3.58 \(\int \frac {(d+e x)^2 (A+B x+C x^2)}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=156 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right )}{8 a^{5/2} c^{5/2}}-\frac {(d+e x) (a e (3 a C+A c)-c x (2 a B e+a C d+3 A c d))}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-1/4*(a*B-(A*c-C*a)*x)*(e*x+d)^2/a/c/(c*x^2+a)^2-1/8*(e*x+d)*(a*(A*c+3*C*a)*e-c*(3*A*c*d+2*B*a*e+C*a*d)*x)/a^2
/c^2/(c*x^2+a)+1/8*(a*(A*c+3*C*a)*e^2+c*d*(3*A*c*d+2*B*a*e+C*a*d))*arctan(x*c^(1/2)/a^(1/2))/a^(5/2)/c^(5/2)

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Rubi [A]  time = 0.23, antiderivative size = 175, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1645, 778, 205} \[ -\frac {x \left (a e^2 (3 a C+A c)-c d (2 a B e+a C d+3 A c d)\right )+2 a e (a B e+2 a C d+2 A c d)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right )}{8 a^{5/2} c^{5/2}}-\frac {(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*B - (A*c - a*C)*x)*(d + e*x)^2)/(4*a*c*(a + c*x^2)^2) - (2*a*e*(2*A*c*d + 2*a*C*d + a*B*e) + (a*(A*c + 3*
a*C)*e^2 - c*d*(3*A*c*d + a*C*d + 2*a*B*e))*x)/(8*a^2*c^2*(a + c*x^2)) + ((a*(A*c + 3*a*C)*e^2 + c*d*(3*A*c*d
+ a*C*d + 2*a*B*e))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 1645

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*c*(p
+ 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p
+ 3) + c*e*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac {\int \frac {(d+e x) (-3 A c d-a C d-2 a B e-(A c+3 a C) e x)}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac {2 a e (2 A c d+2 a C d+a B e)+\left (a (A c+3 a C) e^2-c d (3 A c d+a C d+2 a B e)\right ) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (a (A c+3 a C) e^2+c d (3 A c d+a C d+2 a B e)\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {(a B-(A c-a C) x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac {2 a e (2 A c d+2 a C d+a B e)+\left (a (A c+3 a C) e^2-c d (3 A c d+a C d+2 a B e)\right ) x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (a (A c+3 a C) e^2+c d (3 A c d+a C d+2 a B e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 211, normalized size = 1.35 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c \left (a e^2+3 c d^2\right )+a \left (3 a C e^2+c d (2 B e+C d)\right )\right )}{8 a^{5/2} c^{5/2}}+\frac {a^2 (-e) (4 B e+8 C d+5 C e x)+a c x \left (e (A e+2 B d)+C d^2\right )+3 A c^2 d^2 x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {a^2 e (B e+2 C d+C e x)-a c \left (A e (2 d+e x)+B d (d+2 e x)+C d^2 x\right )+A c^2 d^2 x}{4 a c^2 \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*A*c^2*d^2*x + a*c*(C*d^2 + e*(2*B*d + A*e))*x - a^2*e*(8*C*d + 4*B*e + 5*C*e*x))/(8*a^2*c^2*(a + c*x^2)) +
(A*c^2*d^2*x + a^2*e*(2*C*d + B*e + C*e*x) - a*c*(C*d^2*x + A*e*(2*d + e*x) + B*d*(d + 2*e*x)))/(4*a*c^2*(a +
c*x^2)^2) + ((A*c*(3*c*d^2 + a*e^2) + a*(3*a*C*e^2 + c*d*(C*d + 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/
2)*c^(5/2))

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fricas [B]  time = 1.22, size = 806, normalized size = 5.17 \[ \left [-\frac {4 \, B a^{3} c^{2} d^{2} + 4 \, B a^{4} c e^{2} - 2 \, {\left (2 \, B a^{2} c^{3} d e + {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{2} - {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} e^{2}\right )} x^{3} + 8 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d e + 8 \, {\left (2 \, C a^{3} c^{2} d e + B a^{3} c^{2} e^{2}\right )} x^{2} + {\left (2 \, B a^{3} c d e + {\left (2 \, B a c^{3} d e + {\left (C a c^{3} + 3 \, A c^{4}\right )} d^{2} + {\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} e^{2}\right )} x^{4} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} e^{2} + 2 \, {\left (2 \, B a^{2} c^{2} d e + {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{2} + {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (2 \, B a^{3} c^{2} d e + {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{2} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} e^{2}\right )} x}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{2} + 2 \, B a^{4} c e^{2} - {\left (2 \, B a^{2} c^{3} d e + {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{2} - {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} e^{2}\right )} x^{3} + 4 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d e + 4 \, {\left (2 \, C a^{3} c^{2} d e + B a^{3} c^{2} e^{2}\right )} x^{2} - {\left (2 \, B a^{3} c d e + {\left (2 \, B a c^{3} d e + {\left (C a c^{3} + 3 \, A c^{4}\right )} d^{2} + {\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} e^{2}\right )} x^{4} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} e^{2} + 2 \, {\left (2 \, B a^{2} c^{2} d e + {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{2} + {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (2 \, B a^{3} c^{2} d e + {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{2} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} e^{2}\right )} x}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*B*a^3*c^2*d^2 + 4*B*a^4*c*e^2 - 2*(2*B*a^2*c^3*d*e + (C*a^2*c^3 + 3*A*a*c^4)*d^2 - (5*C*a^3*c^2 - A*
a^2*c^3)*e^2)*x^3 + 8*(C*a^4*c + A*a^3*c^2)*d*e + 8*(2*C*a^3*c^2*d*e + B*a^3*c^2*e^2)*x^2 + (2*B*a^3*c*d*e + (
2*B*a*c^3*d*e + (C*a*c^3 + 3*A*c^4)*d^2 + (3*C*a^2*c^2 + A*a*c^3)*e^2)*x^4 + (C*a^3*c + 3*A*a^2*c^2)*d^2 + (3*
C*a^4 + A*a^3*c)*e^2 + 2*(2*B*a^2*c^2*d*e + (C*a^2*c^2 + 3*A*a*c^3)*d^2 + (3*C*a^3*c + A*a^2*c^2)*e^2)*x^2)*sq
rt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(2*B*a^3*c^2*d*e + (C*a^3*c^2 - 5*A*a^2*c^3)*d^2 +
(3*C*a^4*c + A*a^3*c^2)*e^2)*x)/(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c^3), -1/8*(2*B*a^3*c^2*d^2 + 2*B*a^4*c*e^2
- (2*B*a^2*c^3*d*e + (C*a^2*c^3 + 3*A*a*c^4)*d^2 - (5*C*a^3*c^2 - A*a^2*c^3)*e^2)*x^3 + 4*(C*a^4*c + A*a^3*c^
2)*d*e + 4*(2*C*a^3*c^2*d*e + B*a^3*c^2*e^2)*x^2 - (2*B*a^3*c*d*e + (2*B*a*c^3*d*e + (C*a*c^3 + 3*A*c^4)*d^2 +
(3*C*a^2*c^2 + A*a*c^3)*e^2)*x^4 + (C*a^3*c + 3*A*a^2*c^2)*d^2 + (3*C*a^4 + A*a^3*c)*e^2 + 2*(2*B*a^2*c^2*d*e
+ (C*a^2*c^2 + 3*A*a*c^3)*d^2 + (3*C*a^3*c + A*a^2*c^2)*e^2)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (2*B*a^3*
c^2*d*e + (C*a^3*c^2 - 5*A*a^2*c^3)*d^2 + (3*C*a^4*c + A*a^3*c^2)*e^2)*x)/(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c
^3)]

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giac [A]  time = 0.16, size = 254, normalized size = 1.63 \[ \frac {{\left (C a c d^{2} + 3 \, A c^{2} d^{2} + 2 \, B a c d e + 3 \, C a^{2} e^{2} + A a c e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {C a c^{2} d^{2} x^{3} + 3 \, A c^{3} d^{2} x^{3} + 2 \, B a c^{2} d x^{3} e - 5 \, C a^{2} c x^{3} e^{2} + A a c^{2} x^{3} e^{2} - 8 \, C a^{2} c d x^{2} e - C a^{2} c d^{2} x + 5 \, A a c^{2} d^{2} x - 4 \, B a^{2} c x^{2} e^{2} - 2 \, B a^{2} c d x e - 2 \, B a^{2} c d^{2} - 3 \, C a^{3} x e^{2} - A a^{2} c x e^{2} - 4 \, C a^{3} d e - 4 \, A a^{2} c d e - 2 \, B a^{3} e^{2}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(C*a*c*d^2 + 3*A*c^2*d^2 + 2*B*a*c*d*e + 3*C*a^2*e^2 + A*a*c*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2
) + 1/8*(C*a*c^2*d^2*x^3 + 3*A*c^3*d^2*x^3 + 2*B*a*c^2*d*x^3*e - 5*C*a^2*c*x^3*e^2 + A*a*c^2*x^3*e^2 - 8*C*a^2
*c*d*x^2*e - C*a^2*c*d^2*x + 5*A*a*c^2*d^2*x - 4*B*a^2*c*x^2*e^2 - 2*B*a^2*c*d*x*e - 2*B*a^2*c*d^2 - 3*C*a^3*x
*e^2 - A*a^2*c*x*e^2 - 4*C*a^3*d*e - 4*A*a^2*c*d*e - 2*B*a^3*e^2)/((c*x^2 + a)^2*a^2*c^2)

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maple [A]  time = 0.01, size = 283, normalized size = 1.81 \[ \frac {A \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}+\frac {3 A \,d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}+\frac {B d e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, a c}+\frac {C \,d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}+\frac {3 C \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, c^{2}}+\frac {-\frac {\left (B e +2 C d \right ) e \,x^{2}}{2 c}+\frac {\left (A a c \,e^{2}+3 A \,c^{2} d^{2}+2 B a c d e -5 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{3}}{8 a^{2} c}-\frac {\left (A a c \,e^{2}-5 A \,c^{2} d^{2}+2 B a c d e +3 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{8 a \,c^{2}}-\frac {2 A c d e +B a \,e^{2}+B c \,d^{2}+2 C a d e}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(C*x^2+B*x+A)/(c*x^2+a)^3,x)

[Out]

(1/8*(A*a*c*e^2+3*A*c^2*d^2+2*B*a*c*d*e-5*C*a^2*e^2+C*a*c*d^2)/a^2/c*x^3-1/2*e*(B*e+2*C*d)*x^2/c-1/8*(A*a*c*e^
2-5*A*c^2*d^2+2*B*a*c*d*e+3*C*a^2*e^2+C*a*c*d^2)/a/c^2*x-1/4*(2*A*c*d*e+B*a*e^2+B*c*d^2+2*C*a*d*e)/c^2)/(c*x^2
+a)^2+1/8/a/c/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*e^2+3/8/a^2/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^2+
1/4/a/c/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*d*e+3/8/c^2/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*e^2+1/8/a/
c/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*d^2

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maxima [A]  time = 0.99, size = 253, normalized size = 1.62 \[ -\frac {2 \, B a^{2} c d^{2} + 2 \, B a^{3} e^{2} - {\left (2 \, B a c^{2} d e + {\left (C a c^{2} + 3 \, A c^{3}\right )} d^{2} - {\left (5 \, C a^{2} c - A a c^{2}\right )} e^{2}\right )} x^{3} + 4 \, {\left (C a^{3} + A a^{2} c\right )} d e + 4 \, {\left (2 \, C a^{2} c d e + B a^{2} c e^{2}\right )} x^{2} + {\left (2 \, B a^{2} c d e + {\left (C a^{2} c - 5 \, A a c^{2}\right )} d^{2} + {\left (3 \, C a^{3} + A a^{2} c\right )} e^{2}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {{\left (2 \, B a c d e + {\left (C a c + 3 \, A c^{2}\right )} d^{2} + {\left (3 \, C a^{2} + A a c\right )} e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(C*x^2+B*x+A)/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/8*(2*B*a^2*c*d^2 + 2*B*a^3*e^2 - (2*B*a*c^2*d*e + (C*a*c^2 + 3*A*c^3)*d^2 - (5*C*a^2*c - A*a*c^2)*e^2)*x^3
+ 4*(C*a^3 + A*a^2*c)*d*e + 4*(2*C*a^2*c*d*e + B*a^2*c*e^2)*x^2 + (2*B*a^2*c*d*e + (C*a^2*c - 5*A*a*c^2)*d^2 +
(3*C*a^3 + A*a^2*c)*e^2)*x)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2) + 1/8*(2*B*a*c*d*e + (C*a*c + 3*A*c^2)*d^
2 + (3*C*a^2 + A*a*c)*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2)

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mupad [B]  time = 3.96, size = 230, normalized size = 1.47 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+3\,A\,c^2\,d^2\right )}{8\,a^{5/2}\,c^{5/2}}-\frac {\frac {B\,a\,e^2+B\,c\,d^2+2\,A\,c\,d\,e+2\,C\,a\,d\,e}{4\,c^2}+\frac {x^2\,\left (B\,e^2+2\,C\,d\,e\right )}{2\,c}+\frac {x\,\left (3\,C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2-5\,A\,c^2\,d^2\right )}{8\,a\,c^2}-\frac {x^3\,\left (-5\,C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+3\,A\,c^2\,d^2\right )}{8\,a^2\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)^2*(A + B*x + C*x^2))/(a + c*x^2)^3,x)

[Out]

(atan((c^(1/2)*x)/a^(1/2))*(3*A*c^2*d^2 + 3*C*a^2*e^2 + A*a*c*e^2 + C*a*c*d^2 + 2*B*a*c*d*e))/(8*a^(5/2)*c^(5/
2)) - ((B*a*e^2 + B*c*d^2 + 2*A*c*d*e + 2*C*a*d*e)/(4*c^2) + (x^2*(B*e^2 + 2*C*d*e))/(2*c) + (x*(3*C*a^2*e^2 -
5*A*c^2*d^2 + A*a*c*e^2 + C*a*c*d^2 + 2*B*a*c*d*e))/(8*a*c^2) - (x^3*(3*A*c^2*d^2 - 5*C*a^2*e^2 + A*a*c*e^2 +
C*a*c*d^2 + 2*B*a*c*d*e))/(8*a^2*c))/(a^2 + c^2*x^4 + 2*a*c*x^2)

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sympy [B]  time = 141.18, size = 391, normalized size = 2.51 \[ - \frac {\sqrt {- \frac {1}{a^{5} c^{5}}} \left (A a c e^{2} + 3 A c^{2} d^{2} + 2 B a c d e + 3 C a^{2} e^{2} + C a c d^{2}\right ) \log {\left (- a^{3} c^{2} \sqrt {- \frac {1}{a^{5} c^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} c^{5}}} \left (A a c e^{2} + 3 A c^{2} d^{2} + 2 B a c d e + 3 C a^{2} e^{2} + C a c d^{2}\right ) \log {\left (a^{3} c^{2} \sqrt {- \frac {1}{a^{5} c^{5}}} + x \right )}}{16} + \frac {- 4 A a^{2} c d e - 2 B a^{3} e^{2} - 2 B a^{2} c d^{2} - 4 C a^{3} d e + x^{3} \left (A a c^{2} e^{2} + 3 A c^{3} d^{2} + 2 B a c^{2} d e - 5 C a^{2} c e^{2} + C a c^{2} d^{2}\right ) + x^{2} \left (- 4 B a^{2} c e^{2} - 8 C a^{2} c d e\right ) + x \left (- A a^{2} c e^{2} + 5 A a c^{2} d^{2} - 2 B a^{2} c d e - 3 C a^{3} e^{2} - C a^{2} c d^{2}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(C*x**2+B*x+A)/(c*x**2+a)**3,x)

[Out]

-sqrt(-1/(a**5*c**5))*(A*a*c*e**2 + 3*A*c**2*d**2 + 2*B*a*c*d*e + 3*C*a**2*e**2 + C*a*c*d**2)*log(-a**3*c**2*s
qrt(-1/(a**5*c**5)) + x)/16 + sqrt(-1/(a**5*c**5))*(A*a*c*e**2 + 3*A*c**2*d**2 + 2*B*a*c*d*e + 3*C*a**2*e**2 +
C*a*c*d**2)*log(a**3*c**2*sqrt(-1/(a**5*c**5)) + x)/16 + (-4*A*a**2*c*d*e - 2*B*a**3*e**2 - 2*B*a**2*c*d**2 -
4*C*a**3*d*e + x**3*(A*a*c**2*e**2 + 3*A*c**3*d**2 + 2*B*a*c**2*d*e - 5*C*a**2*c*e**2 + C*a*c**2*d**2) + x**2
*(-4*B*a**2*c*e**2 - 8*C*a**2*c*d*e) + x*(-A*a**2*c*e**2 + 5*A*a*c**2*d**2 - 2*B*a**2*c*d*e - 3*C*a**3*e**2 -
C*a**2*c*d**2))/(8*a**4*c**2 + 16*a**3*c**3*x**2 + 8*a**2*c**4*x**4)

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