### 3.342 $$\int \frac{(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^7} \, dx$$

Optimal. Leaf size=195 $-\frac{14087245 \left (2 x^2-x+3\right )^{5/2}}{71663616 (2 x+5)^4}+\frac{182165 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}-\frac{(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{13759414272 (2 x+5)^3}+\frac{(27596573612 x+151764102421) \sqrt{2 x^2-x+3}}{55037657088 (2 x+5)}-\frac{1903976002333 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{660451885056 \sqrt{2}}+\frac{369 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}$

[Out]

((151764102421 + 27596573612*x)*Sqrt[3 - x + 2*x^2])/(55037657088*(5 + 2*x)) - ((9802984711 + 6793718806*x)*(3
- x + 2*x^2)^(3/2))/(13759414272*(5 + 2*x)^3) - (3667*(3 - x + 2*x^2)^(5/2))/(3456*(5 + 2*x)^6) + (182165*(3
- x + 2*x^2)^(5/2))/(248832*(5 + 2*x)^5) - (14087245*(3 - x + 2*x^2)^(5/2))/(71663616*(5 + 2*x)^4) + (369*ArcS
inh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2]) - (1903976002333*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])
/(660451885056*Sqrt[2])

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Rubi [A]  time = 0.267546, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {1650, 810, 812, 843, 619, 215, 724, 206} $-\frac{14087245 \left (2 x^2-x+3\right )^{5/2}}{71663616 (2 x+5)^4}+\frac{182165 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}-\frac{(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{13759414272 (2 x+5)^3}+\frac{(27596573612 x+151764102421) \sqrt{2 x^2-x+3}}{55037657088 (2 x+5)}-\frac{1903976002333 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{660451885056 \sqrt{2}}+\frac{369 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}$

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)^7,x]

[Out]

((151764102421 + 27596573612*x)*Sqrt[3 - x + 2*x^2])/(55037657088*(5 + 2*x)) - ((9802984711 + 6793718806*x)*(3
- x + 2*x^2)^(3/2))/(13759414272*(5 + 2*x)^3) - (3667*(3 - x + 2*x^2)^(5/2))/(3456*(5 + 2*x)^6) + (182165*(3
- x + 2*x^2)^(5/2))/(248832*(5 + 2*x)^5) - (14087245*(3 - x + 2*x^2)^(5/2))/(71663616*(5 + 2*x)^4) + (369*ArcS
inh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2]) - (1903976002333*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])
/(660451885056*Sqrt[2])

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]

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 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 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 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 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 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 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])

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)^7} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}-\frac{1}{432} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{68375}{16}-\frac{28085 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{10561025}{16}-1010880 x+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{\int \frac{\left (\frac{420053845}{16}-\frac{182410625 x}{4}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac{\int \frac{\left (-\frac{206718515415}{8}+\frac{103487151045 x}{2}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^2} \, dx}{51597803520}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{\int \frac{-\frac{4760153161395}{4}+2379948687360 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{412782428160}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{369}{128} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{1903976002333 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{110075314176}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{1903976002333 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{55037657088}-\frac{369 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt{46}}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac{369 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}-\frac{1903976002333 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{660451885056 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.258643, size = 108, normalized size = 0.55 $\frac{\frac{24 \sqrt{2 x^2-x+3} \left (275188285440 x^6+11854023276320 x^5+103803827945872 x^4+422554114856528 x^3+910256842473992 x^2+1011372787716826 x+458411625354581\right )}{(2 x+5)^6}-1903976002333 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+1903958949888 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1320903770112}$

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)^7,x]

[Out]

((24*Sqrt[3 - x + 2*x^2]*(458411625354581 + 1011372787716826*x + 910256842473992*x^2 + 422554114856528*x^3 + 1
03803827945872*x^4 + 11854023276320*x^5 + 275188285440*x^6))/(5 + 2*x)^6 + 1903958949888*Sqrt[2]*ArcSinh[(1 -
4*x)/Sqrt[23]] - 1903976002333*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/1320903770112

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Maple [A]  time = 0.073, size = 246, normalized size = 1.3 \begin{align*}{\frac{182165}{7962624} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}}-{\frac{-82772668391+331090673564\,x}{3962711310336}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3607708597}{2972033482752} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}-{\frac{1903976002333\,\sqrt{2}}{1320903770112}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }-{\frac{3667}{221184} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}-{\frac{369\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{1903976002333}{213986410758144} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{1903976002333}{3962711310336}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{149610673}{41278242816} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{-125860542215+503442168860\,x}{213986410758144} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{125860542215}{106993205379072} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}-{\frac{14087245}{1146617856} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}} \end{align*}

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)^7,x)

[Out]

182165/7962624/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(5/2)-82772668391/3962711310336*(-1+4*x)*(2*(x+5/2)^2-11*x-19
/2)^(1/2)-3607708597/2972033482752/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(5/2)-1903976002333/1320903770112*2^(1/2)
*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))-3667/221184/(x+5/2)^6*(2*(x+5/2)^2-11*x-19/2)
^(5/2)-369/256*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+1903976002333/213986410758144*(2*(x+5/2)^2-11*x-19/2)^(3
/2)+1903976002333/3962711310336*(2*(x+5/2)^2-11*x-19/2)^(1/2)+149610673/41278242816/(x+5/2)^3*(2*(x+5/2)^2-11*
x-19/2)^(5/2)-125860542215/213986410758144*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(3/2)+125860542215/106993205379072
/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(5/2)-14087245/1146617856/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(5/2)

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Maxima [A]  time = 1.5413, size = 401, normalized size = 2.06 \begin{align*} \frac{3607708597}{1486016741376} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3456 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac{182165 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{248832 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac{14087245 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{71663616 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{149610673 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5159780352 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{3607708597 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{743008370688 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{82772668391}{990677827584} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{369}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{1903976002333}{1320903770112} \, \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{165562389227}{330225942528} \, \sqrt{2 \, x^{2} - x + 3} + \frac{125860542215 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2972033482752 \,{\left (2 \, x + 5\right )}} \end{align*}

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)^7,x, algorithm="maxima")

[Out]

3607708597/1486016741376*(2*x^2 - x + 3)^(3/2) - 3667/3456*(2*x^2 - x + 3)^(5/2)/(64*x^6 + 960*x^5 + 6000*x^4
+ 20000*x^3 + 37500*x^2 + 37500*x + 15625) + 182165/248832*(2*x^2 - x + 3)^(5/2)/(32*x^5 + 400*x^4 + 2000*x^3
+ 5000*x^2 + 6250*x + 3125) - 14087245/71663616*(2*x^2 - x + 3)^(5/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 6
25) + 149610673/5159780352*(2*x^2 - x + 3)^(5/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 3607708597/743008370688*(2*x
^2 - x + 3)^(5/2)/(4*x^2 + 20*x + 25) - 82772668391/990677827584*sqrt(2*x^2 - x + 3)*x - 369/256*sqrt(2)*arcsi
nh(4/23*sqrt(23)*x - 1/23*sqrt(23)) + 1903976002333/1320903770112*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5
) - 17/23*sqrt(23)/abs(2*x + 5)) + 165562389227/330225942528*sqrt(2*x^2 - x + 3) + 125860542215/2972033482752*
(2*x^2 - x + 3)^(3/2)/(2*x + 5)

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Fricas [A]  time = 1.44627, size = 830, normalized size = 4.26 \begin{align*} \frac{1903958949888 \, \sqrt{2}{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 1903976002333 \, \sqrt{2}{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\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 (275188285440 \, x^{6} + 11854023276320 \, x^{5} + 103803827945872 \, x^{4} + 422554114856528 \, x^{3} + 910256842473992 \, x^{2} + 1011372787716826 \, x + 458411625354581\right )} \sqrt{2 \, x^{2} - x + 3}}{2641807540224 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} \end{align*}

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)^7,x, algorithm="fricas")

[Out]

1/2641807540224*(1903958949888*sqrt(2)*(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)
*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 1903976002333*sqrt(2)*(64*x^6 + 960*x^5 +
6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)*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*(275188285440*x^6 + 11854023276320*x^5 + 103803827945872*x^4 + 4
22554114856528*x^3 + 910256842473992*x^2 + 1011372787716826*x + 458411625354581)*sqrt(2*x^2 - x + 3))/(64*x^6
+ 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{7}}\, dx \end{align*}

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)**7,x)

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**7, x)

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Giac [B]  time = 1.26596, size = 610, normalized size = 3.13 \begin{align*} \frac{369}{256} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{1903976002333}{1320903770112} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{1903976002333}{1320903770112} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{5}{64} \, \sqrt{2 \, x^{2} - x + 3} + \frac{\sqrt{2}{\left (159278433934432 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} + 6347903280912544 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} + 48544526840833424 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 305716670132783088 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 88313821135911024 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} - 2423668581998843376 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 397211131697032056 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 11708897232532299576 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 12803484860728491138 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 12593033197867577234 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 3042533760672408875 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 589526263249780195\right )}}{110075314176 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{6}} \end{align*}

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)^7,x, algorithm="giac")

[Out]

369/256*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 1903976002333/1320903770112*sqrt(2)*lo
g(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 1903976002333/1320903770112*sqrt(2)*log(abs(-2*sqrt(2
)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 5/64*sqrt(2*x^2 - x + 3) + 1/110075314176*sqrt(2)*(15927843393443
2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^11 + 6347903280912544*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^10 + 48544
526840833424*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 305716670132783088*(sqrt(2)*x - sqrt(2*x^2 - x + 3)
)^8 + 88313821135911024*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 - 2423668581998843376*(sqrt(2)*x - sqrt(2*
x^2 - x + 3))^6 - 397211131697032056*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 11708897232532299576*(sqrt(
2)*x - sqrt(2*x^2 - x + 3))^4 - 12803484860728491138*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 12593033197
867577234*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 - 3042533760672408875*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))
+ 589526263249780195)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) -
11)^6