3.1.70 \(\int \frac {x^3}{(a+\frac {c}{x^2}+\frac {b}{x}) (d+e x)^2} \, dx\) [70]

Optimal. Leaf size=343 \[ -\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 a d^2-c e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2} \]

[Out]

-(2*a*d+b*e)*x/a^2/e^3+1/2*x^2/a/e^2+d^5/e^4/(a*d^2-e*(b*d-c*e))/(e*x+d)+d^4*(3*a*d^2-e*(4*b*d-5*c*e))*ln(e*x+
d)/e^4/(a*d^2-e*(b*d-c*e))^2+1/2*(b^4*d^2-2*b^3*c*d*e+4*a*b*c^2*d*e+a*c^2*(a*d^2-c*e^2)-b^2*c*(3*a*d^2-c*e^2))
*ln(a*x^2+b*x+c)/a^3/(a*d^2-e*(b*d-c*e))^2+(b^5*d^2-2*b^4*c*d*e+8*a*b^2*c^2*d*e-4*a^2*c^3*d*e+a*b*c^2*(5*a*d^2
-3*c*e^2)-b^3*c*(5*a*d^2-c*e^2))*arctanh((2*a*x+b)/(-4*a*c+b^2)^(1/2))/a^3/(a*d^2-e*(b*d-c*e))^2/(-4*a*c+b^2)^
(1/2)

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Rubi [A]
time = 0.60, antiderivative size = 343, 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 {\left (-b^2 c \left (3 a d^2-c e^2\right )+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {x (2 a d+b e)}{a^2 e^3}+\frac {\left (-4 a^2 c^3 d e-b^3 c \left (5 a d^2-c e^2\right )+8 a b^2 c^2 d e+a b c^2 \left (5 a d^2-3 c e^2\right )+b^5 d^2-2 b^4 c d e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d^5}{e^4 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {d^4 \log (d+e x) \left (3 a d^2-e (4 b d-5 c e)\right )}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {x^2}{2 a e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((a + c/x^2 + b/x)*(d + e*x)^2),x]

[Out]

-(((2*a*d + b*e)*x)/(a^2*e^3)) + x^2/(2*a*e^2) + d^5/(e^4*(a*d^2 - e*(b*d - c*e))*(d + e*x)) + ((b^5*d^2 - 2*b
^4*c*d*e + 8*a*b^2*c^2*d*e - 4*a^2*c^3*d*e + a*b*c^2*(5*a*d^2 - 3*c*e^2) - b^3*c*(5*a*d^2 - c*e^2))*ArcTanh[(b
+ 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a^3*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) + (d^4*(3*a*d^2 - e*(4*b*d - 5
*c*e))*Log[d + e*x])/(e^4*(a*d^2 - e*(b*d - c*e))^2) + ((b^4*d^2 - 2*b^3*c*d*e + 4*a*b*c^2*d*e + a*c^2*(a*d^2
- c*e^2) - b^2*c*(3*a*d^2 - c*e^2))*Log[c + b*x + a*x^2])/(2*a^3*(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^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx &=\int \frac {x^5}{(d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {-2 a d-b e}{a^2 e^3}+\frac {x}{a e^2}+\frac {d^5}{e^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right )}{e^3 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac {c (b d-c e) \left (b^2 d-2 a c d-b c e\right )+\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) x}{a^2 \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\int \frac {c (b d-c e) \left (b^2 d-2 a c d-b c e\right )+\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) x}{c+b x+a x^2} \, dx}{a^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 a d^2-c e^2\right )\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 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^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 a d^2-c e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 338, normalized size = 0.99 \begin {gather*} -\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2+e (-b d+c e)\right ) (d+e x)}-\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )+b^3 c \left (-5 a d^2+c e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a^3 \sqrt {-b^2+4 a c} \left (a d^2+e (-b d+c e)\right )^2}+\frac {\left (3 a d^6+d^4 e (-4 b d+5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2+e (-b d+c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )+b^2 c \left (-3 a d^2+c e^2\right )\right ) \log (c+x (b+a x))}{2 a^3 \left (a d^2+e (-b d+c e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/((a + c/x^2 + b/x)*(d + e*x)^2),x]

[Out]

-(((2*a*d + b*e)*x)/(a^2*e^3)) + x^2/(2*a*e^2) + d^5/(e^4*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) - ((b^5*d^2 -
2*b^4*c*d*e + 8*a*b^2*c^2*d*e - 4*a^2*c^3*d*e + a*b*c^2*(5*a*d^2 - 3*c*e^2) + b^3*c*(-5*a*d^2 + c*e^2))*ArcTan
[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^3*Sqrt[-b^2 + 4*a*c]*(a*d^2 + e*(-(b*d) + c*e))^2) + ((3*a*d^6 + d^4*e*(-
4*b*d + 5*c*e))*Log[d + e*x])/(e^4*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b^4*d^2 - 2*b^3*c*d*e + 4*a*b*c^2*d*e + a
*c^2*(a*d^2 - c*e^2) + b^2*c*(-3*a*d^2 + c*e^2))*Log[c + x*(b + a*x)])/(2*a^3*(a*d^2 + e*(-(b*d) + c*e))^2)

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Maple [A]
time = 0.28, size = 360, normalized size = 1.05

method	result	size
default	\(\frac {d^{4} \left (3 a \,d^{2}-4 d e b +5 c \,e^{2}\right ) \ln \left (e x +d \right )}{e^{4} \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}+\frac {d^{5}}{e^{4} \left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right )}-\frac {-\frac {1}{2} a e \,x^{2}+2 a d x +e b x}{a^{2} e^{3}}+\frac {\frac {\left (a^{2} c^{2} d^{2}-3 a \,b^{2} c \,d^{2}+4 e d \,c^{2} b a -a \,c^{3} e^{2}+b^{4} d^{2}-2 b^{3} c d e +b^{2} c^{2} e^{2}\right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-2 a b \,c^{2} d^{2}+2 a \,c^{3} d e +b^{3} c \,d^{2}-2 b^{2} c^{2} d e +b \,c^{3} e^{2}-\frac {\left (a^{2} c^{2} d^{2}-3 a \,b^{2} c \,d^{2}+4 e d \,c^{2} b a -a \,c^{3} e^{2}+b^{4} d^{2}-2 b^{3} c d e +b^{2} c^{2} e^{2}\right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{2}}\)	\(360\)
risch	\(\text {Expression too large to display}\)	\(1684\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+c/x^2+b/x)/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/e^4*d^4*(3*a*d^2-4*b*d*e+5*c*e^2)/(a*d^2-b*d*e+c*e^2)^2*ln(e*x+d)+1/e^4*d^5/(a*d^2-b*d*e+c*e^2)/(e*x+d)-1/a^
2/e^3*(-1/2*a*e*x^2+2*a*d*x+e*b*x)+1/(a*d^2-b*d*e+c*e^2)^2/a^2*(1/2*(a^2*c^2*d^2-3*a*b^2*c*d^2+4*a*b*c^2*d*e-a
*c^3*e^2+b^4*d^2-2*b^3*c*d*e+b^2*c^2*e^2)/a*ln(a*x^2+b*x+c)+2*(-2*a*b*c^2*d^2+2*a*c^3*d*e+b^3*c*d^2-2*b^2*c^2*
d*e+b*c^3*e^2-1/2*(a^2*c^2*d^2-3*a*b^2*c*d^2+4*a*b*c^2*d*e-a*c^3*e^2+b^4*d^2-2*b^3*c*d*e+b^2*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^3/(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. 1334 vs. \(2 (345) = 690\).
time = 62.84, size = 2689, normalized size = 7.84 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+c/x^2+b/x)/(e*x+d)^2,x, algorithm="fricas")

[Out]

[1/2*(2*(a^4*b^2 - 4*a^5*c)*d^7 - ((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d^3*e^4 + (b^3*c^2 - 3*a*b*c^3)*x*e^7 - (2*
(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*x - (b^3*c^2 - 3*a*b*c^3)*d)*e^6 + ((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d^2*x
- 2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d^2)*e^5)*sqrt(b^2 - 4*a*c)*log((2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c - sqr
t(b^2 - 4*a*c)*(2*a*x + b))/(a*x^2 + b*x + c)) + ((a^2*b^2*c^2 - 4*a^3*c^3)*x^3 - 2*(a*b^3*c^2 - 4*a^2*b*c^3)*
x^2)*e^7 - (2*(a^2*b^3*c - 4*a^3*b*c^2)*d*x^3 - (4*a*b^4*c - 19*a^2*b^2*c^2 + 12*a^3*c^3)*d*x^2 + 2*(a*b^3*c^2
- 4*a^2*b*c^3)*d*x)*e^6 + ((a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^2*x^3 - 2*(a*b^5 - 5*a^2*b^3*c + 4*a^3*b*c^2
)*d^2*x^2 + 4*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*x)*e^5 - (2*(a^3*b^3 - 4*a^4*b*c)*d^3*x^3 - (a^2*b^4 -
10*a^3*b^2*c + 24*a^4*c^2)*d^3*x^2 + 2*(a*b^5 - 6*a^2*b^3*c + 8*a^3*b*c^2)*d^3*x)*e^4 + ((a^4*b^2 - 4*a^5*c)*
d^4*x^3 + 4*(a^3*b^3 - 4*a^4*b*c)*d^4*x^2 - 8*(a^3*b^2*c - 4*a^4*c^2)*d^4*x)*e^3 - (3*(a^4*b^2 - 4*a^5*c)*d^5*
x^2 - 6*(a^3*b^3 - 4*a^4*b*c)*d^5*x - 2*(a^3*b^2*c - 4*a^4*c^2)*d^5)*e^2 - 2*(2*(a^4*b^2 - 4*a^5*c)*d^6*x + (a
^3*b^3 - 4*a^4*b*c)*d^6)*e + ((b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^4 + (b^4*c^2 - 5*a*b^2*c^3
+ 4*a^2*c^4)*x*e^7 - (2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*x - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d)*e^6 +
((b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*x - 2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d^2)*e^5)*log(a
*x^2 + b*x + c) + 2*(3*(a^4*b^2 - 4*a^5*c)*d^7 + 5*(a^3*b^2*c - 4*a^4*c^2)*d^4*x*e^3 - (4*(a^3*b^3 - 4*a^4*b*c
)*d^5*x - 5*(a^3*b^2*c - 4*a^4*c^2)*d^5)*e^2 + (3*(a^4*b^2 - 4*a^5*c)*d^6*x - 4*(a^3*b^3 - 4*a^4*b*c)*d^6)*e)*
log(x*e + d))/((a^5*b^2 - 4*a^6*c)*d^5*e^4 + (a^3*b^2*c^2 - 4*a^4*c^3)*x*e^9 - (2*(a^3*b^3*c - 4*a^4*b*c^2)*d*
x - (a^3*b^2*c^2 - 4*a^4*c^3)*d)*e^8 + ((a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^2*x - 2*(a^3*b^3*c - 4*a^4*b*c^2
)*d^2)*e^7 - (2*(a^4*b^3 - 4*a^5*b*c)*d^3*x - (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^3)*e^6 + ((a^5*b^2 - 4*a^6
*c)*d^4*x - 2*(a^4*b^3 - 4*a^5*b*c)*d^4)*e^5), 1/2*(2*(a^4*b^2 - 4*a^5*c)*d^7 + 2*((b^5 - 5*a*b^3*c + 5*a^2*b*
c^2)*d^3*e^4 + (b^3*c^2 - 3*a*b*c^3)*x*e^7 - (2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*x - (b^3*c^2 - 3*a*b*c^3)*
d)*e^6 + ((b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*d^2*x - 2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d^2)*e^5)*sqrt(-b^2 + 4*
a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) + ((a^2*b^2*c^2 - 4*a^3*c^3)*x^3 - 2*(a*b^3*c^2 - 4
*a^2*b*c^3)*x^2)*e^7 - (2*(a^2*b^3*c - 4*a^3*b*c^2)*d*x^3 - (4*a*b^4*c - 19*a^2*b^2*c^2 + 12*a^3*c^3)*d*x^2 +
2*(a*b^3*c^2 - 4*a^2*b*c^3)*d*x)*e^6 + ((a^2*b^4 - 2*a^3*b^2*c - 8*a^4*c^2)*d^2*x^3 - 2*(a*b^5 - 5*a^2*b^3*c +
4*a^3*b*c^2)*d^2*x^2 + 4*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d^2*x)*e^5 - (2*(a^3*b^3 - 4*a^4*b*c)*d^3*x^3
- (a^2*b^4 - 10*a^3*b^2*c + 24*a^4*c^2)*d^3*x^2 + 2*(a*b^5 - 6*a^2*b^3*c + 8*a^3*b*c^2)*d^3*x)*e^4 + ((a^4*b^2
- 4*a^5*c)*d^4*x^3 + 4*(a^3*b^3 - 4*a^4*b*c)*d^4*x^2 - 8*(a^3*b^2*c - 4*a^4*c^2)*d^4*x)*e^3 - (3*(a^4*b^2 - 4
*a^5*c)*d^5*x^2 - 6*(a^3*b^3 - 4*a^4*b*c)*d^5*x - 2*(a^3*b^2*c - 4*a^4*c^2)*d^5)*e^2 - 2*(2*(a^4*b^2 - 4*a^5*c
)*d^6*x + (a^3*b^3 - 4*a^4*b*c)*d^6)*e + ((b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*d^3*e^4 + (b^4*c^2 -
5*a*b^2*c^3 + 4*a^2*c^4)*x*e^7 - (2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*x - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c
^4)*d)*e^6 + ((b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*x - 2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d^2
)*e^5)*log(a*x^2 + b*x + c) + 2*(3*(a^4*b^2 - 4*a^5*c)*d^7 + 5*(a^3*b^2*c - 4*a^4*c^2)*d^4*x*e^3 - (4*(a^3*b^3
- 4*a^4*b*c)*d^5*x - 5*(a^3*b^2*c - 4*a^4*c^2)*d^5)*e^2 + (3*(a^4*b^2 - 4*a^5*c)*d^6*x - 4*(a^3*b^3 - 4*a^4*b
*c)*d^6)*e)*log(x*e + d))/((a^5*b^2 - 4*a^6*c)*d^5*e^4 + (a^3*b^2*c^2 - 4*a^4*c^3)*x*e^9 - (2*(a^3*b^3*c - 4*a
^4*b*c^2)*d*x - (a^3*b^2*c^2 - 4*a^4*c^3)*d)*e^8 + ((a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^2*x - 2*(a^3*b^3*c -
4*a^4*b*c^2)*d^2)*e^7 - (2*(a^4*b^3 - 4*a^5*b*c)*d^3*x - (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^3)*e^6 + ((a^5
*b^2 - 4*a^6*c)*d^4*x - 2*(a^4*b^3 - 4*a^5*b*c)*d^4)*e^5)]

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

[Out]

Timed out

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Giac [A]
time = 3.54, size = 565, normalized size = 1.65 \begin {gather*} \frac {d^{5} e^{4}}{{\left (a d^{2} e^{8} - b d e^{9} + c e^{10}\right )} {\left (x e + d\right )}} + \frac {{\left (b^{5} d^{2} e^{2} - 5 \, a b^{3} c d^{2} e^{2} + 5 \, a^{2} b c^{2} d^{2} e^{2} - 2 \, b^{4} c d e^{3} + 8 \, a b^{2} c^{2} d e^{3} - 4 \, a^{2} c^{3} d e^{3} + b^{3} c^{2} e^{4} - 3 \, a b 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^{5} d^{4} - 2 \, a^{4} b d^{3} e + a^{3} b^{2} d^{2} e^{2} + 2 \, a^{4} c d^{2} e^{2} - 2 \, a^{3} b c d e^{3} + a^{3} c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (a^{2} - \frac {2 \, {\left (3 \, a^{2} d e + a b e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-4\right )}}{2 \, a^{3}} + \frac {{\left (b^{4} d^{2} - 3 \, a b^{2} c d^{2} + a^{2} c^{2} d^{2} - 2 \, b^{3} c d e + 4 \, a b c^{2} d e + b^{2} c^{2} e^{2} - a c^{3} 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^{5} d^{4} - 2 \, a^{4} b d^{3} e + a^{3} b^{2} d^{2} e^{2} + 2 \, a^{4} c d^{2} e^{2} - 2 \, a^{3} b c d e^{3} + a^{3} c^{2} e^{4}\right )}} - \frac {{\left (3 \, a^{2} d^{2} + 2 \, a b d e + b^{2} e^{2} - a c e^{2}\right )} e^{\left (-4\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+c/x^2+b/x)/(e*x+d)^2,x, algorithm="giac")

[Out]

d^5*e^4/((a*d^2*e^8 - b*d*e^9 + c*e^10)*(x*e + d)) + (b^5*d^2*e^2 - 5*a*b^3*c*d^2*e^2 + 5*a^2*b*c^2*d^2*e^2 -
2*b^4*c*d*e^3 + 8*a*b^2*c^2*d*e^3 - 4*a^2*c^3*d*e^3 + b^3*c^2*e^4 - 3*a*b*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^5*d^4 - 2*a^4*b*d
^3*e + a^3*b^2*d^2*e^2 + 2*a^4*c*d^2*e^2 - 2*a^3*b*c*d*e^3 + a^3*c^2*e^4)*sqrt(-b^2 + 4*a*c)) + 1/2*(a^2 - 2*(
3*a^2*d*e + a*b*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-4)/a^3 + 1/2*(b^4*d^2 - 3*a*b^2*c*d^2 + a^2*c^2*d^2 - 2
*b^3*c*d*e + 4*a*b*c^2*d*e + b^2*c^2*e^2 - a*c^3*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^5*d^4 - 2*a^4*b*d^3*e + a^3*b^2*d^2*e^2 + 2*a^4*c*d^2*e^2 - 2
*a^3*b*c*d*e^3 + a^3*c^2*e^4) - (3*a^2*d^2 + 2*a*b*d*e + b^2*e^2 - a*c*e^2)*e^(-4)*log(abs(x*e + d)*e^(-1)/(x*
e + d)^2)/a^3

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Mupad [B]
time = 8.04, size = 2500, normalized size = 7.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((d + e*x)^2*(a + b/x + c/x^2)),x)

[Out]

(log(d + e*x)*(3*a*d^6 + 5*c*d^4*e^2 - 4*b*d^5*e))/(c^2*e^8 + a^2*d^4*e^4 + b^2*d^2*e^6 - 2*b*c*d*e^7 - 2*a*b*
d^3*e^5 + 2*a*c*d^2*e^6) - (log(12*a^5*c*d^8 - 2*a*c^5*e^8 - 3*a^4*b^2*d^8 + b^2*c^4*e^8 + b^6*d^4*e^4 + 4*a^3
*b^3*d^7*e - 4*b^3*c^3*d*e^7 - 4*b^5*c*d^3*e^5 + b^5*d^4*e^4*(b^2 - 4*a*c)^(1/2) + 12*a^2*c^4*d^2*e^6 - 22*a^3
*c^3*d^4*e^4 + 8*a^4*c^2*d^6*e^2 + 6*b^4*c^2*d^2*e^6 - 3*a^4*b*d^8*(b^2 - 4*a*c)^(1/2) + b*c^4*e^8*(b^2 - 4*a*
c)^(1/2) - 6*a^5*d^8*x*(b^2 - 4*a*c)^(1/2) + 12*a*b*c^4*d*e^7 - 16*a^4*b*c*d^7*e - 4*a^2*c^3*d^3*e^5*(b^2 - 4*
a*c)^(1/2) + 20*a^3*c^2*d^5*e^3*(b^2 - 4*a*c)^(1/2) + 6*b^3*c^2*d^2*e^6*(b^2 - 4*a*c)^(1/2) + a*b*c^4*e^8*x +
24*a^5*c*d^7*e*x + 14*a^2*b^2*c^2*d^4*e^4 + 4*a*c^4*d*e^7*(b^2 - 4*a*c)^(1/2) + 12*a^4*c*d^7*e*(b^2 - 4*a*c)^(
1/2) + a*c^4*e^8*x*(b^2 - 4*a*c)^(1/2) - 6*a*b^4*c*d^4*e^4 + a*b^5*d^4*e^4*x - 6*a^4*b^2*d^7*e*x + 8*a^2*c^4*d
*e^7*x + 4*a^3*b^2*d^7*e*(b^2 - 4*a*c)^(1/2) - 4*b^2*c^3*d*e^7*(b^2 - 4*a*c)^(1/2) - 4*b^4*c*d^3*e^5*(b^2 - 4*
a*c)^(1/2) - 24*a*b^2*c^3*d^2*e^6 + 20*a*b^3*c^2*d^3*e^5 - 20*a^2*b*c^3*d^3*e^5 - 4*a^2*b^3*c*d^5*e^3 + 16*a^3
*b*c^2*d^5*e^3 - 2*a^3*b^2*c*d^6*e^2 - 4*a^2*b^4*d^5*e^3*x + 11*a^3*b^3*d^6*e^2*x - 8*a^3*c^3*d^3*e^5*x + 40*a
^4*c^2*d^5*e^3*x - 12*a*b*c^3*d^2*e^6*(b^2 - 4*a*c)^(1/2) - 4*a*b^3*c*d^4*e^4*(b^2 - 4*a*c)^(1/2) - 24*a^3*b*c
*d^6*e^2*(b^2 - 4*a*c)^(1/2) + a*b^4*d^4*e^4*x*(b^2 - 4*a*c)^(1/2) - 4*a^4*c*d^6*e^2*x*(b^2 - 4*a*c)^(1/2) + 6
*a*b^3*c^2*d^2*e^6*x - 18*a^2*b*c^3*d^2*e^6*x - 15*a^3*b*c^2*d^4*e^4*x + 6*a^3*b^2*c*d^5*e^3*x + 12*a*b^2*c^2*
d^3*e^5*(b^2 - 4*a*c)^(1/2) - 2*a^2*b*c^2*d^4*e^4*(b^2 - 4*a*c)^(1/2) + 4*a^2*b^2*c*d^5*e^3*(b^2 - 4*a*c)^(1/2
) + 4*a^2*b^3*d^5*e^3*x*(b^2 - 4*a*c)^(1/2) - 11*a^3*b^2*d^6*e^2*x*(b^2 - 4*a*c)^(1/2) - 6*a^2*c^3*d^2*e^6*x*(
b^2 - 4*a*c)^(1/2) + 11*a^3*c^2*d^4*e^4*x*(b^2 - 4*a*c)^(1/2) + 16*a^2*b^2*c^2*d^3*e^5*x + 14*a^4*b*d^7*e*x*(b
^2 - 4*a*c)^(1/2) - 4*a*b^2*c^3*d*e^7*x - 4*a*b^4*c*d^3*e^5*x - 44*a^4*b*c*d^6*e^2*x - 4*a*b*c^3*d*e^7*x*(b^2
- 4*a*c)^(1/2) - 4*a*b^3*c*d^3*e^5*x*(b^2 - 4*a*c)^(1/2) + 2*a^3*b*c*d^5*e^3*x*(b^2 - 4*a*c)^(1/2) + 6*a*b^2*c
^2*d^2*e^6*x*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c^2*d^3*e^5*x*(b^2 - 4*a*c)^(1/2) - 8*a^2*b^2*c*d^4*e^4*x*(b^2 - 4*
a*c)^(1/2))*(b^6*d^2 + b^5*d^2*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^3*d^2 + 4*a^2*c^4*e^2 + b^4*c^2*e^2 - 5*a*b^2*c^3
*e^2 + b^3*c^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*b^5*c*d*e + 13*a^2*b^2*c^2*d^2 - 7*a*b^4*c*d^2 + 12*a*b^3*c^2*d*e -
16*a^2*b*c^3*d*e - 5*a*b^3*c*d^2*(b^2 - 4*a*c)^(1/2) - 3*a*b*c^3*e^2*(b^2 - 4*a*c)^(1/2) - 4*a^2*c^3*d*e*(b^2
- 4*a*c)^(1/2) + 5*a^2*b*c^2*d^2*(b^2 - 4*a*c)^(1/2) - 2*b^4*c*d*e*(b^2 - 4*a*c)^(1/2) + 8*a*b^2*c^2*d*e*(b^2
- 4*a*c)^(1/2)))/(2*(4*a^6*c*d^4 - a^5*b^2*d^4 + 4*a^4*c^3*e^4 + 2*a^4*b^3*d^3*e - a^3*b^2*c^2*e^4 - a^3*b^4*
d^2*e^2 + 8*a^5*c^2*d^2*e^2 - 8*a^5*b*c*d^3*e + 2*a^3*b^3*c*d*e^3 - 8*a^4*b*c^2*d*e^3 + 2*a^4*b^2*c*d^2*e^2))
- (log(2*a*c^5*e^8 - 12*a^5*c*d^8 + 3*a^4*b^2*d^8 - b^2*c^4*e^8 - b^6*d^4*e^4 - 4*a^3*b^3*d^7*e + 4*b^3*c^3*d*
e^7 + 4*b^5*c*d^3*e^5 + b^5*d^4*e^4*(b^2 - 4*a*c)^(1/2) - 12*a^2*c^4*d^2*e^6 + 22*a^3*c^3*d^4*e^4 - 8*a^4*c^2*
d^6*e^2 - 6*b^4*c^2*d^2*e^6 - 3*a^4*b*d^8*(b^2 - 4*a*c)^(1/2) + b*c^4*e^8*(b^2 - 4*a*c)^(1/2) - 6*a^5*d^8*x*(b
^2 - 4*a*c)^(1/2) - 12*a*b*c^4*d*e^7 + 16*a^4*b*c*d^7*e - 4*a^2*c^3*d^3*e^5*(b^2 - 4*a*c)^(1/2) + 20*a^3*c^2*d
^5*e^3*(b^2 - 4*a*c)^(1/2) + 6*b^3*c^2*d^2*e^6*(b^2 - 4*a*c)^(1/2) - a*b*c^4*e^8*x - 24*a^5*c*d^7*e*x - 14*a^2
*b^2*c^2*d^4*e^4 + 4*a*c^4*d*e^7*(b^2 - 4*a*c)^(1/2) + 12*a^4*c*d^7*e*(b^2 - 4*a*c)^(1/2) + a*c^4*e^8*x*(b^2 -
4*a*c)^(1/2) + 6*a*b^4*c*d^4*e^4 - a*b^5*d^4*e^4*x + 6*a^4*b^2*d^7*e*x - 8*a^2*c^4*d*e^7*x + 4*a^3*b^2*d^7*e*
(b^2 - 4*a*c)^(1/2) - 4*b^2*c^3*d*e^7*(b^2 - 4*a*c)^(1/2) - 4*b^4*c*d^3*e^5*(b^2 - 4*a*c)^(1/2) + 24*a*b^2*c^3
*d^2*e^6 - 20*a*b^3*c^2*d^3*e^5 + 20*a^2*b*c^3*d^3*e^5 + 4*a^2*b^3*c*d^5*e^3 - 16*a^3*b*c^2*d^5*e^3 + 2*a^3*b^
2*c*d^6*e^2 + 4*a^2*b^4*d^5*e^3*x - 11*a^3*b^3*d^6*e^2*x + 8*a^3*c^3*d^3*e^5*x - 40*a^4*c^2*d^5*e^3*x - 12*a*b
*c^3*d^2*e^6*(b^2 - 4*a*c)^(1/2) - 4*a*b^3*c*d^4*e^4*(b^2 - 4*a*c)^(1/2) - 24*a^3*b*c*d^6*e^2*(b^2 - 4*a*c)^(1
/2) + a*b^4*d^4*e^4*x*(b^2 - 4*a*c)^(1/2) - 4*a^4*c*d^6*e^2*x*(b^2 - 4*a*c)^(1/2) - 6*a*b^3*c^2*d^2*e^6*x + 18
*a^2*b*c^3*d^2*e^6*x + 15*a^3*b*c^2*d^4*e^4*x - 6*a^3*b^2*c*d^5*e^3*x + 12*a*b^2*c^2*d^3*e^5*(b^2 - 4*a*c)^(1/
2) - 2*a^2*b*c^2*d^4*e^4*(b^2 - 4*a*c)^(1/2) + 4*a^2*b^2*c*d^5*e^3*(b^2 - 4*a*c)^(1/2) + 4*a^2*b^3*d^5*e^3*x*(
b^2 - 4*a*c)^(1/2) - 11*a^3*b^2*d^6*e^2*x*(b^2 - 4*a*c)^(1/2) - 6*a^2*c^3*d^2*e^6*x*(b^2 - 4*a*c)^(1/2) + 11*a
^3*c^2*d^4*e^4*x*(b^2 - 4*a*c)^(1/2) - 16*a^2*b^2*c^2*d^3*e^5*x + 14*a^4*b*d^7*e*x*(b^2 - 4*a*c)^(1/2) + 4*a*b
^2*c^3*d*e^7*x + 4*a*b^4*c*d^3*e^5*x + 44*a^4*b*c*d^6*e^2*x - 4*a*b*c^3*d*e^7*x*(b^2 - 4*a*c)^(1/2) - 4*a*b^3*
c*d^3*e^5*x*(b^2 - 4*a*c)^(1/2) + 2*a^3*b*c*d^5*e^3*x*(b^2 - 4*a*c)^(1/2) + 6*a*b^2*c^2*d^2*e^6*x*(b^2 - 4*a*c
)^(1/2) + 8*a^2*b*c^2*d^3*e^5*x*(b^2 - 4*a*c)^(1/2) - 8*a^2*b^2*c*d^4*e^4*x*(b^2 - 4*a*c)^(1/2))*(b^6*d^2 - b^
5*d^2*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^3*d^2 + 4*a...

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