### 3.29 $$\int \frac{(a+c x^2)^2 (A+B x+C x^2)}{d+e x} \, dx$$

Optimal. Leaf size=297 $\frac{x^2 \left (a^2 C e^4+2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 d^2 \left (C d^2-e (B d-A e)\right )\right )}{2 e^5}-\frac{x \left (a^2 e^4 (C d-B e)+2 a c d e^2 \left (C d^2-e (B d-A e)\right )+c^2 d^3 \left (C d^2-e (B d-A e)\right )\right )}{e^6}+\frac{c x^4 \left (2 a C e^2+c \left (C d^2-e (B d-A e)\right )\right )}{4 e^3}-\frac{c x^3 \left (2 a e^2 (C d-B e)+c d \left (C d^2-e (B d-A e)\right )\right )}{3 e^4}+\frac{\left (a e^2+c d^2\right )^2 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^7}-\frac{c^2 x^5 (C d-B e)}{5 e^2}+\frac{c^2 C x^6}{6 e}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.6399, antiderivative size = 295, 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} $\frac{x^2 \left (a^2 C e^4+2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^4-d^2 e (B d-A e)\right )\right )}{2 e^5}-\frac{x \left (a^2 e^4 (C d-B e)+2 a c d e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^5-d^3 e (B d-A e)\right )\right )}{e^6}+\frac{c x^4 \left (2 a C e^2-c e (B d-A e)+c C d^2\right )}{4 e^3}-\frac{c x^3 \left (2 a e^2 (C d-B e)-c d e (B d-A e)+c C d^3\right )}{3 e^4}+\frac{\left (a e^2+c d^2\right )^2 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^7}-\frac{c^2 x^5 (C d-B e)}{5 e^2}+\frac{c^2 C x^6}{6 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.16785, size = 285, normalized size = 0.96 $\frac{e x \left (30 a^2 e^4 (2 B e-2 C d+C e x)+10 a c e^2 \left (2 e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+C \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+c^2 \left (e \left (5 A e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+B \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+C \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )+60 \left (a e^2+c d^2\right )^2 \log (d+e x) \left (e (A e-B d)+C d^2\right )}{60 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*x*(30*a^2*e^4*(-2*C*d + 2*B*e + C*e*x) + 10*a*c*e^2*(C*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*
e*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + c^2*(C*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15
*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + e*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d
^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)))) + 60*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A
*e))*Log[d + e*x])/(60*e^7)

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 490, normalized size = 1.7 \begin{align*} -2\,{\frac{\ln \left ( ex+d \right ) Bac{d}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{4}}{{e}^{5}}}-{\frac{C{c}^{2}{d}^{5}x}{{e}^{6}}}-{\frac{\ln \left ( ex+d \right ) B{a}^{2}d}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{5}}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ) C{a}^{2}{d}^{2}}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) C{c}^{2}{d}^{6}}{{e}^{7}}}-{\frac{{a}^{2}Cdx}{{e}^{2}}}+{\frac{C{x}^{2}{c}^{2}{d}^{4}}{2\,{e}^{5}}}-{\frac{A{d}^{3}{c}^{2}x}{{e}^{4}}}+{\frac{A{x}^{2}ac}{e}}+{\frac{B{c}^{2}{x}^{3}{d}^{2}}{3\,{e}^{3}}}+{\frac{B{c}^{2}{d}^{4}x}{{e}^{5}}}-{\frac{B{c}^{2}{x}^{2}{d}^{3}}{2\,{e}^{4}}}+{\frac{A{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{A{x}^{3}{c}^{2}d}{3\,{e}^{2}}}+{\frac{C{x}^{4}{c}^{2}{d}^{2}}{4\,{e}^{3}}}-{\frac{B{c}^{2}{x}^{4}d}{4\,{e}^{2}}}+{\frac{2\,B{x}^{3}ac}{3\,e}}-{\frac{C{x}^{3}{c}^{2}{d}^{3}}{3\,{e}^{4}}}+2\,{\frac{\ln \left ( ex+d \right ) Cac{d}^{4}}{{e}^{5}}}-2\,{\frac{acC{d}^{3}x}{{e}^{4}}}+{\frac{C{x}^{4}ac}{2\,e}}-{\frac{C{x}^{5}{c}^{2}d}{5\,{e}^{2}}}+{\frac{{a}^{2}Bx}{e}}+{\frac{A{c}^{2}{x}^{4}}{4\,e}}+{\frac{B{c}^{2}{x}^{5}}{5\,e}}+{\frac{\ln \left ( ex+d \right ) A{a}^{2}}{e}}+{\frac{C{x}^{2}{a}^{2}}{2\,e}}+2\,{\frac{aBc{d}^{2}x}{{e}^{3}}}+2\,{\frac{\ln \left ( ex+d \right ) Aac{d}^{2}}{{e}^{3}}}-{\frac{2\,C{x}^{3}acd}{3\,{e}^{2}}}-{\frac{aB{x}^{2}cd}{{e}^{2}}}+{\frac{C{x}^{2}ac{d}^{2}}{{e}^{3}}}-2\,{\frac{aAcdx}{{e}^{2}}}+{\frac{C{c}^{2}{x}^{6}}{6\,e}} \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),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.998783, size = 509, normalized size = 1.71 \begin{align*} \frac{10 \, C c^{2} e^{5} x^{6} - 12 \,{\left (C c^{2} d e^{4} - B c^{2} e^{5}\right )} x^{5} + 15 \,{\left (C c^{2} d^{2} e^{3} - B c^{2} d e^{4} +{\left (2 \, C a c + A c^{2}\right )} e^{5}\right )} x^{4} - 20 \,{\left (C c^{2} d^{3} e^{2} - B c^{2} d^{2} e^{3} - 2 \, B a c e^{5} +{\left (2 \, C a c + A c^{2}\right )} d e^{4}\right )} x^{3} + 30 \,{\left (C c^{2} d^{4} e - B c^{2} d^{3} e^{2} - 2 \, B a c d e^{4} +{\left (2 \, C a c + A c^{2}\right )} d^{2} e^{3} +{\left (C a^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} - 60 \,{\left (C c^{2} d^{5} - B c^{2} d^{4} e - 2 \, B a c d^{2} e^{3} - B a^{2} e^{5} +{\left (2 \, C a c + A c^{2}\right )} d^{3} e^{2} +{\left (C a^{2} + 2 \, A a c\right )} d e^{4}\right )} x}{60 \, e^{6}} + \frac{{\left (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}\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),x, algorithm="maxima")

[Out]

1/60*(10*C*c^2*e^5*x^6 - 12*(C*c^2*d*e^4 - B*c^2*e^5)*x^5 + 15*(C*c^2*d^2*e^3 - B*c^2*d*e^4 + (2*C*a*c + A*c^2
)*e^5)*x^4 - 20*(C*c^2*d^3*e^2 - B*c^2*d^2*e^3 - 2*B*a*c*e^5 + (2*C*a*c + A*c^2)*d*e^4)*x^3 + 30*(C*c^2*d^4*e
- B*c^2*d^3*e^2 - 2*B*a*c*d*e^4 + (2*C*a*c + A*c^2)*d^2*e^3 + (C*a^2 + 2*A*a*c)*e^5)*x^2 - 60*(C*c^2*d^5 - B*c
^2*d^4*e - 2*B*a*c*d^2*e^3 - B*a^2*e^5 + (2*C*a*c + A*c^2)*d^3*e^2 + (C*a^2 + 2*A*a*c)*d*e^4)*x)/e^6 + (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)*log(e*x + d)/e^7

________________________________________________________________________________________

Fricas [A]  time = 1.75467, size = 786, normalized size = 2.65 \begin{align*} \frac{10 \, C c^{2} e^{6} x^{6} - 12 \,{\left (C c^{2} d e^{5} - B c^{2} e^{6}\right )} x^{5} + 15 \,{\left (C c^{2} d^{2} e^{4} - B c^{2} d e^{5} +{\left (2 \, C a c + A c^{2}\right )} e^{6}\right )} x^{4} - 20 \,{\left (C c^{2} d^{3} e^{3} - B c^{2} d^{2} e^{4} - 2 \, B a c e^{6} +{\left (2 \, C a c + A c^{2}\right )} d e^{5}\right )} x^{3} + 30 \,{\left (C c^{2} d^{4} e^{2} - B c^{2} d^{3} e^{3} - 2 \, B a c d e^{5} +{\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4} +{\left (C a^{2} + 2 \, A a c\right )} e^{6}\right )} x^{2} - 60 \,{\left (C c^{2} d^{5} e - B c^{2} d^{4} e^{2} - 2 \, B a c d^{2} e^{4} - B a^{2} e^{6} +{\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 (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}\right )} \log \left (e x + d\right )}{60 \, 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),x, algorithm="fricas")

[Out]

1/60*(10*C*c^2*e^6*x^6 - 12*(C*c^2*d*e^5 - B*c^2*e^6)*x^5 + 15*(C*c^2*d^2*e^4 - B*c^2*d*e^5 + (2*C*a*c + A*c^2
)*e^6)*x^4 - 20*(C*c^2*d^3*e^3 - B*c^2*d^2*e^4 - 2*B*a*c*e^6 + (2*C*a*c + A*c^2)*d*e^5)*x^3 + 30*(C*c^2*d^4*e^
2 - B*c^2*d^3*e^3 - 2*B*a*c*d*e^5 + (2*C*a*c + A*c^2)*d^2*e^4 + (C*a^2 + 2*A*a*c)*e^6)*x^2 - 60*(C*c^2*d^5*e -
B*c^2*d^4*e^2 - 2*B*a*c*d^2*e^4 - B*a^2*e^6 + (2*C*a*c + A*c^2)*d^3*e^3 + (C*a^2 + 2*A*a*c)*d*e^5)*x + 60*(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)*log(e*x + d))/e^7

________________________________________________________________________________________

Sympy [A]  time = 1.46908, size = 350, normalized size = 1.18 \begin{align*} \frac{C c^{2} x^{6}}{6 e} - \frac{x^{5} \left (- B c^{2} e + C c^{2} d\right )}{5 e^{2}} + \frac{x^{4} \left (A c^{2} e^{2} - B c^{2} d e + 2 C a c e^{2} + C c^{2} d^{2}\right )}{4 e^{3}} - \frac{x^{3} \left (A c^{2} d e^{2} - 2 B a c e^{3} - B c^{2} d^{2} e + 2 C a c d e^{2} + C c^{2} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (2 A a c e^{4} + A c^{2} d^{2} e^{2} - 2 B a c d e^{3} - B c^{2} d^{3} e + C a^{2} e^{4} + 2 C a c d^{2} e^{2} + C c^{2} d^{4}\right )}{2 e^{5}} - \frac{x \left (2 A a c d e^{4} + A c^{2} d^{3} e^{2} - B a^{2} e^{5} - 2 B a c d^{2} e^{3} - B c^{2} d^{4} e + C a^{2} d e^{4} + 2 C a c d^{3} e^{2} + C c^{2} d^{5}\right )}{e^{6}} + \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (A e^{2} - B d e + C d^{2}\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),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13939, size = 562, normalized size = 1.89 \begin{align*}{\left (C c^{2} d^{6} - B c^{2} d^{5} e + 2 \, C a c d^{4} e^{2} + A c^{2} d^{4} e^{2} - 2 \, B a c d^{3} e^{3} + C a^{2} d^{2} e^{4} + 2 \, A a c d^{2} e^{4} - B a^{2} d e^{5} + A a^{2} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, C c^{2} x^{6} e^{5} - 12 \, C c^{2} d x^{5} e^{4} + 15 \, C c^{2} d^{2} x^{4} e^{3} - 20 \, C c^{2} d^{3} x^{3} e^{2} + 30 \, C c^{2} d^{4} x^{2} e - 60 \, C c^{2} d^{5} x + 12 \, B c^{2} x^{5} e^{5} - 15 \, B c^{2} d x^{4} e^{4} + 20 \, B c^{2} d^{2} x^{3} e^{3} - 30 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 30 \, C a c x^{4} e^{5} + 15 \, A c^{2} x^{4} e^{5} - 40 \, C a c d x^{3} e^{4} - 20 \, A c^{2} d x^{3} e^{4} + 60 \, C a c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} - 120 \, C a c d^{3} x e^{2} - 60 \, A c^{2} d^{3} x e^{2} + 40 \, B a c x^{3} e^{5} - 60 \, B a c d x^{2} e^{4} + 120 \, B a c d^{2} x e^{3} + 30 \, C a^{2} x^{2} e^{5} + 60 \, A a c x^{2} e^{5} - 60 \, C a^{2} d x e^{4} - 120 \, A a c d x e^{4} + 60 \, B a^{2} x e^{5}\right )} e^{\left (-6\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),x, algorithm="giac")

[Out]

(C*c^2*d^6 - B*c^2*d^5*e + 2*C*a*c*d^4*e^2 + A*c^2*d^4*e^2 - 2*B*a*c*d^3*e^3 + C*a^2*d^2*e^4 + 2*A*a*c*d^2*e^4
- B*a^2*d*e^5 + A*a^2*e^6)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*C*c^2*x^6*e^5 - 12*C*c^2*d*x^5*e^4 + 15*C*c^2*
d^2*x^4*e^3 - 20*C*c^2*d^3*x^3*e^2 + 30*C*c^2*d^4*x^2*e - 60*C*c^2*d^5*x + 12*B*c^2*x^5*e^5 - 15*B*c^2*d*x^4*e
^4 + 20*B*c^2*d^2*x^3*e^3 - 30*B*c^2*d^3*x^2*e^2 + 60*B*c^2*d^4*x*e + 30*C*a*c*x^4*e^5 + 15*A*c^2*x^4*e^5 - 40
*C*a*c*d*x^3*e^4 - 20*A*c^2*d*x^3*e^4 + 60*C*a*c*d^2*x^2*e^3 + 30*A*c^2*d^2*x^2*e^3 - 120*C*a*c*d^3*x*e^2 - 60
*A*c^2*d^3*x*e^2 + 40*B*a*c*x^3*e^5 - 60*B*a*c*d*x^2*e^4 + 120*B*a*c*d^2*x*e^3 + 30*C*a^2*x^2*e^5 + 60*A*a*c*x
^2*e^5 - 60*C*a^2*d*x*e^4 - 120*A*a*c*d*x*e^4 + 60*B*a^2*x*e^5)*e^(-6)