3.2373 \(\int \frac{(d+e x)^2 (f+g x)}{(a+b x+c x^2)^3} \, dx\)
Optimal. Leaf size=305 \[ -\frac{-x \left (-2 c^2 (2 a e (e f-2 d g)+3 b d (d g+2 e f))-2 b c e (a e g-2 b (d g+e f))+b^3 \left (-e^2\right ) g+12 c^3 d^2 f\right )+b^2 \left (a e^2 g+c d (3 d g+2 e f)\right )-6 b c \left (a e (2 d g+e f)+c d^2 f\right )+8 a c e (a e g+2 c d f)}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-c (3 b d (d g+2 e f)-2 a e (2 d g+e f))+b e (-3 a e g+2 b d g+b e f)+6 c^2 d^2 f\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^2 (-2 a g+x (2 c f-b g)+b f)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
[Out]
-((d + e*x)^2*(b*f - 2*a*g + (2*c*f - b*g)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (8*a*c*e*(2*c*d*f + a*e
*g) - 6*b*c*(c*d^2*f + a*e*(e*f + 2*d*g)) + b^2*(a*e^2*g + c*d*(2*e*f + 3*d*g)) - (12*c^3*d^2*f - b^3*e^2*g -
2*b*c*e*(a*e*g - 2*b*(e*f + d*g)) - 2*c^2*(2*a*e*(e*f - 2*d*g) + 3*b*d*(2*e*f + d*g)))*x)/(2*c*(b^2 - 4*a*c)^2
*(a + b*x + c*x^2)) - (2*(6*c^2*d^2*f + b*e*(b*e*f + 2*b*d*g - 3*a*e*g) - c*(3*b*d*(2*e*f + d*g) - 2*a*e*(e*f
+ 2*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)
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Rubi [A] time = 0.379716, antiderivative size = 303, normalized size of antiderivative =
0.99, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules
used = {820, 777, 618, 206} \[ -\frac{-x \left (-2 c^2 (2 a e (e f-2 d g)+3 b d (d g+2 e f))-2 b c e (a e g-2 b (d g+e f))+b^3 \left (-e^2\right ) g+12 c^3 d^2 f\right )+b^2 \left (a e^2 g+c d (3 d g+2 e f)\right )-6 b c \left (a e (2 d g+e f)+c d^2 f\right )+8 a c e (a e g+2 c d f)}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (b e (-3 a e g+2 b d g+b e f)+2 a c e (2 d g+e f)-3 b c d (d g+2 e f)+6 c^2 d^2 f\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^2 (-2 a g+x (2 c f-b g)+b f)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x)^2*(f + g*x))/(a + b*x + c*x^2)^3,x]
[Out]
-((d + e*x)^2*(b*f - 2*a*g + (2*c*f - b*g)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (8*a*c*e*(2*c*d*f + a*e
*g) - 6*b*c*(c*d^2*f + a*e*(e*f + 2*d*g)) + b^2*(a*e^2*g + c*d*(2*e*f + 3*d*g)) - (12*c^3*d^2*f - b^3*e^2*g -
2*b*c*e*(a*e*g - 2*b*(e*f + d*g)) - 2*c^2*(2*a*e*(e*f - 2*d*g) + 3*b*d*(2*e*f + d*g)))*x)/(2*c*(b^2 - 4*a*c)^2
*(a + b*x + c*x^2)) - (2*(6*c^2*d^2*f - 3*b*c*d*(2*e*f + d*g) + 2*a*c*e*(e*f + 2*d*g) + b*e*(b*e*f + 2*b*d*g -
3*a*e*g))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)
Rule 820
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*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, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])
Rule 777
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Rule 618
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^2 (b f-2 a g+(2 c f-b g) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{(d+e x) (6 c d f-2 b e f-3 b d g+4 a e g+e (2 c f-b g) x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(d+e x)^2 (b f-2 a g+(2 c f-b g) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{8 a c e (2 c d f+a e g)-6 b c \left (c d^2 f+a e (e f+2 d g)\right )+b^2 \left (a e^2 g+c d (2 e f+3 d g)\right )-\left (12 c^3 d^2 f-b^3 e^2 g-2 b c e (a e g-2 b (e f+d g))-2 c^2 (2 a e (e f-2 d g)+3 b d (2 e f+d g))\right ) x}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\left (6 c^2 d^2 f-3 b c d (2 e f+d g)+2 a c e (e f+2 d g)+b e (b e f+2 b d g-3 a e g)\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x)^2 (b f-2 a g+(2 c f-b g) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{8 a c e (2 c d f+a e g)-6 b c \left (c d^2 f+a e (e f+2 d g)\right )+b^2 \left (a e^2 g+c d (2 e f+3 d g)\right )-\left (12 c^3 d^2 f-b^3 e^2 g-2 b c e (a e g-2 b (e f+d g))-2 c^2 (2 a e (e f-2 d g)+3 b d (2 e f+d g))\right ) x}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (2 \left (6 c^2 d^2 f-3 b c d (2 e f+d g)+2 a c e (e f+2 d g)+b e (b e f+2 b d g-3 a e g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x)^2 (b f-2 a g+(2 c f-b g) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{8 a c e (2 c d f+a e g)-6 b c \left (c d^2 f+a e (e f+2 d g)\right )+b^2 \left (a e^2 g+c d (2 e f+3 d g)\right )-\left (12 c^3 d^2 f-b^3 e^2 g-2 b c e (a e g-2 b (e f+d g))-2 c^2 (2 a e (e f-2 d g)+3 b d (2 e f+d g))\right ) x}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (6 c^2 d^2 f-3 b c d (2 e f+d g)+2 a c e (e f+2 d g)+b e (b e f+2 b d g-3 a e g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.86209, size = 408, normalized size = 1.34 \[ \frac{1}{2} \left (\frac{4 c^2 \left (-4 a^2 e^2 g+a c e x (2 d g+e f)+3 c^2 d^2 f x\right )+b^2 c \left (5 a e^2 g+c \left (-3 d^2 g-6 d e f+4 d e g x+2 e^2 f x\right )\right )+2 b c^2 (a e (2 d g+e f-3 e g x)+3 c d (d f-d g x-2 e f x))+b^3 c e (2 d g+e f)+b^4 \left (-e^2\right ) g}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{2 c \left (a^2 e^2 g-a c \left (d^2 g+2 d e (f+g x)+e^2 f x\right )+c^2 d^2 f x\right )+b^2 e (c x (2 d g+e f)-a e g)+b c (a e (2 d g+e f+3 e g x)+c d (d f-d g x-2 e f x))+b^3 \left (-e^2\right ) g x}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{4 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (b e (-3 a e g+2 b d g+b e f)+2 a c e (2 d g+e f)-3 b c d (d g+2 e f)+6 c^2 d^2 f\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x)^2*(f + g*x))/(a + b*x + c*x^2)^3,x]
[Out]
((-(b^4*e^2*g) + b^3*c*e*(e*f + 2*d*g) + 4*c^2*(-4*a^2*e^2*g + 3*c^2*d^2*f*x + a*c*e*(e*f + 2*d*g)*x) + 2*b*c^
2*(3*c*d*(d*f - 2*e*f*x - d*g*x) + a*e*(e*f + 2*d*g - 3*e*g*x)) + b^2*c*(5*a*e^2*g + c*(-6*d*e*f - 3*d^2*g + 2
*e^2*f*x + 4*d*e*g*x)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^3*e^2*g*x) + b^2*e*(-(a*e*g) + c*(e*f +
2*d*g)*x) + b*c*(c*d*(d*f - 2*e*f*x - d*g*x) + a*e*(e*f + 2*d*g + 3*e*g*x)) + 2*c*(a^2*e^2*g + c^2*d^2*f*x -
a*c*(d^2*g + e^2*f*x + 2*d*e*(f + g*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (4*(6*c^2*d^2*f - 3*b*c*d
*(2*e*f + d*g) + 2*a*c*e*(e*f + 2*d*g) + b*e*(b*e*f + 2*b*d*g - 3*a*e*g))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c
]])/(-b^2 + 4*a*c)^(5/2))/2
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Maple [B] time = 0.013, size = 1014, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^3,x)
[Out]
(-c*(3*a*b*e^2*g-4*a*c*d*e*g-2*a*c*e^2*f-2*b^2*d*e*g-b^2*e^2*f+3*b*c*d^2*g+6*b*c*d*e*f-6*c^2*d^2*f)/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^3-1/2*(16*a^2*c^2*e^2*g+a*b^2*c*e^2*g-12*a*b*c^2*d*e*g-6*a*b*c^2*e^2*f+b^4*e^2*g-6*b^3*c*d*
e*g-3*b^3*c*e^2*f+9*b^2*c^2*d^2*g+18*b^2*c^2*d*e*f-18*b*c^3*d^2*f)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/c*(5*a^2
*b*c*e^2*g+4*a^2*c^2*d*e*g+2*a^2*c^2*e^2*f+a*b^3*e^2*g-10*a*b^2*c*d*e*g-5*a*b^2*c*e^2*f+5*a*b*c^2*d^2*g+10*a*b
*c^2*d*e*f-10*a*c^3*d^2*f+b^3*c*d^2*g+2*b^3*c*d*e*f-2*b^2*c^2*d^2*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/2*(8*a^3*c
*e^2*g+a^2*b^2*e^2*g-12*a^2*b*c*d*e*g-6*a^2*b*c*e^2*f+8*a^2*c^2*d^2*g+16*a^2*c^2*d*e*f+a*b^2*c*d^2*g+2*a*b^2*c
*d*e*f-10*a*b*c^2*d^2*f+b^3*c*d^2*f)/c/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2-6/(16*a^2*c^2-8*a*b^2*c+b^4
)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e^2*g+8/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c*d*e*g+4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*a*c*e^2*f+4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*b^2*d*e*g+2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e^2*f-6/
(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d^2*g-12/(16*a^2*c^2-8*a*
b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d*e*f+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*
c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d^2*f
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.13812, size = 5937, normalized size = 19.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
[1/2*(2*((6*(b^2*c^4 - 4*a*c^5)*d^2 - 6*(b^3*c^3 - 4*a*b*c^4)*d*e + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*e^2)*f
- (3*(b^3*c^3 - 4*a*b*c^4)*d^2 - 2*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d*e + 3*(a*b^3*c^2 - 4*a^2*b*c^3)*e^2)
*g)*x^3 + (3*(6*(b^3*c^3 - 4*a*b*c^4)*d^2 - 6*(b^4*c^2 - 4*a*b^2*c^3)*d*e + (b^5*c - 2*a*b^3*c^2 - 8*a^2*b*c^3
)*e^2)*f - (9*(b^4*c^2 - 4*a*b^2*c^3)*d^2 - 6*(b^5*c - 2*a*b^3*c^2 - 8*a^2*b*c^3)*d*e + (b^6 - 3*a*b^4*c + 12*
a^2*b^2*c^2 - 64*a^3*c^3)*e^2)*g)*x^2 - 2*(((6*c^5*d^2 - 6*b*c^4*d*e + (b^2*c^3 + 2*a*c^4)*e^2)*f - (3*b*c^4*d
^2 + 3*a*b*c^3*e^2 - 2*(b^2*c^3 + 2*a*c^4)*d*e)*g)*x^4 + 2*((6*b*c^4*d^2 - 6*b^2*c^3*d*e + (b^3*c^2 + 2*a*b*c^
3)*e^2)*f - (3*b^2*c^3*d^2 + 3*a*b^2*c^2*e^2 - 2*(b^3*c^2 + 2*a*b*c^3)*d*e)*g)*x^3 + ((6*(b^2*c^3 + 2*a*c^4)*d
^2 - 6*(b^3*c^2 + 2*a*b*c^3)*d*e + (b^4*c + 4*a*b^2*c^2 + 4*a^2*c^3)*e^2)*f - (3*(b^3*c^2 + 2*a*b*c^3)*d^2 - 2
*(b^4*c + 4*a*b^2*c^2 + 4*a^2*c^3)*d*e + 3*(a*b^3*c + 2*a^2*b*c^2)*e^2)*g)*x^2 + (6*a^2*c^3*d^2 - 6*a^2*b*c^2*
d*e + (a^2*b^2*c + 2*a^3*c^2)*e^2)*f - (3*a^2*b*c^2*d^2 + 3*a^3*b*c*e^2 - 2*(a^2*b^2*c + 2*a^3*c^2)*d*e)*g + 2
*((6*a*b*c^3*d^2 - 6*a*b^2*c^2*d*e + (a*b^3*c + 2*a^2*b*c^2)*e^2)*f - (3*a*b^2*c^2*d^2 + 3*a^2*b^2*c*e^2 - 2*(
a*b^3*c + 2*a^2*b*c^2)*d*e)*g)*x)*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)) - ((b^5*c - 14*a*b^3*c^2 + 40*a^2*b*c^3)*d^2 + 2*(a*b^4*c + 4*a^2*b^2*c^2 - 3
2*a^3*c^3)*d*e - 6*(a^2*b^3*c - 4*a^3*b*c^2)*e^2)*f - ((a*b^4*c + 4*a^2*b^2*c^2 - 32*a^3*c^3)*d^2 - 12*(a^2*b^
3*c - 4*a^3*b*c^2)*d*e + (a^2*b^4 + 4*a^3*b^2*c - 32*a^4*c^2)*e^2)*g + 2*((2*(b^4*c^2 + a*b^2*c^3 - 20*a^2*c^4
)*d^2 - 2*(b^5*c + a*b^3*c^2 - 20*a^2*b*c^3)*d*e + (5*a*b^4*c - 22*a^2*b^2*c^2 + 8*a^3*c^3)*e^2)*f - ((b^5*c +
a*b^3*c^2 - 20*a^2*b*c^3)*d^2 - 2*(5*a*b^4*c - 22*a^2*b^2*c^2 + 8*a^3*c^3)*d*e + (a*b^5 + a^2*b^3*c - 20*a^3*
b*c^2)*e^2)*g)*x)/(a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5*c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2
*b^2*c^5 - 64*a^3*c^6)*x^4 + 2*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^3 + (b^8*c - 10*a*b^
6*c^2 + 24*a^2*b^4*c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5)*x^2 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64
*a^4*b*c^4)*x), 1/2*(2*((6*(b^2*c^4 - 4*a*c^5)*d^2 - 6*(b^3*c^3 - 4*a*b*c^4)*d*e + (b^4*c^2 - 2*a*b^2*c^3 - 8*
a^2*c^4)*e^2)*f - (3*(b^3*c^3 - 4*a*b*c^4)*d^2 - 2*(b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d*e + 3*(a*b^3*c^2 - 4*
a^2*b*c^3)*e^2)*g)*x^3 + (3*(6*(b^3*c^3 - 4*a*b*c^4)*d^2 - 6*(b^4*c^2 - 4*a*b^2*c^3)*d*e + (b^5*c - 2*a*b^3*c^
2 - 8*a^2*b*c^3)*e^2)*f - (9*(b^4*c^2 - 4*a*b^2*c^3)*d^2 - 6*(b^5*c - 2*a*b^3*c^2 - 8*a^2*b*c^3)*d*e + (b^6 -
3*a*b^4*c + 12*a^2*b^2*c^2 - 64*a^3*c^3)*e^2)*g)*x^2 - 4*(((6*c^5*d^2 - 6*b*c^4*d*e + (b^2*c^3 + 2*a*c^4)*e^2)
*f - (3*b*c^4*d^2 + 3*a*b*c^3*e^2 - 2*(b^2*c^3 + 2*a*c^4)*d*e)*g)*x^4 + 2*((6*b*c^4*d^2 - 6*b^2*c^3*d*e + (b^3
*c^2 + 2*a*b*c^3)*e^2)*f - (3*b^2*c^3*d^2 + 3*a*b^2*c^2*e^2 - 2*(b^3*c^2 + 2*a*b*c^3)*d*e)*g)*x^3 + ((6*(b^2*c
^3 + 2*a*c^4)*d^2 - 6*(b^3*c^2 + 2*a*b*c^3)*d*e + (b^4*c + 4*a*b^2*c^2 + 4*a^2*c^3)*e^2)*f - (3*(b^3*c^2 + 2*a
*b*c^3)*d^2 - 2*(b^4*c + 4*a*b^2*c^2 + 4*a^2*c^3)*d*e + 3*(a*b^3*c + 2*a^2*b*c^2)*e^2)*g)*x^2 + (6*a^2*c^3*d^2
- 6*a^2*b*c^2*d*e + (a^2*b^2*c + 2*a^3*c^2)*e^2)*f - (3*a^2*b*c^2*d^2 + 3*a^3*b*c*e^2 - 2*(a^2*b^2*c + 2*a^3*
c^2)*d*e)*g + 2*((6*a*b*c^3*d^2 - 6*a*b^2*c^2*d*e + (a*b^3*c + 2*a^2*b*c^2)*e^2)*f - (3*a*b^2*c^2*d^2 + 3*a^2*
b^2*c*e^2 - 2*(a*b^3*c + 2*a^2*b*c^2)*d*e)*g)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^
2 - 4*a*c)) - ((b^5*c - 14*a*b^3*c^2 + 40*a^2*b*c^3)*d^2 + 2*(a*b^4*c + 4*a^2*b^2*c^2 - 32*a^3*c^3)*d*e - 6*(a
^2*b^3*c - 4*a^3*b*c^2)*e^2)*f - ((a*b^4*c + 4*a^2*b^2*c^2 - 32*a^3*c^3)*d^2 - 12*(a^2*b^3*c - 4*a^3*b*c^2)*d*
e + (a^2*b^4 + 4*a^3*b^2*c - 32*a^4*c^2)*e^2)*g + 2*((2*(b^4*c^2 + a*b^2*c^3 - 20*a^2*c^4)*d^2 - 2*(b^5*c + a*
b^3*c^2 - 20*a^2*b*c^3)*d*e + (5*a*b^4*c - 22*a^2*b^2*c^2 + 8*a^3*c^3)*e^2)*f - ((b^5*c + a*b^3*c^2 - 20*a^2*b
*c^3)*d^2 - 2*(5*a*b^4*c - 22*a^2*b^2*c^2 + 8*a^3*c^3)*d*e + (a*b^5 + a^2*b^3*c - 20*a^3*b*c^2)*e^2)*g)*x)/(a^
2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5*c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6
)*x^4 + 2*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^3 + (b^8*c - 10*a*b^6*c^2 + 24*a^2*b^4*c^
3 + 32*a^3*b^2*c^4 - 128*a^4*c^5)*x^2 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*x)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**2*(g*x+f)/(c*x**2+b*x+a)**3,x)
[Out]
Timed out
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Giac [B] time = 1.12938, size = 871, normalized size = 2.86 \begin{align*} \frac{2 \,{\left (6 \, c^{2} d^{2} f - 3 \, b c d^{2} g - 6 \, b c d f e + 2 \, b^{2} d g e + 4 \, a c d g e + b^{2} f e^{2} + 2 \, a c f e^{2} - 3 \, a b g e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{4} d^{2} f x^{3} - 6 \, b c^{3} d^{2} g x^{3} - 12 \, b c^{3} d f x^{3} e + 4 \, b^{2} c^{2} d g x^{3} e + 8 \, a c^{3} d g x^{3} e + 18 \, b c^{3} d^{2} f x^{2} - 9 \, b^{2} c^{2} d^{2} g x^{2} + 2 \, b^{2} c^{2} f x^{3} e^{2} + 4 \, a c^{3} f x^{3} e^{2} - 6 \, a b c^{2} g x^{3} e^{2} - 18 \, b^{2} c^{2} d f x^{2} e + 6 \, b^{3} c d g x^{2} e + 12 \, a b c^{2} d g x^{2} e + 4 \, b^{2} c^{2} d^{2} f x + 20 \, a c^{3} d^{2} f x - 2 \, b^{3} c d^{2} g x - 10 \, a b c^{2} d^{2} g x + 3 \, b^{3} c f x^{2} e^{2} + 6 \, a b c^{2} f x^{2} e^{2} - b^{4} g x^{2} e^{2} - a b^{2} c g x^{2} e^{2} - 16 \, a^{2} c^{2} g x^{2} e^{2} - 4 \, b^{3} c d f x e - 20 \, a b c^{2} d f x e + 20 \, a b^{2} c d g x e - 8 \, a^{2} c^{2} d g x e - b^{3} c d^{2} f + 10 \, a b c^{2} d^{2} f - a b^{2} c d^{2} g - 8 \, a^{2} c^{2} d^{2} g + 10 \, a b^{2} c f x e^{2} - 4 \, a^{2} c^{2} f x e^{2} - 2 \, a b^{3} g x e^{2} - 10 \, a^{2} b c g x e^{2} - 2 \, a b^{2} c d f e - 16 \, a^{2} c^{2} d f e + 12 \, a^{2} b c d g e + 6 \, a^{2} b c f e^{2} - a^{2} b^{2} g e^{2} - 8 \, a^{3} c g e^{2}}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
2*(6*c^2*d^2*f - 3*b*c*d^2*g - 6*b*c*d*f*e + 2*b^2*d*g*e + 4*a*c*d*g*e + b^2*f*e^2 + 2*a*c*f*e^2 - 3*a*b*g*e^2
)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*c^4*d^2
*f*x^3 - 6*b*c^3*d^2*g*x^3 - 12*b*c^3*d*f*x^3*e + 4*b^2*c^2*d*g*x^3*e + 8*a*c^3*d*g*x^3*e + 18*b*c^3*d^2*f*x^2
- 9*b^2*c^2*d^2*g*x^2 + 2*b^2*c^2*f*x^3*e^2 + 4*a*c^3*f*x^3*e^2 - 6*a*b*c^2*g*x^3*e^2 - 18*b^2*c^2*d*f*x^2*e
+ 6*b^3*c*d*g*x^2*e + 12*a*b*c^2*d*g*x^2*e + 4*b^2*c^2*d^2*f*x + 20*a*c^3*d^2*f*x - 2*b^3*c*d^2*g*x - 10*a*b*c
^2*d^2*g*x + 3*b^3*c*f*x^2*e^2 + 6*a*b*c^2*f*x^2*e^2 - b^4*g*x^2*e^2 - a*b^2*c*g*x^2*e^2 - 16*a^2*c^2*g*x^2*e^
2 - 4*b^3*c*d*f*x*e - 20*a*b*c^2*d*f*x*e + 20*a*b^2*c*d*g*x*e - 8*a^2*c^2*d*g*x*e - b^3*c*d^2*f + 10*a*b*c^2*d
^2*f - a*b^2*c*d^2*g - 8*a^2*c^2*d^2*g + 10*a*b^2*c*f*x*e^2 - 4*a^2*c^2*f*x*e^2 - 2*a*b^3*g*x*e^2 - 10*a^2*b*c
*g*x*e^2 - 2*a*b^2*c*d*f*e - 16*a^2*c^2*d*f*e + 12*a^2*b*c*d*g*e + 6*a^2*b*c*f*e^2 - a^2*b^2*g*e^2 - 8*a^3*c*g
*e^2)/((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*(c*x^2 + b*x + a)^2)