3.4.32 \(\int \frac {\sqrt {3-x+2 x^2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^7} \, dx\) [332]

Optimal. Leaf size=169 \[ -\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac {26972675 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{3962711310336 \sqrt {2}} \]

[Out]

-3667/3456*(2*x^2-x+3)^(3/2)/(5+2*x)^6+92239/138240*(2*x^2-x+3)^(3/2)/(5+2*x)^5-5703277/39813120*(2*x^2-x+3)^(
3/2)/(5+2*x)^4+87677717/8599633920*(2*x^2-x+3)^(3/2)/(5+2*x)^3-26972675/7925422620672*arctanh(1/24*(17-22*x)*2
^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-1172725/330225942528*(17-22*x)*(2*x^2-x+3)^(1/2)/(5+2*x)^2

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1664, 820, 734, 738, 212} \begin {gather*} \frac {87677717 \left (2 x^2-x+3\right )^{3/2}}{8599633920 (2 x+5)^3}-\frac {5703277 \left (2 x^2-x+3\right )^{3/2}}{39813120 (2 x+5)^4}+\frac {92239 \left (2 x^2-x+3\right )^{3/2}}{138240 (2 x+5)^5}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{3456 (2 x+5)^6}-\frac {1172725 (17-22 x) \sqrt {2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac {26972675 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{3962711310336 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^7,x]

[Out]

(-1172725*(17 - 22*x)*Sqrt[3 - x + 2*x^2])/(330225942528*(5 + 2*x)^2) - (3667*(3 - x + 2*x^2)^(3/2))/(3456*(5
+ 2*x)^6) + (92239*(3 - x + 2*x^2)^(3/2))/(138240*(5 + 2*x)^5) - (5703277*(3 - x + 2*x^2)^(3/2))/(39813120*(5
+ 2*x)^4) + (87677717*(3 - x + 2*x^2)^(3/2))/(8599633920*(5 + 2*x)^3) - (26972675*ArcTanh[(17 - 22*x)/(12*Sqrt
[2]*Sqrt[3 - x + 2*x^2])])/(3962711310336*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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 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 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[S
implify[m + 2*p + 3], 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 {\sqrt {3-x+2 x^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 )^{3/2}}{3456 (5+2 x)^6}-\frac {1}{432} \int \frac {\sqrt {3-x+2 x^2} \left (\frac {61041}{16}-\frac {20751 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}+\frac {\int \frac {\sqrt {3-x+2 x^2} \left (\frac {8057313}{16}-\frac {1191609 x}{2}+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}-\frac {\int \frac {\left (\frac {182650383}{16}-\frac {60644907 x}{4}\right ) \sqrt {3-x+2 x^2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac {1172725 \int \frac {\sqrt {3-x+2 x^2}}{(5+2 x)^3} \, dx}{1146617856}\\ &=-\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac {26972675 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{660451885056}\\ &=-\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac {26972675 \text {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{330225942528}\\ &=-\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac {26972675 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{3962711310336 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.77, size = 86, normalized size = 0.51 \begin {gather*} \frac {\frac {12 \sqrt {3-x+2 x^2} \left (-219337079305+27245373694 x+158340720344 x^2+397498825328 x^3+12256250416 x^4+271409942624 x^5\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 )}{19813556551680} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^7,x]

[Out]

((12*Sqrt[3 - x + 2*x^2]*(-219337079305 + 27245373694*x + 158340720344*x^2 + 397498825328*x^3 + 12256250416*x^
4 + 271409942624*x^5))/(5 + 2*x)^6 + 134863375*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6])/198135565
51680

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 195, normalized size = 1.15

method	result	size
risch	\(\frac {542819885248 x^{7}-246897441792 x^{6}[phone] x^{5}-44048633392 x^{4}[phone] x^{3}[phone] x^{2}[phone] x -658011237915}{1651129712640 \left (5+2 x \right )^{6} \sqrt {2 x^{2}-x +3}}-\frac {26972675 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{7925422620672}\)	\(88\)
trager	\(\frac {\left (271409942624 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x -219337079305\right ) \sqrt {2 x^{2}-x +3}}{1651129712640 \left (5+2 x \right )^{6}}-\frac {26972675 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {22 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -17 \RootOf \left (\textit {\_Z}^{2}-2\right )-24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{7925422620672}\)	\(98\)
default	\(\frac {92239 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{4423680 \left (x +\frac {5}{2}\right )^{5}}-\frac {5703277 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{637009920 \left (x +\frac {5}{2}\right )^{4}}+\frac {87677717 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{68797071360 \left (x +\frac {5}{2}\right )^{3}}-\frac {1172725 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{330225942528 \left (x +\frac {5}{2}\right )^{2}}-\frac {12899975 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{11888133931008 \left (x +\frac {5}{2}\right )}+\frac {12899975 \left (4 x -1\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{23776267862016}-\frac {3667 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{221184 \left (x +\frac {5}{2}\right )^{6}}-\frac {26972675 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{7925422620672}+\frac {26972675 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{23776267862016}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x,method=_RETURNVERBOSE)

[Out]

92239/4423680/(x+5/2)^5*(2*(x+5/2)^2-11*x-19/2)^(3/2)-5703277/637009920/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(3/2
)[phone]/68797071360/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(3/2)-1172725/330225942528/(x+5/2)^2*(2*(x+5/2)^2-11*
x-19/2)^(3/2)-12899975/11888133931008/(x+5/2)*(2*(x+5/2)^2-11*x-19/2)^(3/2)[phone]/23776267862016*(4*x-1)*(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)^(3/2)-26972675/7925422620672*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]/23776267862016*(2*(x+5/2)^2-11*x-19/
2)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 250, normalized size = 1.48 \begin {gather*} \frac {26972675}{7925422620672} \, \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 {1172725}{165112971264} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3456 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac {92239 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{138240 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac {5703277 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{39813120 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {87677717 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8599633920 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {1172725 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{82556485632 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {12899975 \, \sqrt {2 \, x^{2} - x + 3}}{330225942528 \, {\left (2 \, x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x, algorithm="maxima")

[Out]

26972675/7925422620672*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 1172725/
165112971264*sqrt(2*x^2 - x + 3) - 3667/3456*(2*x^2 - x + 3)^(3/2)/(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 +
37500*x^2 + 37500*x + 15625) + 92239/138240*(2*x^2 - x + 3)^(3/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 62
50*x + 3125) - 5703277/39813120*(2*x^2 - x + 3)^(3/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 87677717/8
599633920*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 1172725/82556485632*(2*x^2 - x + 3)^(3/2)/(4*
x^2 + 20*x + 25) - 12899975/330225942528*sqrt(2*x^2 - x + 3)/(2*x + 5)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 156, normalized size = 0.92 \begin {gather*} \frac {134863375 \, \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 (271409942624 \, x^{5} + 12256250416 \, x^{4} + 397498825328 \, x^{3} + 158340720344 \, x^{2} + 27245373694 \, x - 219337079305\right )} \sqrt {2 \, x^{2} - x + 3}}{79254226206720 \, {\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((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x, algorithm="fricas")

[Out]

1/79254226206720*(134863375*sqrt(2)*(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 37500*x + 15625)*lo
g(-(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*(27140994
2624*x^5 + 12256250416*x^4 + 397498825328*x^3 + 158340720344*x^2 + 27245373694*x - 219337079305)*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 {\sqrt {2 x^{2} - x + 3} \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((5*x**4-x**3+3*x**2+x+2)*(2*x**2-x+3)**(1/2)/(5+2*x)**7,x)

[Out]

Integral(sqrt(2*x**2 - x + 3)*(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. 405 vs. \(2 (139) = 278\).
time = 3.24, size = 405, normalized size = 2.40 \begin {gather*} -\frac {26972675}{7925422620672} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {26972675}{7925422620672} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {\sqrt {2} {\left (16506981498400 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} + 389429252643040 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} + 2263923918689840 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 11663651054548560 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 902212326134736 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} - 84192729519861840 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 4317200555009448 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 351543414066518760 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 376787166452923830 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 356306707647610982 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 82348353128195465 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 15499394004553969\right )}}{3302259425280 \, {\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((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^7,x, algorithm="giac")

[Out]

-26972675/7925422620672*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 26972675/7925422620
672*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 1/3302259425280*sqrt(2)*(16506981498
400*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^11 + 389429252643040*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^10 + 2263
923918689840*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 11663651054548560*(sqrt(2)*x - sqrt(2*x^2 - x + 3))
^8 + 902212326134736*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 - 84192729519861840*(sqrt(2)*x - sqrt(2*x^2 -
x + 3))^6 - 4317200555009448*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 351543414066518760*(sqrt(2)*x - sq
rt(2*x^2 - x + 3))^4 - 376787166452923830*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 356306707647610982*(sq
rt(2)*x - sqrt(2*x^2 - x + 3))^2 - 82348353128195465*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1549939400455
3969)/(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 {\sqrt {2\,x^2-x+3}\,\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)^(1/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^7,x)

[Out]

int(((2*x^2 - x + 3)^(1/2)*(x + 3*x^2 - x^3 + 5*x^4 + 2))/(2*x + 5)^7, x)

________________________________________________________________________________________