3.4.42 \(\int \frac {(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^7} \, dx\) [342]

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

[Out]

-1/13759414272*(9802984711+6793718806*x)*(2*x^2-x+3)^(3/2)/(5+2*x)^3-3667/3456*(2*x^2-x+3)^(5/2)/(5+2*x)^6+182
165/248832*(2*x^2-x+3)^(5/2)/(5+2*x)^5-14087245/71663616*(2*x^2-x+3)^(5/2)/(5+2*x)^4+369/256*arcsinh(1/23*(1-4
*x)*23^(1/2))*2^(1/2)-1903976002333/1320903770112*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)+1/
55037657088*(151764102421+27596573612*x)*(2*x^2-x+3)^(1/2)/(5+2*x)

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 195, 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 = {1664, 824, 826, 857, 633, 221, 738, 212} \begin {gather*} -\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}} \end {gather*}

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 212

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

Rule 221

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 633

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 738

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 826

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 857

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 1664

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)^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}[phone] 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 \text {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{55037657088}-\frac {369 \text {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 = 0.97, size = 120, normalized size = 0.62 \begin {gather*} \frac {\frac {12 \sqrt {3-x+2 x^2} \left (458411625354581+1011372787716826 x+910256842473992 x^2+422554114856528 x^3+103803827945872 x^4+11854023276320 x^5+275188285440 x^6\right )}{(5+2 x)^6}[phone] \sqrt {2} \tanh ^{-1}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )[phone] \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{660451885056} \end {gather*}

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]

((12*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 + 1903976002333*Sqrt[2]*ArcTanh[(5 +
2*x - Sqrt[6 - 2*x + 4*x^2])/6] + 951979474944*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/660451885056

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 246, normalized size = 1.26 \[-\frac {3667 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{221184 \left (x +\frac {5}{2}\right )^{6}}+\frac {182165 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{7962624 \left (x +\frac {5}{2}\right )^{5}}+\frac {149610673 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{41278242816 \left (x +\frac {5}{2}\right )^{3}}-\frac {3607708597 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{2972033482752 \left (x +\frac {5}{2}\right )^{2}}+\frac {125860542215 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{106993205379072 \left (x +\frac {5}{2}\right )}-\frac {125860542215 \left (4 x -1\right ) \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{213986410758144}-\frac {82772668391 \left (4 x -1\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{3962711310336}-\frac {14087245 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {5}{2}}}{1146617856 \left (x +\frac {5}{2}\right )^{4}}-\frac {1903976002333 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{1320903770112}+\frac {1903976002333 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{3962711310336}-\frac {369 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}+\frac {1903976002333 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{213986410758144}\]

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]

-3667/221184/(x+5/2)^6*(2*(x+5/2)^2-11*x-19/2)^(5/2)+182165/7962624/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(5/2)+14
9610673/41278242816/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(5/2)-3607708597/2972033482752/(x+5/2)^2*(2*(x+5/2)^2-11
*x-19/2)^(5/2)[phone]/106993205379072/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(5/2)-125860542215/213986410758144
*(4*x-1)*(2*(x+5/2)^2-11*x-19/2)^(3/2)-82772668391/3962711310336*(4*x-1)*(2*(x+5/2)^2-11*x-19/2)^(1/2)-1408724
5/1146617856/(x+5/2)^4*(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))[phone]/3962711310336*(2*(x+5/2)^2-11*x-19/2)^(1/2)-369/256*2
^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))[phone]/213986410758144*(2*(x+5/2)^2-11*x-19/2)^(3/2)

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 297, normalized size = 1.52 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 229, normalized size = 1.17 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{7}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (160) = 320\).
time = 3.17, size = 452, normalized size = 2.32 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 )}^7} \,d x \end {gather*}

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)^7,x)

[Out]

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

________________________________________________________________________________________