### 3.30 $$\int \frac{(a+c x^2)^2 (A+B x+C x^2)}{(d+e x)^2} \, dx$$

Optimal. Leaf size=292 $\frac{x \left (a^2 C e^4+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 d^2 \left (5 C d^2-e (4 B d-3 A e)\right )\right )}{e^6}+\frac{c x^3 \left (2 a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right )}{3 e^4}-\frac{c x^2 \left (2 a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{2 e^5}-\frac{\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)+c d \left (6 C d^2-e (5 B d-4 A e)\right )\right )}{e^7}-\frac{c^2 x^4 (2 C d-B e)}{4 e^3}+\frac{c^2 C x^5}{5 e^2}$

[Out]

((a^2*C*e^4 + c^2*d^2*(5*C*d^2 - e*(4*B*d - 3*A*e)) + 2*a*c*e^2*(3*C*d^2 - e*(2*B*d - A*e)))*x)/e^6 - (c*(2*a*
e^2*(2*C*d - B*e) + c*d*(4*C*d^2 - e*(3*B*d - 2*A*e)))*x^2)/(2*e^5) + (c*(2*a*C*e^2 + c*(3*C*d^2 - e*(2*B*d -
A*e)))*x^3)/(3*e^4) - (c^2*(2*C*d - B*e)*x^4)/(4*e^3) + (c^2*C*x^5)/(5*e^2) - ((c*d^2 + a*e^2)^2*(C*d^2 - B*d*
e + A*e^2))/(e^7*(d + e*x)) - ((c*d^2 + a*e^2)*(a*e^2*(2*C*d - B*e) + c*d*(6*C*d^2 - e*(5*B*d - 4*A*e)))*Log[d
+ e*x])/e^7

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Rubi [A]  time = 0.525428, antiderivative size = 289, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.037, Rules used = {1628} $\frac{x \left (a^2 C e^4+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )\right )}{e^6}+\frac{c x^3 \left (2 a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{3 e^4}-\frac{c x^2 \left (2 a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{2 e^5}-\frac{\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)-c d e (5 B d-4 A e)+6 c C d^3\right )}{e^7}-\frac{c^2 x^4 (2 C d-B e)}{4 e^3}+\frac{c^2 C x^5}{5 e^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a^2*C*e^4 + c^2*(5*C*d^4 - d^2*e*(4*B*d - 3*A*e)) + 2*a*c*e^2*(3*C*d^2 - e*(2*B*d - A*e)))*x)/e^6 - (c*(4*c*
C*d^3 - c*d*e*(3*B*d - 2*A*e) + 2*a*e^2*(2*C*d - B*e))*x^2)/(2*e^5) + (c*(3*c*C*d^2 + 2*a*C*e^2 - c*e*(2*B*d -
A*e))*x^3)/(3*e^4) - (c^2*(2*C*d - B*e)*x^4)/(4*e^3) + (c^2*C*x^5)/(5*e^2) - ((c*d^2 + a*e^2)^2*(C*d^2 - B*d*
e + A*e^2))/(e^7*(d + e*x)) - ((c*d^2 + a*e^2)*(6*c*C*d^3 - c*d*e*(5*B*d - 4*A*e) + a*e^2*(2*C*d - B*e))*Log[d
+ e*x])/e^7

Rule 1628

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{\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac{a^2 C e^4+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )}{e^6}+\frac{c \left (-4 c C d^3+c d e (3 B d-2 A e)-2 a e^2 (2 C d-B e)\right ) x}{e^5}+\frac{c \left (3 c C d^2+2 a C e^2-c e (2 B d-A e)\right ) x^2}{e^4}+\frac{c^2 (-2 C d+B e) x^3}{e^3}+\frac{c^2 C x^4}{e^2}+\frac{\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right )}{e^6 (d+e x)^2}+\frac{\left (c d^2+a e^2\right ) \left (-6 c C d^3+c d e (5 B d-4 A e)-a e^2 (2 C d-B e)\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{\left (a^2 C e^4+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )\right ) x}{e^6}-\frac{c \left (4 c C d^3-c d e (3 B d-2 A e)+2 a e^2 (2 C d-B e)\right ) x^2}{2 e^5}+\frac{c \left (3 c C d^2+2 a C e^2-c e (2 B d-A e)\right ) x^3}{3 e^4}-\frac{c^2 (2 C d-B e) x^4}{4 e^3}+\frac{c^2 C x^5}{5 e^2}-\frac{\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right )}{e^7 (d+e x)}-\frac{\left (c d^2+a e^2\right ) \left (6 c C d^3-c d e (5 B d-4 A e)+a e^2 (2 C d-B e)\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.291287, size = 272, normalized size = 0.93 $\frac{60 e x \left (a^2 C e^4+2 a c e^2 \left (e (A e-2 B d)+3 C d^2\right )+c^2 \left (d^2 e (3 A e-4 B d)+5 C d^4\right )\right )+20 c e^3 x^3 \left (2 a C e^2+c e (A e-2 B d)+3 c C d^2\right )-30 c e^2 x^2 \left (-2 a e^2 (B e-2 C d)+c d e (2 A e-3 B d)+4 c C d^3\right )-\frac{60 \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{d+e x}-60 \left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)+c d e (4 A e-5 B d)+6 c C d^3\right )+15 c^2 e^4 x^4 (B e-2 C d)+12 c^2 C e^5 x^5}{60 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(60*e*(a^2*C*e^4 + 2*a*c*e^2*(3*C*d^2 + e*(-2*B*d + A*e)) + c^2*(5*C*d^4 + d^2*e*(-4*B*d + 3*A*e)))*x - 30*c*e
^2*(4*c*C*d^3 + c*d*e*(-3*B*d + 2*A*e) - 2*a*e^2*(-2*C*d + B*e))*x^2 + 20*c*e^3*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(
-2*B*d + A*e))*x^3 + 15*c^2*e^4*(-2*C*d + B*e)*x^4 + 12*c^2*C*e^5*x^5 - (60*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*
d) + A*e)))/(d + e*x) - 60*(c*d^2 + a*e^2)*(6*c*C*d^3 + c*d*e*(-5*B*d + 4*A*e) + a*e^2*(2*C*d - B*e))*Log[d +
e*x])/(60*e^7)

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Maple [A]  time = 0.058, size = 527, normalized size = 1.8 \begin{align*} -4\,{\frac{B{c}^{2}{d}^{3}x}{{e}^{5}}}+5\,{\frac{C{c}^{2}{d}^{4}x}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ) B{a}^{2}}{{e}^{2}}}+{\frac{{a}^{2}Cx}{{e}^{2}}}+{\frac{A{x}^{3}{c}^{2}}{3\,{e}^{2}}}+5\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{4}}{{e}^{6}}}+2\,{\frac{aAcx}{{e}^{2}}}+{\frac{2\,C{x}^{3}ac}{3\,{e}^{2}}}+{\frac{C{x}^{3}{c}^{2}{d}^{2}}{{e}^{4}}}+{\frac{Bd{a}^{2}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{B{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{a}^{2}C{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{C{c}^{2}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{aB{x}^{2}c}{{e}^{2}}}+{\frac{B{c}^{2}{x}^{4}}{4\,{e}^{2}}}+3\,{\frac{A{c}^{2}{d}^{2}x}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{3}}{{e}^{5}}}-{\frac{C{x}^{4}{c}^{2}d}{2\,{e}^{3}}}-{\frac{2\,B{c}^{2}{x}^{3}d}{3\,{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ) C{a}^{2}d}{{e}^{3}}}+{\frac{3\,B{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{4}}}-2\,{\frac{C{x}^{2}{c}^{2}{d}^{3}}{{e}^{5}}}-6\,{\frac{\ln \left ( ex+d \right ) C{c}^{2}{d}^{5}}{{e}^{7}}}-{\frac{A{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{A{x}^{2}{c}^{2}d}{{e}^{3}}}-{\frac{{a}^{2}A}{e \left ( ex+d \right ) }}-8\,{\frac{\ln \left ( ex+d \right ) Cac{d}^{3}}{{e}^{5}}}-2\,{\frac{aAc{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{aBc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-2\,{\frac{aCc{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+6\,{\frac{aCc{d}^{2}x}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ) Aacd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) Bac{d}^{2}}{{e}^{4}}}-2\,{\frac{C{x}^{2}acd}{{e}^{3}}}-4\,{\frac{acdBx}{{e}^{3}}}+{\frac{C{c}^{2}{x}^{5}}{5\,{e}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/e^5*B*c^2*d^3*x+5/e^6*C*c^2*d^4*x+1/e^2*ln(e*x+d)*B*a^2+1/e^2*a^2*C*x+1/3/e^2*A*x^3*c^2+5/e^6*ln(e*x+d)*B*c
^2*d^4+2/e^2*A*a*c*x+2/3/e^2*C*x^3*a*c+1/e^4*C*x^3*c^2*d^2+1/e^2/(e*x+d)*B*d*a^2+1/e^6/(e*x+d)*B*c^2*d^5-1/e^3
/(e*x+d)*C*a^2*d^2-1/e^7/(e*x+d)*C*c^2*d^6+1/e^2*B*x^2*a*c+1/4/e^2*B*x^4*c^2+3/e^4*A*c^2*d^2*x-4/e^5*ln(e*x+d)
*A*c^2*d^3-1/2/e^3*C*x^4*c^2*d-2/3/e^3*B*x^3*c^2*d-2/e^3*ln(e*x+d)*C*a^2*d+3/2/e^4*B*x^2*c^2*d^2-2/e^5*C*x^2*c
^2*d^3-6/e^7*ln(e*x+d)*C*c^2*d^5-1/e^5/(e*x+d)*A*c^2*d^4-1/e^3*A*x^2*c^2*d-1/e/(e*x+d)*A*a^2-8/e^5*ln(e*x+d)*C
*a*c*d^3-2/e^3/(e*x+d)*A*a*c*d^2+2/e^4/(e*x+d)*B*a*c*d^3-2/e^5/(e*x+d)*C*a*c*d^4+6/e^4*C*a*c*d^2*x-4/e^3*ln(e*
x+d)*A*a*c*d+6/e^4*ln(e*x+d)*B*a*c*d^2-2/e^3*C*x^2*a*c*d-4/e^3*B*a*c*d*x+1/5*c^2*C*x^5/e^2

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Maxima [A]  time = 1.01945, size = 529, normalized size = 1.81 \begin{align*} -\frac{C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} - B a^{2} d e^{5} + A a^{2} e^{6} +{\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} +{\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac{12 \, C c^{2} e^{4} x^{5} - 15 \,{\left (2 \, C c^{2} d e^{3} - B c^{2} e^{4}\right )} x^{4} + 20 \,{\left (3 \, C c^{2} d^{2} e^{2} - 2 \, B c^{2} d e^{3} +{\left (2 \, C a c + A c^{2}\right )} e^{4}\right )} x^{3} - 30 \,{\left (4 \, C c^{2} d^{3} e - 3 \, B c^{2} d^{2} e^{2} - 2 \, B a c e^{4} + 2 \,{\left (2 \, C a c + A c^{2}\right )} d e^{3}\right )} x^{2} + 60 \,{\left (5 \, C c^{2} d^{4} - 4 \, B c^{2} d^{3} e - 4 \, B a c d e^{3} + 3 \,{\left (2 \, C a c + A c^{2}\right )} d^{2} e^{2} +{\left (C a^{2} + 2 \, A a c\right )} e^{4}\right )} x}{60 \, e^{6}} - \frac{{\left (6 \, C c^{2} d^{5} - 5 \, B c^{2} d^{4} e - 6 \, B a c d^{2} e^{3} - B a^{2} e^{5} + 4 \,{\left (2 \, C a c + A c^{2}\right )} d^{3} e^{2} + 2 \,{\left (C a^{2} + 2 \, A a c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(C*c^2*d^6 - B*c^2*d^5*e - 2*B*a*c*d^3*e^3 - B*a^2*d*e^5 + A*a^2*e^6 + (2*C*a*c + A*c^2)*d^4*e^2 + (C*a^2 + 2
*A*a*c)*d^2*e^4)/(e^8*x + d*e^7) + 1/60*(12*C*c^2*e^4*x^5 - 15*(2*C*c^2*d*e^3 - B*c^2*e^4)*x^4 + 20*(3*C*c^2*d
^2*e^2 - 2*B*c^2*d*e^3 + (2*C*a*c + A*c^2)*e^4)*x^3 - 30*(4*C*c^2*d^3*e - 3*B*c^2*d^2*e^2 - 2*B*a*c*e^4 + 2*(2
*C*a*c + A*c^2)*d*e^3)*x^2 + 60*(5*C*c^2*d^4 - 4*B*c^2*d^3*e - 4*B*a*c*d*e^3 + 3*(2*C*a*c + A*c^2)*d^2*e^2 + (
C*a^2 + 2*A*a*c)*e^4)*x)/e^6 - (6*C*c^2*d^5 - 5*B*c^2*d^4*e - 6*B*a*c*d^2*e^3 - B*a^2*e^5 + 4*(2*C*a*c + A*c^2
)*d^3*e^2 + 2*(C*a^2 + 2*A*a*c)*d*e^4)*log(e*x + d)/e^7

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Fricas [A]  time = 1.73969, size = 1185, normalized size = 4.06 \begin{align*} \frac{12 \, C c^{2} e^{6} x^{6} - 60 \, C c^{2} d^{6} + 60 \, B c^{2} d^{5} e + 120 \, B a c d^{3} e^{3} + 60 \, B a^{2} d e^{5} - 60 \, A a^{2} e^{6} - 60 \,{\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} - 60 \,{\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} - 3 \,{\left (6 \, C c^{2} d e^{5} - 5 \, B c^{2} e^{6}\right )} x^{5} + 5 \,{\left (6 \, C c^{2} d^{2} e^{4} - 5 \, B c^{2} d e^{5} + 4 \,{\left (2 \, C a c + A c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (6 \, C c^{2} d^{3} e^{3} - 5 \, B c^{2} d^{2} e^{4} - 6 \, B a c e^{6} + 4 \,{\left (2 \, C a c + A c^{2}\right )} d e^{5}\right )} x^{3} + 30 \,{\left (6 \, C c^{2} d^{4} e^{2} - 5 \, B c^{2} d^{3} e^{3} - 6 \, B a c d e^{5} + 4 \,{\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4} + 2 \,{\left (C a^{2} + 2 \, A a c\right )} e^{6}\right )} x^{2} + 60 \,{\left (5 \, C c^{2} d^{5} e - 4 \, B c^{2} d^{4} e^{2} - 4 \, B a c d^{2} e^{4} + 3 \,{\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} +{\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x - 60 \,{\left (6 \, C c^{2} d^{6} - 5 \, B c^{2} d^{5} e - 6 \, B a c d^{3} e^{3} - B a^{2} d e^{5} + 4 \,{\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 2 \,{\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} +{\left (6 \, C c^{2} d^{5} e - 5 \, B c^{2} d^{4} e^{2} - 6 \, B a c d^{2} e^{4} - B a^{2} e^{6} + 4 \,{\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + 2 \,{\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{60 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*C*c^2*e^6*x^6 - 60*C*c^2*d^6 + 60*B*c^2*d^5*e + 120*B*a*c*d^3*e^3 + 60*B*a^2*d*e^5 - 60*A*a^2*e^6 - 6
0*(2*C*a*c + A*c^2)*d^4*e^2 - 60*(C*a^2 + 2*A*a*c)*d^2*e^4 - 3*(6*C*c^2*d*e^5 - 5*B*c^2*e^6)*x^5 + 5*(6*C*c^2*
d^2*e^4 - 5*B*c^2*d*e^5 + 4*(2*C*a*c + A*c^2)*e^6)*x^4 - 10*(6*C*c^2*d^3*e^3 - 5*B*c^2*d^2*e^4 - 6*B*a*c*e^6 +
4*(2*C*a*c + A*c^2)*d*e^5)*x^3 + 30*(6*C*c^2*d^4*e^2 - 5*B*c^2*d^3*e^3 - 6*B*a*c*d*e^5 + 4*(2*C*a*c + A*c^2)*
d^2*e^4 + 2*(C*a^2 + 2*A*a*c)*e^6)*x^2 + 60*(5*C*c^2*d^5*e - 4*B*c^2*d^4*e^2 - 4*B*a*c*d^2*e^4 + 3*(2*C*a*c +
A*c^2)*d^3*e^3 + (C*a^2 + 2*A*a*c)*d*e^5)*x - 60*(6*C*c^2*d^6 - 5*B*c^2*d^5*e - 6*B*a*c*d^3*e^3 - B*a^2*d*e^5
+ 4*(2*C*a*c + A*c^2)*d^4*e^2 + 2*(C*a^2 + 2*A*a*c)*d^2*e^4 + (6*C*c^2*d^5*e - 5*B*c^2*d^4*e^2 - 6*B*a*c*d^2*e
^4 - B*a^2*e^6 + 4*(2*C*a*c + A*c^2)*d^3*e^3 + 2*(C*a^2 + 2*A*a*c)*d*e^5)*x)*log(e*x + d))/(e^8*x + d*e^7)

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Sympy [A]  time = 3.76405, size = 411, normalized size = 1.41 \begin{align*} \frac{C c^{2} x^{5}}{5 e^{2}} - \frac{A a^{2} e^{6} + 2 A a c d^{2} e^{4} + A c^{2} d^{4} e^{2} - B a^{2} d e^{5} - 2 B a c d^{3} e^{3} - B c^{2} d^{5} e + C a^{2} d^{2} e^{4} + 2 C a c d^{4} e^{2} + C c^{2} d^{6}}{d e^{7} + e^{8} x} - \frac{x^{4} \left (- B c^{2} e + 2 C c^{2} d\right )}{4 e^{3}} + \frac{x^{3} \left (A c^{2} e^{2} - 2 B c^{2} d e + 2 C a c e^{2} + 3 C c^{2} d^{2}\right )}{3 e^{4}} - \frac{x^{2} \left (2 A c^{2} d e^{2} - 2 B a c e^{3} - 3 B c^{2} d^{2} e + 4 C a c d e^{2} + 4 C c^{2} d^{3}\right )}{2 e^{5}} + \frac{x \left (2 A a c e^{4} + 3 A c^{2} d^{2} e^{2} - 4 B a c d e^{3} - 4 B c^{2} d^{3} e + C a^{2} e^{4} + 6 C a c d^{2} e^{2} + 5 C c^{2} d^{4}\right )}{e^{6}} - \frac{\left (a e^{2} + c d^{2}\right ) \left (4 A c d e^{2} - B a e^{3} - 5 B c d^{2} e + 2 C a d e^{2} + 6 C c d^{3}\right ) \log{\left (d + e x \right )}}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*c**2*x**5/(5*e**2) - (A*a**2*e**6 + 2*A*a*c*d**2*e**4 + A*c**2*d**4*e**2 - B*a**2*d*e**5 - 2*B*a*c*d**3*e**3
- B*c**2*d**5*e + C*a**2*d**2*e**4 + 2*C*a*c*d**4*e**2 + C*c**2*d**6)/(d*e**7 + e**8*x) - x**4*(-B*c**2*e + 2
*C*c**2*d)/(4*e**3) + x**3*(A*c**2*e**2 - 2*B*c**2*d*e + 2*C*a*c*e**2 + 3*C*c**2*d**2)/(3*e**4) - x**2*(2*A*c*
*2*d*e**2 - 2*B*a*c*e**3 - 3*B*c**2*d**2*e + 4*C*a*c*d*e**2 + 4*C*c**2*d**3)/(2*e**5) + x*(2*A*a*c*e**4 + 3*A*
c**2*d**2*e**2 - 4*B*a*c*d*e**3 - 4*B*c**2*d**3*e + C*a**2*e**4 + 6*C*a*c*d**2*e**2 + 5*C*c**2*d**4)/e**6 - (a
*e**2 + c*d**2)*(4*A*c*d*e**2 - B*a*e**3 - 5*B*c*d**2*e + 2*C*a*d*e**2 + 6*C*c*d**3)*log(d + e*x)/e**7

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Giac [A]  time = 1.1782, size = 671, normalized size = 2.3 \begin{align*} \frac{1}{60} \,{\left (12 \, C c^{2} - \frac{15 \,{\left (6 \, C c^{2} d e - B c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (15 \, C c^{2} d^{2} e^{2} - 5 \, B c^{2} d e^{3} + 2 \, C a c e^{4} + A c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{60 \,{\left (10 \, C c^{2} d^{3} e^{3} - 5 \, B c^{2} d^{2} e^{4} + 4 \, C a c d e^{5} + 2 \, A c^{2} d e^{5} - B a c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{60 \,{\left (15 \, C c^{2} d^{4} e^{4} - 10 \, B c^{2} d^{3} e^{5} + 12 \, C a c d^{2} e^{6} + 6 \, A c^{2} d^{2} e^{6} - 6 \, B a c d e^{7} + C a^{2} e^{8} + 2 \, A a c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} +{\left (6 \, C c^{2} d^{5} - 5 \, B c^{2} d^{4} e + 8 \, C a c d^{3} e^{2} + 4 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} + 2 \, C a^{2} d e^{4} + 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{C c^{2} d^{6} e^{5}}{x e + d} - \frac{B c^{2} d^{5} e^{6}}{x e + d} + \frac{2 \, C a c d^{4} e^{7}}{x e + d} + \frac{A c^{2} d^{4} e^{7}}{x e + d} - \frac{2 \, B a c d^{3} e^{8}}{x e + d} + \frac{C a^{2} d^{2} e^{9}}{x e + d} + \frac{2 \, A a c d^{2} e^{9}}{x e + d} - \frac{B a^{2} d e^{10}}{x e + d} + \frac{A a^{2} e^{11}}{x e + d}\right )} e^{\left (-12\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*C*c^2 - 15*(6*C*c^2*d*e - B*c^2*e^2)*e^(-1)/(x*e + d) + 20*(15*C*c^2*d^2*e^2 - 5*B*c^2*d*e^3 + 2*C*a*
c*e^4 + A*c^2*e^4)*e^(-2)/(x*e + d)^2 - 60*(10*C*c^2*d^3*e^3 - 5*B*c^2*d^2*e^4 + 4*C*a*c*d*e^5 + 2*A*c^2*d*e^5
- B*a*c*e^6)*e^(-3)/(x*e + d)^3 + 60*(15*C*c^2*d^4*e^4 - 10*B*c^2*d^3*e^5 + 12*C*a*c*d^2*e^6 + 6*A*c^2*d^2*e^
6 - 6*B*a*c*d*e^7 + C*a^2*e^8 + 2*A*a*c*e^8)*e^(-4)/(x*e + d)^4)*(x*e + d)^5*e^(-7) + (6*C*c^2*d^5 - 5*B*c^2*d
^4*e + 8*C*a*c*d^3*e^2 + 4*A*c^2*d^3*e^2 - 6*B*a*c*d^2*e^3 + 2*C*a^2*d*e^4 + 4*A*a*c*d*e^4 - B*a^2*e^5)*e^(-7)
*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (C*c^2*d^6*e^5/(x*e + d) - B*c^2*d^5*e^6/(x*e + d) + 2*C*a*c*d^4*e^7/(
x*e + d) + A*c^2*d^4*e^7/(x*e + d) - 2*B*a*c*d^3*e^8/(x*e + d) + C*a^2*d^2*e^9/(x*e + d) + 2*A*a*c*d^2*e^9/(x*
e + d) - B*a^2*d*e^10/(x*e + d) + A*a^2*e^11/(x*e + d))*e^(-12)