac {8969688643 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{21233664 \sqrt {2}}+\frac {432565 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1024 \sqrt {2}} \]

[Out]

1/95551488*(762984903+67865260*x)*(2*x^2-x+3)^(3/2)/(5+2*x)-3667/2304*(2*x^2-x+3)^(5/2)/(5+2*x)^4+224815/16588
8*(2*x^2-x+3)^(5/2)/(5+2*x)^3-14477995/23887872*(2*x^2-x+3)^(5/2)/(5+2*x)^2+432565/2048*arcsinh(1/23*(1-4*x)*2
3^(1/2))*2^(1/2)-8969688643/42467328*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)+1/31850496*(233
9916063-389975609*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1650, 812, 814, 843, 619, 215, 724, 206} \[ -\frac {14477995 \left (2 x^2-x+3\right )^{5/2}}{23887872 (2 x+5)^2}+\frac {224815 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^3}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{2304 (2 x+5)^4}+\frac {(67865260 x+762984903) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)}+\frac {(2339916063-389975609 x) \sqrt {2 x^2-x+3}}{31850496}-\frac {8969688643 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{21233664 \sqrt {2}}+\frac {432565 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1024 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^5,x]

[Out]

((2339916063 - 389975609*x)*Sqrt[3 - x + 2*x^2])/31850496 + ((762984903 + 67865260*x)*(3 - x + 2*x^2)^(3/2))/(
95551488*(5 + 2*x)) - (3667*(3 - x + 2*x^2)^(5/2))/(2304*(5 + 2*x)^4) + (224815*(3 - x + 2*x^2)^(5/2))/(165888
*(5 + 2*x)^3) - (14477995*(3 - x + 2*x^2)^(5/2))/(23887872*(5 + 2*x)^2) + (432565*ArcSinh[(1 - 4*x)/Sqrt[23]])
/(1024*Sqrt[2]) - (8969688643*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(21233664*Sqrt[2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^5} \, dx &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}-\frac {1}{288} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {51695}{16}-\frac {24835 x}{4}+1944 x^2-720 x^3\right )}{(5+2 x)^4} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}+\frac {\int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {5995005}{16}-\frac {1483149 x}{2}+77760 x^2\right )}{(5+2 x)^3} \, dx}{62208}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac {14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac {\int \frac {\left (\frac {252996909}{16}-\frac {152696835 x}{4}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^2} \, dx}{8957952}\\ &=\frac {(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac {14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}+\frac {\int \frac {\left (\frac {10531588167}{8}-3509780481 x\right ) \sqrt {3-x+2 x^2}}{5+2 x} \, dx}{71663616}\\ &=\frac {(2339916063-389975609 x) \sqrt {3-x+2 x^2}}{31850496}+\frac {(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac {14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac {\int \frac {-968737607064+1937448253440 x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{2293235712}\\ &=\frac {(2339916063-389975609 x) \sqrt {3-x+2 x^2}}{31850496}+\frac {(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac {14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac {432565 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{1024}+\frac {8969688643 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{3538944}\\ &=\frac {(2339916063-389975609 x) \sqrt {3-x+2 x^2}}{31850496}+\frac {(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac {14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac {8969688643 \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{1769472}-\frac {432565 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{1024 \sqrt {46}}\\ &=\frac {(2339916063-389975609 x) \sqrt {3-x+2 x^2}}{31850496}+\frac {(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac {224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac {14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}+\frac {432565 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1024 \sqrt {2}}-\frac {8969688643 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{21233664 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 108, normalized size = 0.57 \[ \frac {-8969688643 \sqrt {2} \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )+\frac {24 \sqrt {2 x^2-x+3} \left (2949120 x^6-29270016 x^5+468043776 x^4+11761910072 x^3+60528581892 x^2+121473790266 x+86386856771\right )}{(2 x+5)^4}[phone] \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{42467328} \]

Antiderivative was successfully verified.

[In]

Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^5,x]

[Out]

((24*Sqrt[3 - x + 2*x^2]*(86386856771 + 121473790266*x + 60528581892*x^2 + 11761910072*x^3 + 468043776*x^4 - 2
9270016*x^5 + 2949120*x^6))/(5 + 2*x)^4 + 8969667840*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 8969688643*Sqrt[2]*
ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/42467328

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fricas [A]  time = 0.96, size = 199, normalized size = 1.06 \[ \frac {8969667840 \, \sqrt {2} {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8969688643 \, \sqrt {2} {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (2949120 \, x^{6} - 29270016 \, x^{5} + 468043776 \, x^{4} + 11761910072 \, x^{3} + 60528581892 \, x^{2} + 121473790266 \, x + 86386856771\right )} \sqrt {2 \, x^{2} - x + 3}}{84934656 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^5,x, algorithm="fricas")

[Out]

1/84934656*(8969667840*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(
4*x - 1) - 32*x^2 + 16*x - 25) + 8969688643*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)*log(-(24*sqrt(
2)*sqrt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) + 48*(2949120*x^6 - 292700
16*x^5 + 468043776*x^4 + 11761910072*x^3 + 60528581892*x^2 + 121473790266*x + 86386856771)*sqrt(2*x^2 - x + 3)
)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)

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giac [B]  time = 0.42, size = 503, normalized size = 2.68 \[ -\frac {1}{42467328} \, \sqrt {2} {\left (8969688643 \, \log \left (12 \, \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {72}{2 \, x + 5} - 11\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) + 8969667840 \, \log \left ({\left | \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5} + 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) - 8969667840 \, \log \left ({\left | \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) + 12 \, {\left (\frac {24 \, {\left (\frac {1296 \, {\left (\frac {29336 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}{2 \, x + 5} - 42907 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )}}{2 \, x + 5} + 39923563 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )}}{2 \, x + 5} - 541312039 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )} \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {13824 \, {\left (806241 \, {\left (\sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) - 1152288 \, {\left (\sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) - 957352 \, {\left (\sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) + 1529280 \, {\left (\sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) + 394431 \, {\left (\sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right ) - 620352 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )}}{{\left ({\left (\sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {6}{2 \, x + 5}\right )}^{2} - 1\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^5,x, algorithm="giac")

[Out]

-1/42467328*sqrt(2)*(8969688643*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)*sgn(1/(2*
x + 5)) + 8969667840*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) + 1))*sgn(1/(2*x + 5)) - 8
969667840*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) - 1))*sgn(1/(2*x + 5)) + 12*(24*(1296
*(29336*sgn(1/(2*x + 5))/(2*x + 5) - 42907*sgn(1/(2*x + 5)))/(2*x + 5) + 39923563*sgn(1/(2*x + 5)))/(2*x + 5)
- 541312039*sgn(1/(2*x + 5)))*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 13824*(806241*(sqrt(-11/(2*x + 5) + 3
6/(2*x + 5)^2 + 1) + 6/(2*x + 5))^5*sgn(1/(2*x + 5)) - 1152288*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(
2*x + 5))^4*sgn(1/(2*x + 5)) - 957352*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^3*sgn(1/(2*x +
5)) + 1529280*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2*sgn(1/(2*x + 5)) + 394431*(sqrt(-11/(
2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))*sgn(1/(2*x + 5)) - 620352*sgn(1/(2*x + 5)))/((sqrt(-11/(2*x + 5)
+ 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2 - 1)^3)

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maple [A]  time = 0.02, size = 204, normalized size = 1.09 \[ -\frac {432565 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2048}-\frac {8969688643 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{42467328}+\frac {8969688643 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{127401984}+\frac {8969688643 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{6879707136}-\frac {3667 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{36864 \left (x +\frac {5}{2}\right )^{4}}+\frac {224815 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{1327104 \left (x +\frac {5}{2}\right )^{3}}-\frac {14477995 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{95551488 \left (x +\frac {5}{2}\right )^{2}}-\frac {593321753 \left (4 x -1\right ) \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{6879707136}+\frac {593321753 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{3439853568 \left (x +\frac {5}{2}\right )}-\frac {389975609 \left (4 x -1\right ) \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{127401984} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^5,x)

[Out]

8969688643/127401984*(-11*x+2*(x+5/2)^2-19/2)^(1/2)[phone]/6879707136*(-11*x+2*(x+5/2)^2-19/2)^(3/2)-3667/
36864/(x+5/2)^4*(-11*x+2*(x+5/2)^2-19/2)^(5/2)+224815/1327104/(x+5/2)^3*(-11*x+2*(x+5/2)^2-19/2)^(5/2)-1447799
5/95551488/(x+5/2)^2*(-11*x+2*(x+5/2)^2-19/2)^(5/2)-593321753/6879707136*(4*x-1)*(-11*x+2*(x+5/2)^2-19/2)^(3/2
)[phone]/3439853568/(x+5/2)*(-11*x+2*(x+5/2)^2-19/2)^(5/2)-389975609/127401984*(4*x-1)*(-11*x+2*(x+5/2)^2-1
9/2)^(1/2)-8969688643/42467328*2^(1/2)*arctanh(1/12*(-11*x+17/2)*2^(1/2)/(-11*x+2*(x+5/2)^2-19/2)^(1/2))-43256
5/2048*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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maxima [A]  time = 1.04, size = 210, normalized size = 1.12 \[ \frac {16966315}{47775744} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2304 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {224815 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{165888 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {14477995 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{23887872 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {389975609}{31850496} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {432565}{2048} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {8969688643}{42467328} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {779972021}{10616832} \, \sqrt {2 \, x^{2} - x + 3} + \frac {593321753 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{95551488 \, {\left (2 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^5,x, algorithm="maxima")

[Out]

16966315/47775744*(2*x^2 - x + 3)^(3/2) - 3667/2304*(2*x^2 - x + 3)^(5/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x
+ 625) + 224815/165888*(2*x^2 - x + 3)^(5/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 14477995/23887872*(2*x^2 - x +
3)^(5/2)/(4*x^2 + 20*x + 25) - 389975609/31850496*sqrt(2*x^2 - x + 3)*x - 432565/2048*sqrt(2)*arcsinh(4/23*sqr
t(23)*x - 1/23*sqrt(23)) + 8969688643/42467328*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/
abs(2*x + 5)) + 779972021/10616832*sqrt(2*x^2 - x + 3) + 593321753/95551488*(2*x^2 - x + 3)^(3/2)/(2*x + 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2 - x + 3)^(3/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^5,x)

[Out]

int(((2*x^2 - x + 3)^(3/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-x+3)**(3/2)*(5*x**4-x**3+3*x**2+x+2)/(5+2*x)**5,x)

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**5, x)

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