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

Optimal. Leaf size=295 \[ \frac{\log (d+e x) \left (a^2 C e^4+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (15 C d^2-2 e (5 B d-3 A e)\right )\right )}{e^7}+\frac{c x^2 \left (2 a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right )}{2 e^5}-\frac{c x \left (2 a e^2 (3 C d-B e)+c d \left (10 C d^2-3 e (2 B d-A e)\right )\right )}{e^6}+\frac{\left (a e^2+c d^2\right ) \left (a e^2 (2 C d-B e)+c d \left (6 C d^2-e (5 B d-4 A e)\right )\right )}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (3 C d-B e)}{3 e^4}+\frac{c^2 C x^4}{4 e^3} \]

[Out]

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

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Rubi [A]  time = 0.494568, antiderivative size = 292, 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} \[ \frac{\log (d+e x) \left (a^2 C e^4+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )\right )}{e^7}+\frac{c x^2 \left (2 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{2 e^5}-\frac{c x \left (2 a e^2 (3 C d-B e)-3 c d e (2 B d-A e)+10 c C d^3\right )}{e^6}+\frac{\left (a e^2+c d^2\right ) \left (a e^2 (2 C d-B e)-c d e (5 B d-4 A e)+6 c C d^3\right )}{e^7 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (3 C d-B e)}{3 e^4}+\frac{c^2 C x^4}{4 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*(10*c*C*d^3 - 3*c*d*e*(2*B*d - A*e) + 2*a*e^2*(3*C*d - B*e))*x)/e^6) + (c*(6*c*C*d^2 + 2*a*C*e^2 - c*e*(3
*B*d - A*e))*x^2)/(2*e^5) - (c^2*(3*C*d - B*e)*x^3)/(3*e^4) + (c^2*C*x^4)/(4*e^3) - ((c*d^2 + a*e^2)^2*(C*d^2
- B*d*e + A*e^2))/(2*e^7*(d + e*x)^2) + ((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)))/(e^7*(d + e*x)) + ((a^2*C*e^4 + c^2*(15*C*d^4 - 2*d^2*e*(5*B*d - 3*A*e)) + 2*a*c*e^2*(6*C*d^2 - e*(3*B*d
- A*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)^3} \, dx &=\int \left (\frac{c \left (-10 c C d^3+3 c d e (2 B d-A e)-2 a e^2 (3 C d-B e)\right )}{e^6}+\frac{c \left (6 c C d^2+2 a C e^2-c e (3 B d-A e)\right ) x}{e^5}+\frac{c^2 (-3 C d+B e) x^2}{e^4}+\frac{c^2 C x^3}{e^3}+\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)^3}+\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)^2}+\frac{a^2 C e^4+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{c \left (10 c C d^3-3 c d e (2 B d-A e)+2 a e^2 (3 C d-B e)\right ) x}{e^6}+\frac{c \left (6 c C d^2+2 a C e^2-c e (3 B d-A e)\right ) x^2}{2 e^5}-\frac{c^2 (3 C d-B e) x^3}{3 e^4}+\frac{c^2 C x^4}{4 e^3}-\frac{\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right )}{2 e^7 (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^7 (d+e x)}+\frac{\left (a^2 C e^4+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )+2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.135966, size = 274, normalized size = 0.93 \[ \frac{12 \log (d+e x) \left (a^2 C e^4+2 a c e^2 \left (e (A e-3 B d)+6 C d^2\right )+c^2 \left (2 d^2 e (3 A e-5 B d)+15 C d^4\right )\right )+6 c e^2 x^2 \left (2 a C e^2+c e (A e-3 B d)+6 c C d^2\right )-12 c e x \left (-2 a e^2 (B e-3 C d)+3 c d e (A e-2 B d)+10 c C d^3\right )+\frac{12 \left (a e^2+c d^2\right ) \left (a e^2 (2 C d-B e)+c d e (4 A e-5 B d)+6 c C d^3\right )}{d+e x}-\frac{6 \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}+4 c^2 e^3 x^3 (B e-3 C d)+3 c^2 C e^4 x^4}{12 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.061, size = 563, normalized size = 1.9 \begin{align*} -{\frac{3\,B{c}^{2}{x}^{2}d}{2\,{e}^{4}}}+15\,{\frac{\ln \left ( ex+d \right ) C{c}^{2}{d}^{4}}{{e}^{7}}}+4\,{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-5\,{\frac{B{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+2\,{\frac{{a}^{2}Cd}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{C{c}^{2}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }}-10\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{3}}{{e}^{6}}}+{\frac{cC{x}^{2}a}{{e}^{3}}}+3\,{\frac{C{c}^{2}{x}^{2}{d}^{2}}{{e}^{5}}}+2\,{\frac{\ln \left ( ex+d \right ) Aac}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{2}}{{e}^{5}}}-{\frac{B{a}^{2}}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{A{c}^{2}dx}{{e}^{4}}}+2\,{\frac{aBcx}{{e}^{3}}}+6\,{\frac{B{c}^{2}{d}^{2}x}{{e}^{5}}}-10\,{\frac{C{c}^{2}{d}^{3}x}{{e}^{6}}}-{\frac{C{c}^{2}{x}^{3}d}{{e}^{4}}}-{\frac{{a}^{2}A}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{2}{x}^{3}}{3\,{e}^{3}}}+{\frac{A{x}^{2}{c}^{2}}{2\,{e}^{3}}}+{\frac{\ln \left ( ex+d \right ){a}^{2}C}{{e}^{3}}}-{\frac{A{c}^{2}{d}^{4}}{2\,{e}^{5} \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 ) ^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) Bacd}{{e}^{4}}}+12\,{\frac{\ln \left ( ex+d \right ) Cac{d}^{2}}{{e}^{5}}}+4\,{\frac{aAcd}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{aBc{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+8\,{\frac{aCc{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-6\,{\frac{acdCx}{{e}^{4}}}-{\frac{A{d}^{2}ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bd{a}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{C{d}^{2}{a}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{C{c}^{2}{d}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{C{c}^{2}{x}^{4}}{4\,{e}^{3}}} \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)^3,x)

[Out]

-3/2*c^2/e^4*B*x^2*d+15/e^7*ln(e*x+d)*C*c^2*d^4+4/e^5/(e*x+d)*A*c^2*d^3-5/e^6/(e*x+d)*B*c^2*d^4+2/e^3/(e*x+d)*
C*a^2*d+6/e^7/(e*x+d)*C*c^2*d^5-10/e^6*ln(e*x+d)*B*c^2*d^3+c/e^3*C*x^2*a+3*c^2/e^5*C*x^2*d^2+2/e^3*ln(e*x+d)*A
*a*c+6/e^5*ln(e*x+d)*A*c^2*d^2-1/e^2/(e*x+d)*B*a^2-3*c^2/e^4*A*d*x+2*c/e^3*B*a*x+6*c^2/e^5*B*d^2*x-10*c^2/e^6*
C*d^3*x-c^2/e^4*C*x^3*d-1/2/e/(e*x+d)^2*A*a^2+1/3*c^2/e^3*B*x^3+1/2*c^2/e^3*A*x^2+1/e^3*ln(e*x+d)*a^2*C-1/2/e^
5/(e*x+d)^2*A*c^2*d^4+1/e^4/(e*x+d)^2*B*a*c*d^3-1/e^5/(e*x+d)^2*C*a*c*d^4-6/e^4*ln(e*x+d)*B*a*c*d+12/e^5*ln(e*
x+d)*C*a*c*d^2+4/e^3/(e*x+d)*A*a*c*d-6/e^4/(e*x+d)*B*a*c*d^2+8/e^5/(e*x+d)*C*a*c*d^3-6*c/e^4*C*a*d*x-1/e^3/(e*
x+d)^2*A*d^2*a*c+1/2/e^2/(e*x+d)^2*B*d*a^2+1/2/e^6/(e*x+d)^2*B*c^2*d^5-1/2/e^3/(e*x+d)^2*C*d^2*a^2-1/2/e^7/(e*
x+d)^2*C*c^2*d^6+1/4*c^2*C*x^4/e^3

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Maxima [A]  time = 1.0299, size = 543, normalized size = 1.84 \begin{align*} \frac{11 \, C c^{2} d^{6} - 9 \, B c^{2} d^{5} e - 10 \, B a c d^{3} e^{3} - B a^{2} d e^{5} - A a^{2} e^{6} + 7 \,{\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 3 \,{\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} + 2 \,{\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}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{3 \, C c^{2} e^{3} x^{4} - 4 \,{\left (3 \, C c^{2} d e^{2} - B c^{2} e^{3}\right )} x^{3} + 6 \,{\left (6 \, C c^{2} d^{2} e - 3 \, B c^{2} d e^{2} +{\left (2 \, C a c + A c^{2}\right )} e^{3}\right )} x^{2} - 12 \,{\left (10 \, C c^{2} d^{3} - 6 \, B c^{2} d^{2} e - 2 \, B a c e^{3} + 3 \,{\left (2 \, C a c + A c^{2}\right )} d e^{2}\right )} x}{12 \, e^{6}} + \frac{{\left (15 \, C c^{2} d^{4} - 10 \, B c^{2} d^{3} e - 6 \, B a c d e^{3} + 6 \,{\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 )} \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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.7306, size = 1305, normalized size = 4.42 \begin{align*} \frac{3 \, C c^{2} e^{6} x^{6} + 66 \, C c^{2} d^{6} - 54 \, B c^{2} d^{5} e - 60 \, B a c d^{3} e^{3} - 6 \, B a^{2} d e^{5} - 6 \, A a^{2} e^{6} + 42 \,{\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 18 \,{\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} - 2 \,{\left (3 \, C c^{2} d e^{5} - 2 \, B c^{2} e^{6}\right )} x^{5} +{\left (15 \, C c^{2} d^{2} e^{4} - 10 \, B c^{2} d e^{5} + 6 \,{\left (2 \, C a c + A c^{2}\right )} e^{6}\right )} x^{4} - 4 \,{\left (15 \, C c^{2} d^{3} e^{3} - 10 \, B c^{2} d^{2} e^{4} - 6 \, B a c e^{6} + 6 \,{\left (2 \, C a c + A c^{2}\right )} d e^{5}\right )} x^{3} - 6 \,{\left (34 \, C c^{2} d^{4} e^{2} - 21 \, B c^{2} d^{3} e^{3} - 8 \, B a c d e^{5} + 11 \,{\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4}\right )} x^{2} - 12 \,{\left (4 \, C c^{2} d^{5} e - B c^{2} d^{4} e^{2} + 4 \, 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} - 2 \,{\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x + 12 \,{\left (15 \, C c^{2} d^{6} - 10 \, B c^{2} d^{5} e - 6 \, B a c d^{3} e^{3} + 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} +{\left (15 \, C c^{2} d^{4} e^{2} - 10 \, B c^{2} d^{3} e^{3} - 6 \, B a c d e^{5} + 6 \,{\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} + 2 \,{\left (15 \, C c^{2} d^{5} e - 10 \, B c^{2} d^{4} e^{2} - 6 \, B a c d^{2} e^{4} + 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\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/12*(3*C*c^2*e^6*x^6 + 66*C*c^2*d^6 - 54*B*c^2*d^5*e - 60*B*a*c*d^3*e^3 - 6*B*a^2*d*e^5 - 6*A*a^2*e^6 + 42*(2
*C*a*c + A*c^2)*d^4*e^2 + 18*(C*a^2 + 2*A*a*c)*d^2*e^4 - 2*(3*C*c^2*d*e^5 - 2*B*c^2*e^6)*x^5 + (15*C*c^2*d^2*e
^4 - 10*B*c^2*d*e^5 + 6*(2*C*a*c + A*c^2)*e^6)*x^4 - 4*(15*C*c^2*d^3*e^3 - 10*B*c^2*d^2*e^4 - 6*B*a*c*e^6 + 6*
(2*C*a*c + A*c^2)*d*e^5)*x^3 - 6*(34*C*c^2*d^4*e^2 - 21*B*c^2*d^3*e^3 - 8*B*a*c*d*e^5 + 11*(2*C*a*c + A*c^2)*d
^2*e^4)*x^2 - 12*(4*C*c^2*d^5*e - B*c^2*d^4*e^2 + 4*B*a*c*d^2*e^4 + B*a^2*e^6 - (2*C*a*c + A*c^2)*d^3*e^3 - 2*
(C*a^2 + 2*A*a*c)*d*e^5)*x + 12*(15*C*c^2*d^6 - 10*B*c^2*d^5*e - 6*B*a*c*d^3*e^3 + 6*(2*C*a*c + A*c^2)*d^4*e^2
+ (C*a^2 + 2*A*a*c)*d^2*e^4 + (15*C*c^2*d^4*e^2 - 10*B*c^2*d^3*e^3 - 6*B*a*c*d*e^5 + 6*(2*C*a*c + A*c^2)*d^2*
e^4 + (C*a^2 + 2*A*a*c)*e^6)*x^2 + 2*(15*C*c^2*d^5*e - 10*B*c^2*d^4*e^2 - 6*B*a*c*d^2*e^4 + 6*(2*C*a*c + A*c^2
)*d^3*e^3 + (C*a^2 + 2*A*a*c)*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 16.4422, size = 471, normalized size = 1.6 \begin{align*} \frac{C c^{2} x^{4}}{4 e^{3}} + \frac{- A a^{2} e^{6} + 6 A a c d^{2} e^{4} + 7 A c^{2} d^{4} e^{2} - B a^{2} d e^{5} - 10 B a c d^{3} e^{3} - 9 B c^{2} d^{5} e + 3 C a^{2} d^{2} e^{4} + 14 C a c d^{4} e^{2} + 11 C c^{2} d^{6} + x \left (8 A a c d e^{5} + 8 A c^{2} d^{3} e^{3} - 2 B a^{2} e^{6} - 12 B a c d^{2} e^{4} - 10 B c^{2} d^{4} e^{2} + 4 C a^{2} d e^{5} + 16 C a c d^{3} e^{3} + 12 C c^{2} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} - \frac{x^{3} \left (- B c^{2} e + 3 C c^{2} d\right )}{3 e^{4}} + \frac{x^{2} \left (A c^{2} e^{2} - 3 B c^{2} d e + 2 C a c e^{2} + 6 C c^{2} d^{2}\right )}{2 e^{5}} - \frac{x \left (3 A c^{2} d e^{2} - 2 B a c e^{3} - 6 B c^{2} d^{2} e + 6 C a c d e^{2} + 10 C c^{2} d^{3}\right )}{e^{6}} + \frac{\left (2 A a c e^{4} + 6 A c^{2} d^{2} e^{2} - 6 B a c d e^{3} - 10 B c^{2} d^{3} e + C a^{2} e^{4} + 12 C a c d^{2} e^{2} + 15 C c^{2} d^{4}\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)**3,x)

[Out]

C*c**2*x**4/(4*e**3) + (-A*a**2*e**6 + 6*A*a*c*d**2*e**4 + 7*A*c**2*d**4*e**2 - B*a**2*d*e**5 - 10*B*a*c*d**3*
e**3 - 9*B*c**2*d**5*e + 3*C*a**2*d**2*e**4 + 14*C*a*c*d**4*e**2 + 11*C*c**2*d**6 + x*(8*A*a*c*d*e**5 + 8*A*c*
*2*d**3*e**3 - 2*B*a**2*e**6 - 12*B*a*c*d**2*e**4 - 10*B*c**2*d**4*e**2 + 4*C*a**2*d*e**5 + 16*C*a*c*d**3*e**3
+ 12*C*c**2*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) - x**3*(-B*c**2*e + 3*C*c**2*d)/(3*e**4) + x**2
*(A*c**2*e**2 - 3*B*c**2*d*e + 2*C*a*c*e**2 + 6*C*c**2*d**2)/(2*e**5) - x*(3*A*c**2*d*e**2 - 2*B*a*c*e**3 - 6*
B*c**2*d**2*e + 6*C*a*c*d*e**2 + 10*C*c**2*d**3)/e**6 + (2*A*a*c*e**4 + 6*A*c**2*d**2*e**2 - 6*B*a*c*d*e**3 -
10*B*c**2*d**3*e + C*a**2*e**4 + 12*C*a*c*d**2*e**2 + 15*C*c**2*d**4)*log(d + e*x)/e**7

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Giac [A]  time = 1.16243, size = 536, normalized size = 1.82 \begin{align*}{\left (15 \, C c^{2} d^{4} - 10 \, B c^{2} d^{3} e + 12 \, C a c d^{2} e^{2} + 6 \, A c^{2} d^{2} e^{2} - 6 \, B a c d e^{3} + C a^{2} e^{4} + 2 \, A a c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, C c^{2} x^{4} e^{9} - 12 \, C c^{2} d x^{3} e^{8} + 36 \, C c^{2} d^{2} x^{2} e^{7} - 120 \, C c^{2} d^{3} x e^{6} + 4 \, B c^{2} x^{3} e^{9} - 18 \, B c^{2} d x^{2} e^{8} + 72 \, B c^{2} d^{2} x e^{7} + 12 \, C a c x^{2} e^{9} + 6 \, A c^{2} x^{2} e^{9} - 72 \, C a c d x e^{8} - 36 \, A c^{2} d x e^{8} + 24 \, B a c x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, C c^{2} d^{6} - 9 \, B c^{2} d^{5} e + 14 \, C a c d^{4} e^{2} + 7 \, A c^{2} d^{4} e^{2} - 10 \, B a c d^{3} e^{3} + 3 \, C a^{2} d^{2} e^{4} + 6 \, A a c d^{2} e^{4} - B a^{2} d e^{5} - A a^{2} e^{6} + 2 \,{\left (6 \, C c^{2} d^{5} e - 5 \, B c^{2} d^{4} e^{2} + 8 \, C a c d^{3} e^{3} + 4 \, A c^{2} d^{3} e^{3} - 6 \, B a c d^{2} e^{4} + 2 \, C a^{2} d e^{5} + 4 \, A a c d e^{5} - B a^{2} e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

(15*C*c^2*d^4 - 10*B*c^2*d^3*e + 12*C*a*c*d^2*e^2 + 6*A*c^2*d^2*e^2 - 6*B*a*c*d*e^3 + C*a^2*e^4 + 2*A*a*c*e^4)
*e^(-7)*log(abs(x*e + d)) + 1/12*(3*C*c^2*x^4*e^9 - 12*C*c^2*d*x^3*e^8 + 36*C*c^2*d^2*x^2*e^7 - 120*C*c^2*d^3*
x*e^6 + 4*B*c^2*x^3*e^9 - 18*B*c^2*d*x^2*e^8 + 72*B*c^2*d^2*x*e^7 + 12*C*a*c*x^2*e^9 + 6*A*c^2*x^2*e^9 - 72*C*
a*c*d*x*e^8 - 36*A*c^2*d*x*e^8 + 24*B*a*c*x*e^9)*e^(-12) + 1/2*(11*C*c^2*d^6 - 9*B*c^2*d^5*e + 14*C*a*c*d^4*e^
2 + 7*A*c^2*d^4*e^2 - 10*B*a*c*d^3*e^3 + 3*C*a^2*d^2*e^4 + 6*A*a*c*d^2*e^4 - B*a^2*d*e^5 - A*a^2*e^6 + 2*(6*C*
c^2*d^5*e - 5*B*c^2*d^4*e^2 + 8*C*a*c*d^3*e^3 + 4*A*c^2*d^3*e^3 - 6*B*a*c*d^2*e^4 + 2*C*a^2*d*e^5 + 4*A*a*c*d*
e^5 - B*a^2*e^6)*x)*e^(-7)/(x*e + d)^2