3.9.91 \(\int \frac {x^3 (d+e x)}{(a+b x+c x^2)^2} \, dx\) [891]

Optimal. Leaf size=192 \[ -\frac {\left (b c d-2 b^2 e+6 a c e\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (b^3 c d-6 a b c^2 d-2 b^4 e+12 a b^2 c e-12 a^2 c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \]

[Out]

-(6*a*c*e-2*b^2*e+b*c*d)*x/c^2/(-4*a*c+b^2)+x^2*(a*(-b*e+2*c*d)+(2*a*c*e-b^2*e+b*c*d)*x)/c/(-4*a*c+b^2)/(c*x^2
+b*x+a)+(-12*a^2*c^2*e+12*a*b^2*c*e-6*a*b*c^2*d-2*b^4*e+b^3*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4
*a*c+b^2)^(3/2)+1/2*(-2*b*e+c*d)*ln(c*x^2+b*x+a)/c^3

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Rubi [A]
time = 0.21, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {832, 787, 648, 632, 212, 642} \begin {gather*} \frac {\left (-12 a^2 c^2 e+12 a b^2 c e-6 a b c^2 d-2 b^4 e+b^3 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {x \left (6 a c e-2 b^2 e+b c d\right )}{c^2 \left (b^2-4 a c\right )}+\frac {x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*c*d - 2*b^2*e + 6*a*c*e)*x)/(c^2*(b^2 - 4*a*c))) + (x^2*(a*(2*c*d - b*e) + (b*c*d - b^2*e + 2*a*c*e)*x))
/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((b^3*c*d - 6*a*b*c^2*d - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*ArcTan
h[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) + ((c*d - 2*b*e)*Log[a + b*x + c*x^2])/(2*c^3)

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 787

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

Rule 832

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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(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] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx &=\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x \left (-2 a (2 c d-b e)-\left (b c d-2 b^2 e+6 a c e\right ) x\right )}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=-\frac {\left (b c d-2 b^2 e+6 a c e\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {-a \left (-b c d+2 b^2 e-6 a c e\right )+\left (-2 a c (2 c d-b e)-b \left (-b c d+2 b^2 e-6 a c e\right )\right ) x}{a+b x+c x^2} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=-\frac {\left (b c d-2 b^2 e+6 a c e\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(c d-2 b e) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}-\frac {\left (b^3 c d-6 a b c^2 d-2 b^4 e+12 a b^2 c e-12 a^2 c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^3 \left (b^2-4 a c\right )}\\ &=-\frac {\left (b c d-2 b^2 e+6 a c e\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac {\left (b^3 c d-6 a b c^2 d-2 b^4 e+12 a b^2 c e-12 a^2 c^2 e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3 \left (b^2-4 a c\right )}\\ &=-\frac {\left (b c d-2 b^2 e+6 a c e\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (b^3 c d-6 a b c^2 d-2 b^4 e+12 a b^2 c e-12 a^2 c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 190, normalized size = 0.99 \begin {gather*} \frac {2 c e x+\frac {2 \left (b^3 (c d-b e) x+a^2 c (3 b e-2 c (d+e x))+a b \left (-b^2 e-3 c^2 d x+b c (d+4 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (-b^3 c d+6 a b c^2 d+2 b^4 e-12 a b^2 c e+12 a^2 c^2 e\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+(c d-2 b e) \log (a+x (b+c x))}{2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*e*x + (2*(b^3*(c*d - b*e)*x + a^2*c*(3*b*e - 2*c*(d + e*x)) + a*b*(-(b^2*e) - 3*c^2*d*x + b*c*(d + 4*e*x)
)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(-(b^3*c*d) + 6*a*b*c^2*d + 2*b^4*e - 12*a*b^2*c*e + 12*a^2*c^2*e)*
ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c*d - 2*b*e)*Log[a + x*(b + c*x)])/(2*c^3)

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Maple [A]
time = 0.83, size = 269, normalized size = 1.40

method	result	size
default	\(\frac {e x}{c^{2}}-\frac {\frac {-\frac {\left (2 a^{2} c^{2} e -4 a \,b^{2} c e +3 a b \,c^{2} d +b^{4} e -b^{3} d c \right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a \left (3 a b c e -2 a \,c^{2} d -e \,b^{3}+b^{2} c d \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (8 a b c e -4 a \,c^{2} d -2 e \,b^{3}+b^{2} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (6 a^{2} c e -2 a \,b^{2} e +a b c d -\frac {\left (8 a b c e -4 a \,c^{2} d -2 e \,b^{3}+b^{2} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{2}}\)	\(269\)
risch	\(\text {Expression too large to display}\)	\(3047\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x/c^2-1/c^2*((-(2*a^2*c^2*e-4*a*b^2*c*e+3*a*b*c^2*d+b^4*e-b^3*c*d)/c/(4*a*c-b^2)*x+a*(3*a*b*c*e-2*a*c^2*d-b^
3*e+b^2*c*d)/c/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)*(1/2*(8*a*b*c*e-4*a*c^2*d-2*b^3*e+b^2*c*d)/c*ln(c*x^2+
b*x+a)+2*(6*a^2*c*e-2*a*b^2*e+a*b*c*d-1/2*(8*a*b*c*e-4*a*c^2*d-2*b^3*e+b^2*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan(
(2*c*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*(e*x+d)/(c*x^2+b*x+a)^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. 622 vs. \(2 (195) = 390\).
time = 1.14, size = 1263, normalized size = 6.58 \begin {gather*} \left [\frac {2 \, {\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} d x + {\left ({\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d - 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d - 2 \, {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x\right )} e + {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d - 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x\right )}}, \frac {2 \, {\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} d x + 2 \, {\left ({\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d - 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d - 2 \, {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x\right )} e + {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d - 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d*x + ((b^3*c^2 - 6*a*b*c^3)*d*x^2 + (b^4*c - 6*a*b^2*c^2)*d*x +
(a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 + (b^5
- 6*a*b^3*c + 6*a^2*b*c^2)*x)*e)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*
(2*c*x + b))/(c*x^2 + b*x + a)) + 2*(a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*
b*c^2 - (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 9*a*b^4*c
+ 26*a^2*b^2*c^2 - 24*a^3*c^3)*x)*e + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^2 + (b^5*c - 8*a*b^3*c^2 + 16
*a^2*b*c^3)*d*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d - 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + (b^5*c -
8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x)*e)*log(c*x^2 + b*x + a))/(a*b^4*c^3 -
8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)
*x), 1/2*(2*(b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*d*x + 2*((b^3*c^2 - 6*a*b*c^3)*d*x^2 + (b^4*c - 6*a*b^2*c^2)*
d*x + (a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 +
(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x)*e)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)
) + 2*(a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2 - (b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4)*x^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3
)*x)*e + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x + (a*b^4*c - 8
*a^2*b^2*c^2 + 16*a^3*c^3)*d - 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^
2 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x)*e)*log(c*x^2 + b*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (
b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1248 vs. \(2 (187) = 374\).
time = 3.18, size = 1248, normalized size = 6.50 \begin {gather*} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right ) \log {\left (x + \frac {- 10 a^{2} b c e - 16 a^{2} c^{4} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right ) + 8 a^{2} c^{2} d + 2 a b^{3} e + 8 a b^{2} c^{3} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right ) - a b^{2} c d - b^{4} c^{2} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right )}{12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d} \right )} + \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right ) \log {\left (x + \frac {- 10 a^{2} b c e - 16 a^{2} c^{4} \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right ) + 8 a^{2} c^{2} d + 2 a b^{3} e + 8 a b^{2} c^{3} \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right ) - a b^{2} c d - b^{4} c^{2} \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right )}{2 c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 b e - c d}{2 c^{3}}\right )}{12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d} \right )} + \frac {- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + x \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{4 a^{2} c^{4} - a b^{2} c^{3} + x^{2} \cdot \left (4 a c^{5} - b^{2} c^{4}\right ) + x \left (4 a b c^{4} - b^{3} c^{3}\right )} + \frac {e x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**
2*c**4*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**
3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**
3*e + 8*a*b**2*c**3*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**
3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) - a*b**2*c*d
- b**4*c**2*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/
(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e -
12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)) + (sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c
*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2
*b*e - c*d)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**2*c**4*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*
b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
) - (2*b*e - c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**3*e + 8*a*b**2*c**3*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c*
*2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) - a*b**2*c*d - b**4*c**2*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 1
2*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b
**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)) + (-3*
a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b**2*c*d + x*(2*a**2*c**2*e - 4*a*b**2*c*e + 3*a*b*c**2*d + b**4*e -
b**3*c*d))/(4*a**2*c**4 - a*b**2*c**3 + x**2*(4*a*c**5 - b**2*c**4) + x*(4*a*b*c**4 - b**3*c**3)) + e*x/c**2

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Giac [A]
time = 0.88, size = 235, normalized size = 1.22 \begin {gather*} -\frac {{\left (b^{3} c d - 6 \, a b c^{2} d - 2 \, b^{4} e + 12 \, a b^{2} c e - 12 \, a^{2} c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x e}{c^{2}} + \frac {{\left (c d - 2 \, b e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {\frac {{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} x}{c} + \frac {a b^{2} c d - 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e}{c}}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^3*c*d - 6*a*b*c^2*d - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*
c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + x*e/c^2 + 1/2*(c*d - 2*b*e)*log(c*x^2 + b*x + a)/c^3 + ((b^3*c*d - 3*a*b*
c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)*x/c + (a*b^2*c*d - 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)/c)/((c*x^2
+ b*x + a)*(b^2 - 4*a*c)*c^2)

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Mupad [B]
time = 1.85, size = 360, normalized size = 1.88 \begin {gather*} \frac {\frac {a\,\left (e\,b^3-d\,b^2\,c-3\,a\,e\,b\,c+2\,a\,d\,c^2\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,e\,a^3\,b\,c^3+64\,d\,a^3\,c^4+96\,e\,a^2\,b^3\,c^2-48\,d\,a^2\,b^2\,c^3-24\,e\,a\,b^5\,c+12\,d\,a\,b^4\,c^2+2\,e\,b^7-d\,b^6\,c\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}+\frac {e\,x}{c^2}-\frac {\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (12\,e\,a^2\,c^2-12\,e\,a\,b^2\,c+6\,d\,a\,b\,c^2+2\,e\,b^4-d\,b^3\,c\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*(b^3*e + 2*a*c^2*d - b^2*c*d - 3*a*b*c*e))/(c*(4*a*c - b^2)) + (x*(b^4*e + 2*a^2*c^2*e - b^3*c*d + 3*a*b*c
^2*d - 4*a*b^2*c*e))/(c*(4*a*c - b^2)))/(a*c^2 + c^3*x^2 + b*c^2*x) + (log(a + b*x + c*x^2)*(2*b^7*e + 64*a^3*
c^4*d - b^6*c*d - 48*a^2*b^2*c^3*d + 96*a^2*b^3*c^2*e - 24*a*b^5*c*e + 12*a*b^4*c^2*d - 128*a^3*b*c^3*e))/(2*(
64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)) + (e*x)/c^2 - (atan((2*c*x)/(4*a*c - b^2)^(1/2) - (b^3*
c^2 - 4*a*b*c^3)/(c^2*(4*a*c - b^2)^(3/2)))*(2*b^4*e + 12*a^2*c^2*e - b^3*c*d + 6*a*b*c^2*d - 12*a*b^2*c*e))/(
c^3*(4*a*c - b^2)^(3/2))

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