3.1.71 \(\int \frac {x^2}{(a+\frac {c}{x^2}+\frac {b}{x}) (d+e x)^2} \, dx\) [71]

Optimal. Leaf size=274 \[ \frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )-b^2 c \left (4 a d^2-c e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {d^3 \left (2 a d^2-e (3 b d-4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {(b d-c e) \left (b^2 d-2 a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2} \]

[Out]

x/a/e^2-d^4/e^3/(a*d^2-e*(b*d-c*e))/(e*x+d)-d^3*(2*a*d^2-e*(3*b*d-4*c*e))*ln(e*x+d)/e^3/(a*d^2-e*(b*d-c*e))^2-
1/2*(b*d-c*e)*(-2*a*c*d+b^2*d-b*c*e)*ln(a*x^2+b*x+c)/a^2/(a*d^2-e*(b*d-c*e))^2-(b^4*d^2-2*b^3*c*d*e+6*a*b*c^2*
d*e+2*a*c^2*(a*d^2-c*e^2)-b^2*c*(4*a*d^2-c*e^2))*arctanh((2*a*x+b)/(-4*a*c+b^2)^(1/2))/a^2/(a*d^2-e*(b*d-c*e))
^2/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1583, 1642, 648, 632, 212, 642} \begin {gather*} -\frac {(b d-c e) \left (-2 a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left (-b^2 c \left (4 a d^2-c e^2\right )+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {d^4}{e^3 (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac {d^3 \log (d+e x) \left (2 a d^2-e (3 b d-4 c e)\right )}{e^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {x}{a e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/((a + c/x^2 + b/x)*(d + e*x)^2),x]

[Out]

x/(a*e^2) - d^4/(e^3*(a*d^2 - e*(b*d - c*e))*(d + e*x)) - ((b^4*d^2 - 2*b^3*c*d*e + 6*a*b*c^2*d*e + 2*a*c^2*(a
*d^2 - c*e^2) - b^2*c*(4*a*d^2 - c*e^2))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]*(a*d^2
- e*(b*d - c*e))^2) - (d^3*(2*a*d^2 - e*(3*b*d - 4*c*e))*Log[d + e*x])/(e^3*(a*d^2 - e*(b*d - c*e))^2) - ((b*
d - c*e)*(b^2*d - 2*a*c*d - b*c*e)*Log[c + b*x + a*x^2])/(2*a^2*(a*d^2 - e*(b*d - c*e))^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 632

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

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 1583

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbo
l] :> Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && E
qQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx &=\int \frac {x^4}{(d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{a e^2}+\frac {d^4}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)^2}+\frac {d^3 \left (-2 a d^2+e (3 b d-4 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac {-c \left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right )-(b d-c e) \left (b^2 d-2 a c d-b c e\right ) x}{a \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=\frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {d^3 \left (2 a d^2-e (3 b d-4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\int \frac {-c \left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right )-(b d-c e) \left (b^2 d-2 a c d-b c e\right ) x}{c+b x+a x^2} \, dx}{a \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {d^3 \left (2 a d^2-e (3 b d-4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left ((b d-c e) \left (b^2 d-2 a c d-b c e\right )\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )-b^2 c \left (4 a d^2-c e^2\right )\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {d^3 \left (2 a d^2-e (3 b d-4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {(b d-c e) \left (b^2 d-2 a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )-b^2 c \left (4 a d^2-c e^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=\frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )-b^2 c \left (4 a d^2-c e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {d^3 \left (2 a d^2-e (3 b d-4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {(b d-c e) \left (b^2 d-2 a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 269, normalized size = 0.98 \begin {gather*} \frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2+e (-b d+c e)\right ) (d+e x)}+\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^2 c \left (-4 a d^2+c e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a^2 \sqrt {-b^2+4 a c} \left (a d^2+e (-b d+c e)\right )^2}-\frac {\left (2 a d^5+d^3 e (-3 b d+4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2+e (-b d+c e)\right )^2}+\frac {(b d-c e) \left (-b^2 d+2 a c d+b c e\right ) \log (c+x (b+a x))}{2 a^2 \left (a d^2+e (-b d+c e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/((a + c/x^2 + b/x)*(d + e*x)^2),x]

[Out]

x/(a*e^2) - d^4/(e^3*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((b^4*d^2 - 2*b^3*c*d*e + 6*a*b*c^2*d*e + 2*a*c^2
*(a*d^2 - c*e^2) + b^2*c*(-4*a*d^2 + c*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^2*Sqrt[-b^2 + 4*a*c]*(
a*d^2 + e*(-(b*d) + c*e))^2) - ((2*a*d^5 + d^3*e*(-3*b*d + 4*c*e))*Log[d + e*x])/(e^3*(a*d^2 + e*(-(b*d) + c*e
))^2) + ((b*d - c*e)*(-(b^2*d) + 2*a*c*d + b*c*e)*Log[c + x*(b + a*x)])/(2*a^2*(a*d^2 + e*(-(b*d) + c*e))^2)

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Maple [A]
time = 0.31, size = 290, normalized size = 1.06

method	result	size
default	\(-\frac {d^{4}}{e^{3} \left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right )}-\frac {d^{3} \left (2 a \,d^{2}-3 d e b +4 c \,e^{2}\right ) \ln \left (e x +d \right )}{e^{3} \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}+\frac {x}{a \,e^{2}}+\frac {\frac {\left (2 a b c \,d^{2}-2 a \,c^{2} d e -b^{3} d^{2}+2 b^{2} c d e -c^{2} e^{2} b \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (a \,c^{2} d^{2}-b^{2} c \,d^{2}+2 e d \,c^{2} b -c^{3} e^{2}-\frac {\left (2 a b c \,d^{2}-2 a \,c^{2} d e -b^{3} d^{2}+2 b^{2} c d e -c^{2} e^{2} b \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a}\)	\(290\)
risch	\(\text {Expression too large to display}\)	\(55425\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+c/x^2+b/x)/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

-1/e^3*d^4/(a*d^2-b*d*e+c*e^2)/(e*x+d)-1/e^3*d^3*(2*a*d^2-3*b*d*e+4*c*e^2)/(a*d^2-b*d*e+c*e^2)^2*ln(e*x+d)+x/a
/e^2+1/(a*d^2-b*d*e+c*e^2)^2/a*(1/2*(2*a*b*c*d^2-2*a*c^2*d*e-b^3*d^2+2*b^2*c*d*e-b*c^2*e^2)/a*ln(a*x^2+b*x+c)+
2*(a*c^2*d^2-b^2*c*d^2+2*e*d*c^2*b-c^3*e^2-1/2*(2*a*b*c*d^2-2*a*c^2*d*e-b^3*d^2+2*b^2*c*d*e-b*c^2*e^2)*b/a)/(4
*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+c/x^2+b/x)/(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1056 vs. \(2 (279) = 558\).
time = 26.56, size = 2131, normalized size = 7.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+c/x^2+b/x)/(e*x+d)^2,x, algorithm="fricas")

[Out]

[-1/2*(2*(a^3*b^2 - 4*a^4*c)*d^6 - 2*(a*b^2*c^2 - 4*a^2*c^3)*x^2*e^6 + ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3*e^3
+ (b^2*c^2 - 2*a*c^3)*x*e^6 - (2*(b^3*c - 3*a*b*c^2)*d*x - (b^2*c^2 - 2*a*c^3)*d)*e^5 + ((b^4 - 4*a*b^2*c + 2*
a^2*c^2)*d^2*x - 2*(b^3*c - 3*a*b*c^2)*d^2)*e^4)*sqrt(b^2 - 4*a*c)*log((2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c + sq
rt(b^2 - 4*a*c)*(2*a*x + b))/(a*x^2 + b*x + c)) + 2*(2*(a*b^3*c - 4*a^2*b*c^2)*d*x^2 - (a*b^2*c^2 - 4*a^2*c^3)
*d*x)*e^5 - 2*((a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*x^2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^2*x)*e^4 + 2*(2*(a^2*b^
3 - 4*a^3*b*c)*d^3*x^2 - (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*x)*e^3 - 2*((a^3*b^2 - 4*a^4*c)*d^4*x^2 - 2*(a^
2*b^3 - 4*a^3*b*c)*d^4*x - (a^2*b^2*c - 4*a^3*c^2)*d^4)*e^2 - 2*((a^3*b^2 - 4*a^4*c)*d^5*x + (a^2*b^3 - 4*a^3*
b*c)*d^5)*e + ((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^3*e^3 + (b^3*c^2 - 4*a*b*c^3)*x*e^6 - (2*(b^4*c - 5*a*b^2*c^2
+ 4*a^2*c^3)*d*x - (b^3*c^2 - 4*a*b*c^3)*d)*e^5 + ((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^2*x - 2*(b^4*c - 5*a*b^2
*c^2 + 4*a^2*c^3)*d^2)*e^4)*log(a*x^2 + b*x + c) + 2*(2*(a^3*b^2 - 4*a^4*c)*d^6 + 4*(a^2*b^2*c - 4*a^3*c^2)*d^
3*x*e^3 - (3*(a^2*b^3 - 4*a^3*b*c)*d^4*x - 4*(a^2*b^2*c - 4*a^3*c^2)*d^4)*e^2 + (2*(a^3*b^2 - 4*a^4*c)*d^5*x -
3*(a^2*b^3 - 4*a^3*b*c)*d^5)*e)*log(x*e + d))/((a^4*b^2 - 4*a^5*c)*d^5*e^3 + (a^2*b^2*c^2 - 4*a^3*c^3)*x*e^8
- (2*(a^2*b^3*c - 4*a^3*b*c^2)*d*x - (a^2*b^2*c^2 - 4*a^3*c^3)*d)*e^7 + ((a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d
^2*x - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^2)*e^6 - (2*(a^3*b^3 - 4*a^4*b*c)*d^3*x - (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*
c^2)*d^3)*e^5 + ((a^4*b^2 - 4*a^5*c)*d^4*x - 2*(a^3*b^3 - 4*a^4*b*c)*d^4)*e^4), -1/2*(2*(a^3*b^2 - 4*a^4*c)*d^
6 - 2*(a*b^2*c^2 - 4*a^2*c^3)*x^2*e^6 + 2*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3*e^3 + (b^2*c^2 - 2*a*c^3)*x*e^6 -
(2*(b^3*c - 3*a*b*c^2)*d*x - (b^2*c^2 - 2*a*c^3)*d)*e^5 + ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*x - 2*(b^3*c - 3
*a*b*c^2)*d^2)*e^4)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) + 2*(2*(a*b^3*c -
4*a^2*b*c^2)*d*x^2 - (a*b^2*c^2 - 4*a^2*c^3)*d*x)*e^5 - 2*((a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*x^2 - 2*(a*b
^3*c - 4*a^2*b*c^2)*d^2*x)*e^4 + 2*(2*(a^2*b^3 - 4*a^3*b*c)*d^3*x^2 - (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*x)
*e^3 - 2*((a^3*b^2 - 4*a^4*c)*d^4*x^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^4*x - (a^2*b^2*c - 4*a^3*c^2)*d^4)*e^2 - 2*(
(a^3*b^2 - 4*a^4*c)*d^5*x + (a^2*b^3 - 4*a^3*b*c)*d^5)*e + ((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^3*e^3 + (b^3*c^2
- 4*a*b*c^3)*x*e^6 - (2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d*x - (b^3*c^2 - 4*a*b*c^3)*d)*e^5 + ((b^5 - 6*a*b^
3*c + 8*a^2*b*c^2)*d^2*x - 2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2)*e^4)*log(a*x^2 + b*x + c) + 2*(2*(a^3*b^2
- 4*a^4*c)*d^6 + 4*(a^2*b^2*c - 4*a^3*c^2)*d^3*x*e^3 - (3*(a^2*b^3 - 4*a^3*b*c)*d^4*x - 4*(a^2*b^2*c - 4*a^3*c
^2)*d^4)*e^2 + (2*(a^3*b^2 - 4*a^4*c)*d^5*x - 3*(a^2*b^3 - 4*a^3*b*c)*d^5)*e)*log(x*e + d))/((a^4*b^2 - 4*a^5*
c)*d^5*e^3 + (a^2*b^2*c^2 - 4*a^3*c^3)*x*e^8 - (2*(a^2*b^3*c - 4*a^3*b*c^2)*d*x - (a^2*b^2*c^2 - 4*a^3*c^3)*d)
*e^7 + ((a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^2*x - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^2)*e^6 - (2*(a^3*b^3 - 4*a^4
*b*c)*d^3*x - (a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^3)*e^5 + ((a^4*b^2 - 4*a^5*c)*d^4*x - 2*(a^3*b^3 - 4*a^4*b
*c)*d^4)*e^4)]

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Sympy [F(-1)] Timed out
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(x**2/(a+c/x**2+b/x)/(e*x+d)**2,x)

[Out]

Timed out

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Giac [A]
time = 3.69, size = 476, normalized size = 1.74 \begin {gather*} -\frac {d^{4} e^{3}}{{\left (a d^{2} e^{6} - b d e^{7} + c e^{8}\right )} {\left (x e + d\right )}} - \frac {{\left (b^{4} d^{2} e^{2} - 4 \, a b^{2} c d^{2} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{2} - 2 \, b^{3} c d e^{3} + 6 \, a b c^{2} d e^{3} + b^{2} c^{2} e^{4} - 2 \, a c^{3} e^{4}\right )} \arctan \left (-\frac {{\left (2 \, a d - \frac {2 \, a d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{4} d^{4} - 2 \, a^{3} b d^{3} e + a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} c d^{2} e^{2} - 2 \, a^{2} b c d e^{3} + a^{2} c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (x e + d\right )} e^{\left (-3\right )}}{a} - \frac {{\left (b^{3} d^{2} - 2 \, a b c d^{2} - 2 \, b^{2} c d e + 2 \, a c^{2} d e + b c^{2} e^{2}\right )} \log \left (-a + \frac {2 \, a d}{x e + d} - \frac {a d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (a^{4} d^{4} - 2 \, a^{3} b d^{3} e + a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} c d^{2} e^{2} - 2 \, a^{2} b c d e^{3} + a^{2} c^{2} e^{4}\right )}} + \frac {{\left (2 \, a d + b e\right )} e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+c/x^2+b/x)/(e*x+d)^2,x, algorithm="giac")

[Out]

-d^4*e^3/((a*d^2*e^6 - b*d*e^7 + c*e^8)*(x*e + d)) - (b^4*d^2*e^2 - 4*a*b^2*c*d^2*e^2 + 2*a^2*c^2*d^2*e^2 - 2*
b^3*c*d*e^3 + 6*a*b*c^2*d*e^3 + b^2*c^2*e^4 - 2*a*c^3*e^4)*arctan(-(2*a*d - 2*a*d^2/(x*e + d) - b*e + 2*b*d*e/
(x*e + d) - 2*c*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/((a^4*d^4 - 2*a^3*b*d^3*e + a^2*b^2*d^2*e^2 +
2*a^3*c*d^2*e^2 - 2*a^2*b*c*d*e^3 + a^2*c^2*e^4)*sqrt(-b^2 + 4*a*c)) + (x*e + d)*e^(-3)/a - 1/2*(b^3*d^2 - 2*
a*b*c*d^2 - 2*b^2*c*d*e + 2*a*c^2*d*e + b*c^2*e^2)*log(-a + 2*a*d/(x*e + d) - a*d^2/(x*e + d)^2 - b*e/(x*e + d
) + b*d*e/(x*e + d)^2 - c*e^2/(x*e + d)^2)/(a^4*d^4 - 2*a^3*b*d^3*e + a^2*b^2*d^2*e^2 + 2*a^3*c*d^2*e^2 - 2*a^
2*b*c*d*e^3 + a^2*c^2*e^4) + (2*a*d + b*e)*e^(-3)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2)/a^2

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Mupad [B]
time = 6.00, size = 2495, normalized size = 9.11 \begin {gather*} \frac {x}{a\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (2\,a\,d^5-3\,b\,d^4\,e+4\,c\,d^3\,e^2\right )}{a^2\,d^4\,e^3-2\,a\,b\,d^3\,e^4+2\,a\,c\,d^2\,e^5+b^2\,d^2\,e^5-2\,b\,c\,d\,e^6+c^2\,e^7}+\frac {\ln \left (8\,a^4\,c\,d^7+b\,c^4\,e^7+c^4\,e^7\,\sqrt {b^2-4\,a\,c}-2\,a^3\,b^2\,d^7+b^5\,d^4\,e^3+3\,a^2\,b^3\,d^6\,e-4\,b^2\,c^3\,d\,e^6-4\,b^4\,c\,d^3\,e^4+b^4\,d^4\,e^3\,\sqrt {b^2-4\,a\,c}-24\,a^2\,c^3\,d^3\,e^4+8\,a^3\,c^2\,d^5\,e^2+6\,b^3\,c^2\,d^2\,e^5+8\,a\,c^4\,d\,e^6+2\,a\,c^4\,e^7\,x-2\,a^3\,b\,d^7\,\sqrt {b^2-4\,a\,c}-4\,a^4\,d^7\,x\,\sqrt {b^2-4\,a\,c}-12\,a^3\,b\,c\,d^6\,e+17\,a^2\,c^2\,d^4\,e^3\,\sqrt {b^2-4\,a\,c}+6\,b^2\,c^2\,d^2\,e^5\,\sqrt {b^2-4\,a\,c}+16\,a^4\,c\,d^6\,e\,x+8\,a^3\,c\,d^6\,e\,\sqrt {b^2-4\,a\,c}-4\,b\,c^3\,d\,e^6\,\sqrt {b^2-4\,a\,c}-18\,a\,b\,c^3\,d^2\,e^5-8\,a\,b^3\,c\,d^4\,e^3-2\,a\,b^4\,d^4\,e^3\,x-4\,a^3\,b^2\,d^6\,e\,x+3\,a^2\,b^2\,d^6\,e\,\sqrt {b^2-4\,a\,c}-6\,a\,c^3\,d^2\,e^5\,\sqrt {b^2-4\,a\,c}-4\,b^3\,c\,d^3\,e^4\,\sqrt {b^2-4\,a\,c}+20\,a\,b^2\,c^2\,d^3\,e^4+17\,a^2\,b\,c^2\,d^4\,e^3-2\,a^2\,b^2\,c\,d^5\,e^2+8\,a^2\,b^3\,d^5\,e^2\,x-12\,a^2\,c^3\,d^2\,e^5\,x+34\,a^3\,c^2\,d^4\,e^3\,x+4\,a\,b\,c^2\,d^3\,e^4\,\sqrt {b^2-4\,a\,c}-18\,a^2\,b\,c\,d^5\,e^2\,\sqrt {b^2-4\,a\,c}+4\,a\,b^3\,d^4\,e^3\,x\,\sqrt {b^2-4\,a\,c}-4\,a^3\,c\,d^5\,e^2\,x\,\sqrt {b^2-4\,a\,c}+6\,a\,b^2\,c^2\,d^2\,e^5\,x-4\,a^2\,b\,c^2\,d^3\,e^4\,x-8\,a^2\,b^2\,d^5\,e^2\,x\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^3\,d\,e^6\,x+12\,a^2\,c^2\,d^3\,e^4\,x\,\sqrt {b^2-4\,a\,c}+10\,a^3\,b\,d^6\,e\,x\,\sqrt {b^2-4\,a\,c}-4\,a\,c^3\,d\,e^6\,x\,\sqrt {b^2-4\,a\,c}-32\,a^3\,b\,c\,d^5\,e^2\,x+6\,a\,b\,c^2\,d^2\,e^5\,x\,\sqrt {b^2-4\,a\,c}-8\,a\,b^2\,c\,d^3\,e^4\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^5\,d^2+b^4\,d^2\,\sqrt {b^2-4\,a\,c}+b^3\,c^2\,e^2+8\,a^2\,b\,c^2\,d^2+2\,a^2\,c^2\,d^2\,\sqrt {b^2-4\,a\,c}+b^2\,c^2\,e^2\,\sqrt {b^2-4\,a\,c}-2\,b^4\,c\,d\,e-6\,a\,b^3\,c\,d^2-4\,a\,b\,c^3\,e^2-8\,a^2\,c^3\,d\,e-2\,a\,c^3\,e^2\,\sqrt {b^2-4\,a\,c}+10\,a\,b^2\,c^2\,d\,e-4\,a\,b^2\,c\,d^2\,\sqrt {b^2-4\,a\,c}-2\,b^3\,c\,d\,e\,\sqrt {b^2-4\,a\,c}+6\,a\,b\,c^2\,d\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^5\,c\,d^4-a^4\,b^2\,d^4-8\,a^4\,b\,c\,d^3\,e+8\,a^4\,c^2\,d^2\,e^2+2\,a^3\,b^3\,d^3\,e+2\,a^3\,b^2\,c\,d^2\,e^2-8\,a^3\,b\,c^2\,d\,e^3+4\,a^3\,c^3\,e^4-a^2\,b^4\,d^2\,e^2+2\,a^2\,b^3\,c\,d\,e^3-a^2\,b^2\,c^2\,e^4\right )}-\frac {\ln \left (c^4\,e^7\,\sqrt {b^2-4\,a\,c}-b\,c^4\,e^7-8\,a^4\,c\,d^7+2\,a^3\,b^2\,d^7-b^5\,d^4\,e^3-3\,a^2\,b^3\,d^6\,e+4\,b^2\,c^3\,d\,e^6+4\,b^4\,c\,d^3\,e^4+b^4\,d^4\,e^3\,\sqrt {b^2-4\,a\,c}+24\,a^2\,c^3\,d^3\,e^4-8\,a^3\,c^2\,d^5\,e^2-6\,b^3\,c^2\,d^2\,e^5-8\,a\,c^4\,d\,e^6-2\,a\,c^4\,e^7\,x-2\,a^3\,b\,d^7\,\sqrt {b^2-4\,a\,c}-4\,a^4\,d^7\,x\,\sqrt {b^2-4\,a\,c}+12\,a^3\,b\,c\,d^6\,e+17\,a^2\,c^2\,d^4\,e^3\,\sqrt {b^2-4\,a\,c}+6\,b^2\,c^2\,d^2\,e^5\,\sqrt {b^2-4\,a\,c}-16\,a^4\,c\,d^6\,e\,x+8\,a^3\,c\,d^6\,e\,\sqrt {b^2-4\,a\,c}-4\,b\,c^3\,d\,e^6\,\sqrt {b^2-4\,a\,c}+18\,a\,b\,c^3\,d^2\,e^5+8\,a\,b^3\,c\,d^4\,e^3+2\,a\,b^4\,d^4\,e^3\,x+4\,a^3\,b^2\,d^6\,e\,x+3\,a^2\,b^2\,d^6\,e\,\sqrt {b^2-4\,a\,c}-6\,a\,c^3\,d^2\,e^5\,\sqrt {b^2-4\,a\,c}-4\,b^3\,c\,d^3\,e^4\,\sqrt {b^2-4\,a\,c}-20\,a\,b^2\,c^2\,d^3\,e^4-17\,a^2\,b\,c^2\,d^4\,e^3+2\,a^2\,b^2\,c\,d^5\,e^2-8\,a^2\,b^3\,d^5\,e^2\,x+12\,a^2\,c^3\,d^2\,e^5\,x-34\,a^3\,c^2\,d^4\,e^3\,x+4\,a\,b\,c^2\,d^3\,e^4\,\sqrt {b^2-4\,a\,c}-18\,a^2\,b\,c\,d^5\,e^2\,\sqrt {b^2-4\,a\,c}+4\,a\,b^3\,d^4\,e^3\,x\,\sqrt {b^2-4\,a\,c}-4\,a^3\,c\,d^5\,e^2\,x\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c^2\,d^2\,e^5\,x+4\,a^2\,b\,c^2\,d^3\,e^4\,x-8\,a^2\,b^2\,d^5\,e^2\,x\,\sqrt {b^2-4\,a\,c}+4\,a\,b\,c^3\,d\,e^6\,x+12\,a^2\,c^2\,d^3\,e^4\,x\,\sqrt {b^2-4\,a\,c}+10\,a^3\,b\,d^6\,e\,x\,\sqrt {b^2-4\,a\,c}-4\,a\,c^3\,d\,e^6\,x\,\sqrt {b^2-4\,a\,c}+32\,a^3\,b\,c\,d^5\,e^2\,x+6\,a\,b\,c^2\,d^2\,e^5\,x\,\sqrt {b^2-4\,a\,c}-8\,a\,b^2\,c\,d^3\,e^4\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,d^2\,\sqrt {b^2-4\,a\,c}-b^5\,d^2-b^3\,c^2\,e^2-8\,a^2\,b\,c^2\,d^2+2\,a^2\,c^2\,d^2\,\sqrt {b^2-4\,a\,c}+b^2\,c^2\,e^2\,\sqrt {b^2-4\,a\,c}+2\,b^4\,c\,d\,e+6\,a\,b^3\,c\,d^2+4\,a\,b\,c^3\,e^2+8\,a^2\,c^3\,d\,e-2\,a\,c^3\,e^2\,\sqrt {b^2-4\,a\,c}-10\,a\,b^2\,c^2\,d\,e-4\,a\,b^2\,c\,d^2\,\sqrt {b^2-4\,a\,c}-2\,b^3\,c\,d\,e\,\sqrt {b^2-4\,a\,c}+6\,a\,b\,c^2\,d\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^5\,c\,d^4-a^4\,b^2\,d^4-8\,a^4\,b\,c\,d^3\,e+8\,a^4\,c^2\,d^2\,e^2+2\,a^3\,b^3\,d^3\,e+2\,a^3\,b^2\,c\,d^2\,e^2-8\,a^3\,b\,c^2\,d\,e^3+4\,a^3\,c^3\,e^4-a^2\,b^4\,d^2\,e^2+2\,a^2\,b^3\,c\,d\,e^3-a^2\,b^2\,c^2\,e^4\right )}-\frac {a\,d^4}{e\,\left (a\,x\,e^3+a\,d\,e^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((d + e*x)^2*(a + b/x + c/x^2)),x)

[Out]

x/(a*e^2) - (log(d + e*x)*(2*a*d^5 + 4*c*d^3*e^2 - 3*b*d^4*e))/(c^2*e^7 + a^2*d^4*e^3 + b^2*d^2*e^5 - 2*b*c*d*
e^6 - 2*a*b*d^3*e^4 + 2*a*c*d^2*e^5) + (log(8*a^4*c*d^7 + b*c^4*e^7 + c^4*e^7*(b^2 - 4*a*c)^(1/2) - 2*a^3*b^2*
d^7 + b^5*d^4*e^3 + 3*a^2*b^3*d^6*e - 4*b^2*c^3*d*e^6 - 4*b^4*c*d^3*e^4 + b^4*d^4*e^3*(b^2 - 4*a*c)^(1/2) - 24
*a^2*c^3*d^3*e^4 + 8*a^3*c^2*d^5*e^2 + 6*b^3*c^2*d^2*e^5 + 8*a*c^4*d*e^6 + 2*a*c^4*e^7*x - 2*a^3*b*d^7*(b^2 -
4*a*c)^(1/2) - 4*a^4*d^7*x*(b^2 - 4*a*c)^(1/2) - 12*a^3*b*c*d^6*e + 17*a^2*c^2*d^4*e^3*(b^2 - 4*a*c)^(1/2) + 6
*b^2*c^2*d^2*e^5*(b^2 - 4*a*c)^(1/2) + 16*a^4*c*d^6*e*x + 8*a^3*c*d^6*e*(b^2 - 4*a*c)^(1/2) - 4*b*c^3*d*e^6*(b
^2 - 4*a*c)^(1/2) - 18*a*b*c^3*d^2*e^5 - 8*a*b^3*c*d^4*e^3 - 2*a*b^4*d^4*e^3*x - 4*a^3*b^2*d^6*e*x + 3*a^2*b^2
*d^6*e*(b^2 - 4*a*c)^(1/2) - 6*a*c^3*d^2*e^5*(b^2 - 4*a*c)^(1/2) - 4*b^3*c*d^3*e^4*(b^2 - 4*a*c)^(1/2) + 20*a*
b^2*c^2*d^3*e^4 + 17*a^2*b*c^2*d^4*e^3 - 2*a^2*b^2*c*d^5*e^2 + 8*a^2*b^3*d^5*e^2*x - 12*a^2*c^3*d^2*e^5*x + 34
*a^3*c^2*d^4*e^3*x + 4*a*b*c^2*d^3*e^4*(b^2 - 4*a*c)^(1/2) - 18*a^2*b*c*d^5*e^2*(b^2 - 4*a*c)^(1/2) + 4*a*b^3*
d^4*e^3*x*(b^2 - 4*a*c)^(1/2) - 4*a^3*c*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + 6*a*b^2*c^2*d^2*e^5*x - 4*a^2*b*c^2*d^
3*e^4*x - 8*a^2*b^2*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) - 4*a*b*c^3*d*e^6*x + 12*a^2*c^2*d^3*e^4*x*(b^2 - 4*a*c)^(1/
2) + 10*a^3*b*d^6*e*x*(b^2 - 4*a*c)^(1/2) - 4*a*c^3*d*e^6*x*(b^2 - 4*a*c)^(1/2) - 32*a^3*b*c*d^5*e^2*x + 6*a*b
*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2) - 8*a*b^2*c*d^3*e^4*x*(b^2 - 4*a*c)^(1/2))*(b^5*d^2 + b^4*d^2*(b^2 - 4*a*c)
^(1/2) + b^3*c^2*e^2 + 8*a^2*b*c^2*d^2 + 2*a^2*c^2*d^2*(b^2 - 4*a*c)^(1/2) + b^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) -
2*b^4*c*d*e - 6*a*b^3*c*d^2 - 4*a*b*c^3*e^2 - 8*a^2*c^3*d*e - 2*a*c^3*e^2*(b^2 - 4*a*c)^(1/2) + 10*a*b^2*c^2*
d*e - 4*a*b^2*c*d^2*(b^2 - 4*a*c)^(1/2) - 2*b^3*c*d*e*(b^2 - 4*a*c)^(1/2) + 6*a*b*c^2*d*e*(b^2 - 4*a*c)^(1/2))
)/(2*(4*a^5*c*d^4 - a^4*b^2*d^4 + 4*a^3*c^3*e^4 + 2*a^3*b^3*d^3*e - a^2*b^2*c^2*e^4 - a^2*b^4*d^2*e^2 + 8*a^4*
c^2*d^2*e^2 - 8*a^4*b*c*d^3*e + 2*a^2*b^3*c*d*e^3 - 8*a^3*b*c^2*d*e^3 + 2*a^3*b^2*c*d^2*e^2)) - (log(c^4*e^7*(
b^2 - 4*a*c)^(1/2) - b*c^4*e^7 - 8*a^4*c*d^7 + 2*a^3*b^2*d^7 - b^5*d^4*e^3 - 3*a^2*b^3*d^6*e + 4*b^2*c^3*d*e^6
+ 4*b^4*c*d^3*e^4 + b^4*d^4*e^3*(b^2 - 4*a*c)^(1/2) + 24*a^2*c^3*d^3*e^4 - 8*a^3*c^2*d^5*e^2 - 6*b^3*c^2*d^2*
e^5 - 8*a*c^4*d*e^6 - 2*a*c^4*e^7*x - 2*a^3*b*d^7*(b^2 - 4*a*c)^(1/2) - 4*a^4*d^7*x*(b^2 - 4*a*c)^(1/2) + 12*a
^3*b*c*d^6*e + 17*a^2*c^2*d^4*e^3*(b^2 - 4*a*c)^(1/2) + 6*b^2*c^2*d^2*e^5*(b^2 - 4*a*c)^(1/2) - 16*a^4*c*d^6*e
*x + 8*a^3*c*d^6*e*(b^2 - 4*a*c)^(1/2) - 4*b*c^3*d*e^6*(b^2 - 4*a*c)^(1/2) + 18*a*b*c^3*d^2*e^5 + 8*a*b^3*c*d^
4*e^3 + 2*a*b^4*d^4*e^3*x + 4*a^3*b^2*d^6*e*x + 3*a^2*b^2*d^6*e*(b^2 - 4*a*c)^(1/2) - 6*a*c^3*d^2*e^5*(b^2 - 4
*a*c)^(1/2) - 4*b^3*c*d^3*e^4*(b^2 - 4*a*c)^(1/2) - 20*a*b^2*c^2*d^3*e^4 - 17*a^2*b*c^2*d^4*e^3 + 2*a^2*b^2*c*
d^5*e^2 - 8*a^2*b^3*d^5*e^2*x + 12*a^2*c^3*d^2*e^5*x - 34*a^3*c^2*d^4*e^3*x + 4*a*b*c^2*d^3*e^4*(b^2 - 4*a*c)^
(1/2) - 18*a^2*b*c*d^5*e^2*(b^2 - 4*a*c)^(1/2) + 4*a*b^3*d^4*e^3*x*(b^2 - 4*a*c)^(1/2) - 4*a^3*c*d^5*e^2*x*(b^
2 - 4*a*c)^(1/2) - 6*a*b^2*c^2*d^2*e^5*x + 4*a^2*b*c^2*d^3*e^4*x - 8*a^2*b^2*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + 4
*a*b*c^3*d*e^6*x + 12*a^2*c^2*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) + 10*a^3*b*d^6*e*x*(b^2 - 4*a*c)^(1/2) - 4*a*c^3*d
*e^6*x*(b^2 - 4*a*c)^(1/2) + 32*a^3*b*c*d^5*e^2*x + 6*a*b*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2) - 8*a*b^2*c*d^3*e^
4*x*(b^2 - 4*a*c)^(1/2))*(b^4*d^2*(b^2 - 4*a*c)^(1/2) - b^5*d^2 - b^3*c^2*e^2 - 8*a^2*b*c^2*d^2 + 2*a^2*c^2*d^
2*(b^2 - 4*a*c)^(1/2) + b^2*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*b^4*c*d*e + 6*a*b^3*c*d^2 + 4*a*b*c^3*e^2 + 8*a^2*
c^3*d*e - 2*a*c^3*e^2*(b^2 - 4*a*c)^(1/2) - 10*a*b^2*c^2*d*e - 4*a*b^2*c*d^2*(b^2 - 4*a*c)^(1/2) - 2*b^3*c*d*e
*(b^2 - 4*a*c)^(1/2) + 6*a*b*c^2*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a^5*c*d^4 - a^4*b^2*d^4 + 4*a^3*c^3*e^4 + 2*a
^3*b^3*d^3*e - a^2*b^2*c^2*e^4 - a^2*b^4*d^2*e^2 + 8*a^4*c^2*d^2*e^2 - 8*a^4*b*c*d^3*e + 2*a^2*b^3*c*d*e^3 - 8
*a^3*b*c^2*d*e^3 + 2*a^3*b^2*c*d^2*e^2)) - (a*d^4)/(e*(a*d*e^2 + a*e^3*x)*(a*d^2 + c*e^2 - b*d*e))

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