### 3.38 $$\int \frac{(a+c x^2)^3 (A+B x+C x^2)}{(d+e x)^3} \, dx$$

Optimal. Leaf size=466 $\frac{c x^2 \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (15 C d^2-2 e (5 B d-3 A e)\right )\right )}{2 e^7}-\frac{c x \left (3 a^2 e^4 (3 C d-B e)+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )+c^2 d^3 \left (21 C d^2-5 e (3 B d-2 A e)\right )\right )}{e^8}+\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 d^2 \left (28 C d^2-3 e (7 B d-5 A e)\right )\right )}{e^9}+\frac{c^2 x^4 \left (3 a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right )}{4 e^5}-\frac{c^2 x^3 \left (3 a e^2 (3 C d-B e)+c d \left (10 C d^2-3 e (2 B d-A e)\right )\right )}{3 e^6}+\frac{\left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right )}{e^9 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{2 e^9 (d+e x)^2}-\frac{c^3 x^5 (3 C d-B e)}{5 e^4}+\frac{c^3 C x^6}{6 e^3}$

[Out]

-((c*(3*a^2*e^4*(3*C*d - B*e) + c^2*d^3*(21*C*d^2 - 5*e*(3*B*d - 2*A*e)) + 3*a*c*d*e^2*(10*C*d^2 - 3*e*(2*B*d
- A*e)))*x)/e^8) + (c*(3*a^2*C*e^4 + c^2*d^2*(15*C*d^2 - 2*e*(5*B*d - 3*A*e)) + 3*a*c*e^2*(6*C*d^2 - e*(3*B*d
- A*e)))*x^2)/(2*e^7) - (c^2*(3*a*e^2*(3*C*d - B*e) + c*d*(10*C*d^2 - 3*e*(2*B*d - A*e)))*x^3)/(3*e^6) + (c^2*
(3*a*C*e^2 + c*(6*C*d^2 - e*(3*B*d - A*e)))*x^4)/(4*e^5) - (c^3*(3*C*d - B*e)*x^5)/(5*e^4) + (c^3*C*x^6)/(6*e^
3) - ((c*d^2 + a*e^2)^3*(C*d^2 - B*d*e + A*e^2))/(2*e^9*(d + e*x)^2) + ((c*d^2 + a*e^2)^2*(a*e^2*(2*C*d - B*e)
+ c*d*(8*C*d^2 - e*(7*B*d - 6*A*e))))/(e^9*(d + e*x)) + ((c*d^2 + a*e^2)*(a^2*C*e^4 + c^2*d^2*(28*C*d^2 - 3*e
*(7*B*d - 5*A*e)) + a*c*e^2*(17*C*d^2 - 3*e*(3*B*d - A*e)))*Log[d + e*x])/e^9

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Rubi [A]  time = 0.966645, antiderivative size = 463, 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} $\frac{c x^2 \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )\right )}{2 e^7}-\frac{c x \left (3 a^2 e^4 (3 C d-B e)+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )+c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )\right )}{e^8}+\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{e^9}+\frac{c^2 x^4 \left (3 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{4 e^5}-\frac{c^2 x^3 \left (3 a e^2 (3 C d-B e)-3 c d e (2 B d-A e)+10 c C d^3\right )}{3 e^6}+\frac{\left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^9 (d+e x)}-\frac{\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{2 e^9 (d+e x)^2}-\frac{c^3 x^5 (3 C d-B e)}{5 e^4}+\frac{c^3 C x^6}{6 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((c*(3*a^2*e^4*(3*C*d - B*e) + c^2*(21*C*d^5 - 5*d^3*e*(3*B*d - 2*A*e)) + 3*a*c*d*e^2*(10*C*d^2 - 3*e*(2*B*d
- A*e)))*x)/e^8) + (c*(3*a^2*C*e^4 + c^2*(15*C*d^4 - 2*d^2*e*(5*B*d - 3*A*e)) + 3*a*c*e^2*(6*C*d^2 - e*(3*B*d
- A*e)))*x^2)/(2*e^7) - (c^2*(10*c*C*d^3 - 3*c*d*e*(2*B*d - A*e) + 3*a*e^2*(3*C*d - B*e))*x^3)/(3*e^6) + (c^2*
(6*c*C*d^2 + 3*a*C*e^2 - c*e*(3*B*d - A*e))*x^4)/(4*e^5) - (c^3*(3*C*d - B*e)*x^5)/(5*e^4) + (c^3*C*x^6)/(6*e^
3) - ((c*d^2 + a*e^2)^3*(C*d^2 - B*d*e + A*e^2))/(2*e^9*(d + e*x)^2) + ((c*d^2 + a*e^2)^2*(8*c*C*d^3 - c*d*e*(
7*B*d - 6*A*e) + a*e^2*(2*C*d - B*e)))/(e^9*(d + e*x)) + ((c*d^2 + a*e^2)*(a^2*C*e^4 + c^2*(28*C*d^4 - 3*d^2*e
*(7*B*d - 5*A*e)) + a*c*e^2*(17*C*d^2 - 3*e*(3*B*d - A*e)))*Log[d + e*x])/e^9

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 )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac{c \left (-3 a^2 e^4 (3 C d-B e)-c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )-3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )\right )}{e^8}+\frac{c \left (3 a^2 C e^4+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) x}{e^7}+\frac{c^2 \left (-10 c C d^3+3 c d e (2 B d-A e)-3 a e^2 (3 C d-B e)\right ) x^2}{e^6}+\frac{c^2 \left (6 c C d^2+3 a C e^2-c e (3 B d-A e)\right ) x^3}{e^5}+\frac{c^3 (-3 C d+B e) x^4}{e^4}+\frac{c^3 C x^5}{e^3}+\frac{\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{e^8 (d+e x)^3}+\frac{\left (c d^2+a e^2\right )^2 \left (-8 c C d^3+c d e (7 B d-6 A e)-a e^2 (2 C d-B e)\right )}{e^8 (d+e x)^2}+\frac{\left (c d^2+a e^2\right ) \left (a^2 C e^4+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )\right )}{e^8 (d+e x)}\right ) \, dx\\ &=-\frac{c \left (3 a^2 e^4 (3 C d-B e)+c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )\right ) x}{e^8}+\frac{c \left (3 a^2 C e^4+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) x^2}{2 e^7}-\frac{c^2 \left (10 c C d^3-3 c d e (2 B d-A e)+3 a e^2 (3 C d-B e)\right ) x^3}{3 e^6}+\frac{c^2 \left (6 c C d^2+3 a C e^2-c e (3 B d-A e)\right ) x^4}{4 e^5}-\frac{c^3 (3 C d-B e) x^5}{5 e^4}+\frac{c^3 C x^6}{6 e^3}-\frac{\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{2 e^9 (d+e x)^2}+\frac{\left (c d^2+a e^2\right )^2 \left (8 c C d^3-c d e (7 B d-6 A e)+a e^2 (2 C d-B e)\right )}{e^9 (d+e x)}+\frac{\left (c d^2+a e^2\right ) \left (a^2 C e^4+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )\right ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.235061, size = 438, normalized size = 0.94 $\frac{30 c e^2 x^2 \left (3 a^2 C e^4+3 a c e^2 \left (e (A e-3 B d)+6 C d^2\right )+c^2 \left (2 d^2 e (3 A e-5 B d)+15 C d^4\right )\right )-60 c e x \left (-3 a^2 e^4 (B e-3 C d)+3 a c d e^2 \left (3 e (A e-2 B d)+10 C d^2\right )+c^2 \left (5 d^3 e (2 A e-3 B d)+21 C d^5\right )\right )+60 \left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (3 e (A e-3 B d)+17 C d^2\right )+c^2 \left (3 d^2 e (5 A e-7 B d)+28 C d^4\right )\right )+15 c^2 e^4 x^4 \left (3 a C e^2+c e (A e-3 B d)+6 c C d^2\right )-20 c^2 e^3 x^3 \left (-3 a e^2 (B e-3 C d)+3 c d e (A e-2 B d)+10 c C d^3\right )+\frac{60 \left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)+c d e (6 A e-7 B d)+8 c C d^3\right )}{d+e x}-\frac{30 \left (a e^2+c d^2\right )^3 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}+12 c^3 e^5 x^5 (B e-3 C d)+10 c^3 C e^6 x^6}{60 e^9}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-60*c*e*(-3*a^2*e^4*(-3*C*d + B*e) + 3*a*c*d*e^2*(10*C*d^2 + 3*e*(-2*B*d + A*e)) + c^2*(21*C*d^5 + 5*d^3*e*(-
3*B*d + 2*A*e)))*x + 30*c*e^2*(3*a^2*C*e^4 + 3*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)))*x^2 - 20*c^2*e^3*(10*c*C*d^3 + 3*c*d*e*(-2*B*d + A*e) - 3*a*e^2*(-3*C*d + B*e))*x^3 + 15*c^
2*e^4*(6*c*C*d^2 + 3*a*C*e^2 + c*e*(-3*B*d + A*e))*x^4 + 12*c^3*e^5*(-3*C*d + B*e)*x^5 + 10*c^3*C*e^6*x^6 - (3
0*(c*d^2 + a*e^2)^3*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 + (60*(c*d^2 + a*e^2)^2*(8*c*C*d^3 + c*d*e*(-7*B*d
+ 6*A*e) + a*e^2*(2*C*d - B*e)))/(d + e*x) + 60*(c*d^2 + a*e^2)*(a^2*C*e^4 + a*c*e^2*(17*C*d^2 + 3*e*(-3*B*d
+ A*e)) + c^2*(28*C*d^4 + 3*d^2*e*(-7*B*d + 5*A*e)))*Log[d + e*x])/(60*e^9)

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Maple [B]  time = 0.061, size = 978, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/e^9/(e*x+d)^2*C*c^3*d^8-10*c^3/e^6*A*d^3*x+15*c^3/e^7*B*d^4*x-5*c^3/e^6*B*x^2*d^3+3*c^3/e^5*A*x^2*d^2-10/
3*c^3/e^6*C*x^3*d^3+2*c^3/e^5*B*x^3*d^2+c^2/e^3*B*x^3*a-c^3/e^4*A*x^3*d-7/e^8/(e*x+d)*B*c^3*d^6+2/e^3/(e*x+d)*
C*a^3*d-3/5*c^3/e^4*C*x^5*d-3/4*c^3/e^4*B*x^4*d-1/2/e^7/(e*x+d)^2*A*c^3*d^6+1/2/e^2/(e*x+d)^2*B*d*a^3+1/2/e^8/
(e*x+d)^2*B*c^3*d^7-1/2/e^3/(e*x+d)^2*C*d^2*a^3-21*c^3/e^8*C*d^5*x+15/2*c^3/e^7*C*x^2*d^4-3/2/e^5/(e*x+d)^2*C*
a^2*c*d^4-3/2/e^7/(e*x+d)^2*C*a*c^2*d^6+18*c^2/e^5*B*a*d^2*x-3*c^2/e^4*C*x^3*a*d-9/2*c^2/e^4*B*x^2*a*d+9*c^2/e
^5*C*x^2*a*d^2+8/e^9/(e*x+d)*C*c^3*d^7+3/e^3*ln(e*x+d)*A*a^2*c+15/e^7*ln(e*x+d)*A*c^3*d^4-21/e^8*ln(e*x+d)*B*c
^3*d^5+28/e^9*ln(e*x+d)*C*c^3*d^6+6/e^7/(e*x+d)*A*c^3*d^5+3/2*c^2/e^3*A*x^2*a+3/2*c/e^3*C*x^2*a^2+3*c/e^3*a^2*
B*x+1/5*c^3/e^3*B*x^5-1/2/e/(e*x+d)^2*A*a^3+1/e^3*ln(e*x+d)*a^3*C-1/e^2/(e*x+d)*B*a^3-3/2/e^3/(e*x+d)^2*A*d^2*
a^2*c-3/2/e^5/(e*x+d)^2*A*a*c^2*d^4+3/2/e^4/(e*x+d)^2*B*a^2*c*d^3-9*c/e^4*C*a^2*d*x-9*c^2/e^4*A*a*d*x+3/2*c^3/
e^5*C*x^4*d^2+3/4*c^2/e^3*C*x^4*a+1/4*c^3/e^3*A*x^4-30*c^2/e^6*C*a*d^3*x+18/e^7/(e*x+d)*C*a*c^2*d^5+12/e^5/(e*
x+d)*C*a^2*c*d^3+45/e^7*ln(e*x+d)*C*a*c^2*d^4-30/e^6*ln(e*x+d)*B*a*c^2*d^3+18/e^5*ln(e*x+d)*A*a*c^2*d^2-9/e^4*
ln(e*x+d)*B*a^2*c*d+3/2/e^6/(e*x+d)^2*B*a*c^2*d^5-9/e^4/(e*x+d)*B*a^2*c*d^2+1/6*c^3*C*x^6/e^3+12/e^5/(e*x+d)*A
*a*c^2*d^3+18/e^5*ln(e*x+d)*C*a^2*c*d^2-15/e^6/(e*x+d)*B*a*c^2*d^4+6/e^3/(e*x+d)*A*a^2*c*d

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Maxima [A]  time = 1.08593, size = 946, normalized size = 2.03 \begin{align*} \frac{15 \, C c^{3} d^{8} - 13 \, B c^{3} d^{7} e - 27 \, B a c^{2} d^{5} e^{3} - 15 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} - A a^{3} e^{8} + 11 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 21 \,{\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + 3 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} + 2 \,{\left (8 \, C c^{3} d^{7} e - 7 \, B c^{3} d^{6} e^{2} - 15 \, B a c^{2} d^{4} e^{4} - 9 \, B a^{2} c d^{2} e^{6} - B a^{3} e^{8} + 6 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 12 \,{\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} + 2 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x}{2 \,{\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac{10 \, C c^{3} e^{5} x^{6} - 12 \,{\left (3 \, C c^{3} d e^{4} - B c^{3} e^{5}\right )} x^{5} + 15 \,{\left (6 \, C c^{3} d^{2} e^{3} - 3 \, B c^{3} d e^{4} +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{5}\right )} x^{4} - 20 \,{\left (10 \, C c^{3} d^{3} e^{2} - 6 \, B c^{3} d^{2} e^{3} - 3 \, B a c^{2} e^{5} + 3 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d e^{4}\right )} x^{3} + 30 \,{\left (15 \, C c^{3} d^{4} e - 10 \, B c^{3} d^{3} e^{2} - 9 \, B a c^{2} d e^{4} + 6 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{3} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{5}\right )} x^{2} - 60 \,{\left (21 \, C c^{3} d^{5} - 15 \, B c^{3} d^{4} e - 18 \, B a c^{2} d^{2} e^{3} - 3 \, B a^{2} c e^{5} + 10 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{2} + 9 \,{\left (C a^{2} c + A a c^{2}\right )} d e^{4}\right )} x}{60 \, e^{8}} + \frac{{\left (28 \, C c^{3} d^{6} - 21 \, B c^{3} d^{5} e - 30 \, B a c^{2} d^{3} e^{3} - 9 \, B a^{2} c d e^{5} + 15 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{2} + 18 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{4} +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(15*C*c^3*d^8 - 13*B*c^3*d^7*e - 27*B*a*c^2*d^5*e^3 - 15*B*a^2*c*d^3*e^5 - B*a^3*d*e^7 - A*a^3*e^8 + 11*(3
*C*a*c^2 + A*c^3)*d^6*e^2 + 21*(C*a^2*c + A*a*c^2)*d^4*e^4 + 3*(C*a^3 + 3*A*a^2*c)*d^2*e^6 + 2*(8*C*c^3*d^7*e
- 7*B*c^3*d^6*e^2 - 15*B*a*c^2*d^4*e^4 - 9*B*a^2*c*d^2*e^6 - B*a^3*e^8 + 6*(3*C*a*c^2 + A*c^3)*d^5*e^3 + 12*(C
*a^2*c + A*a*c^2)*d^3*e^5 + 2*(C*a^3 + 3*A*a^2*c)*d*e^7)*x)/(e^11*x^2 + 2*d*e^10*x + d^2*e^9) + 1/60*(10*C*c^3
*e^5*x^6 - 12*(3*C*c^3*d*e^4 - B*c^3*e^5)*x^5 + 15*(6*C*c^3*d^2*e^3 - 3*B*c^3*d*e^4 + (3*C*a*c^2 + A*c^3)*e^5)
*x^4 - 20*(10*C*c^3*d^3*e^2 - 6*B*c^3*d^2*e^3 - 3*B*a*c^2*e^5 + 3*(3*C*a*c^2 + A*c^3)*d*e^4)*x^3 + 30*(15*C*c^
3*d^4*e - 10*B*c^3*d^3*e^2 - 9*B*a*c^2*d*e^4 + 6*(3*C*a*c^2 + A*c^3)*d^2*e^3 + 3*(C*a^2*c + A*a*c^2)*e^5)*x^2
- 60*(21*C*c^3*d^5 - 15*B*c^3*d^4*e - 18*B*a*c^2*d^2*e^3 - 3*B*a^2*c*e^5 + 10*(3*C*a*c^2 + A*c^3)*d^3*e^2 + 9*
(C*a^2*c + A*a*c^2)*d*e^4)*x)/e^8 + (28*C*c^3*d^6 - 21*B*c^3*d^5*e - 30*B*a*c^2*d^3*e^3 - 9*B*a^2*c*d*e^5 + 15
*(3*C*a*c^2 + A*c^3)*d^4*e^2 + 18*(C*a^2*c + A*a*c^2)*d^2*e^4 + (C*a^3 + 3*A*a^2*c)*e^6)*log(e*x + d)/e^9

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Fricas [B]  time = 1.82771, size = 2198, normalized size = 4.72 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*C*c^3*e^8*x^8 + 450*C*c^3*d^8 - 390*B*c^3*d^7*e - 810*B*a*c^2*d^5*e^3 - 450*B*a^2*c*d^3*e^5 - 30*B*a^
3*d*e^7 - 30*A*a^3*e^8 + 330*(3*C*a*c^2 + A*c^3)*d^6*e^2 + 630*(C*a^2*c + A*a*c^2)*d^4*e^4 + 90*(C*a^3 + 3*A*a
^2*c)*d^2*e^6 - 4*(4*C*c^3*d*e^7 - 3*B*c^3*e^8)*x^7 + (28*C*c^3*d^2*e^6 - 21*B*c^3*d*e^7 + 15*(3*C*a*c^2 + A*c
^3)*e^8)*x^6 - 2*(28*C*c^3*d^3*e^5 - 21*B*c^3*d^2*e^6 - 30*B*a*c^2*e^8 + 15*(3*C*a*c^2 + A*c^3)*d*e^7)*x^5 + 5
*(28*C*c^3*d^4*e^4 - 21*B*c^3*d^3*e^5 - 30*B*a*c^2*d*e^7 + 15*(3*C*a*c^2 + A*c^3)*d^2*e^6 + 18*(C*a^2*c + A*a*
c^2)*e^8)*x^4 - 20*(28*C*c^3*d^5*e^3 - 21*B*c^3*d^4*e^4 - 30*B*a*c^2*d^2*e^6 - 9*B*a^2*c*e^8 + 15*(3*C*a*c^2 +
A*c^3)*d^3*e^5 + 18*(C*a^2*c + A*a*c^2)*d*e^7)*x^3 - 30*(69*C*c^3*d^6*e^2 - 50*B*c^3*d^5*e^3 - 63*B*a*c^2*d^3
*e^5 - 12*B*a^2*c*d*e^7 + 34*(3*C*a*c^2 + A*c^3)*d^4*e^4 + 33*(C*a^2*c + A*a*c^2)*d^2*e^6)*x^2 - 60*(13*C*c^3*
d^7*e - 8*B*c^3*d^6*e^2 - 3*B*a*c^2*d^4*e^4 + 6*B*a^2*c*d^2*e^6 + B*a^3*e^8 + 4*(3*C*a*c^2 + A*c^3)*d^5*e^3 -
3*(C*a^2*c + A*a*c^2)*d^3*e^5 - 2*(C*a^3 + 3*A*a^2*c)*d*e^7)*x + 60*(28*C*c^3*d^8 - 21*B*c^3*d^7*e - 30*B*a*c^
2*d^5*e^3 - 9*B*a^2*c*d^3*e^5 + 15*(3*C*a*c^2 + A*c^3)*d^6*e^2 + 18*(C*a^2*c + A*a*c^2)*d^4*e^4 + (C*a^3 + 3*A
*a^2*c)*d^2*e^6 + (28*C*c^3*d^6*e^2 - 21*B*c^3*d^5*e^3 - 30*B*a*c^2*d^3*e^5 - 9*B*a^2*c*d*e^7 + 15*(3*C*a*c^2
+ A*c^3)*d^4*e^4 + 18*(C*a^2*c + A*a*c^2)*d^2*e^6 + (C*a^3 + 3*A*a^2*c)*e^8)*x^2 + 2*(28*C*c^3*d^7*e - 21*B*c^
3*d^6*e^2 - 30*B*a*c^2*d^4*e^4 - 9*B*a^2*c*d^2*e^6 + 15*(3*C*a*c^2 + A*c^3)*d^5*e^3 + 18*(C*a^2*c + A*a*c^2)*d
^3*e^5 + (C*a^3 + 3*A*a^2*c)*d*e^7)*x)*log(e*x + d))/(e^11*x^2 + 2*d*e^10*x + d^2*e^9)

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Sympy [A]  time = 36.9157, size = 799, normalized size = 1.71 \begin{align*} \frac{C c^{3} x^{6}}{6 e^{3}} + \frac{- A a^{3} e^{8} + 9 A a^{2} c d^{2} e^{6} + 21 A a c^{2} d^{4} e^{4} + 11 A c^{3} d^{6} e^{2} - B a^{3} d e^{7} - 15 B a^{2} c d^{3} e^{5} - 27 B a c^{2} d^{5} e^{3} - 13 B c^{3} d^{7} e + 3 C a^{3} d^{2} e^{6} + 21 C a^{2} c d^{4} e^{4} + 33 C a c^{2} d^{6} e^{2} + 15 C c^{3} d^{8} + x \left (12 A a^{2} c d e^{7} + 24 A a c^{2} d^{3} e^{5} + 12 A c^{3} d^{5} e^{3} - 2 B a^{3} e^{8} - 18 B a^{2} c d^{2} e^{6} - 30 B a c^{2} d^{4} e^{4} - 14 B c^{3} d^{6} e^{2} + 4 C a^{3} d e^{7} + 24 C a^{2} c d^{3} e^{5} + 36 C a c^{2} d^{5} e^{3} + 16 C c^{3} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} - \frac{x^{5} \left (- B c^{3} e + 3 C c^{3} d\right )}{5 e^{4}} + \frac{x^{4} \left (A c^{3} e^{2} - 3 B c^{3} d e + 3 C a c^{2} e^{2} + 6 C c^{3} d^{2}\right )}{4 e^{5}} - \frac{x^{3} \left (3 A c^{3} d e^{2} - 3 B a c^{2} e^{3} - 6 B c^{3} d^{2} e + 9 C a c^{2} d e^{2} + 10 C c^{3} d^{3}\right )}{3 e^{6}} + \frac{x^{2} \left (3 A a c^{2} e^{4} + 6 A c^{3} d^{2} e^{2} - 9 B a c^{2} d e^{3} - 10 B c^{3} d^{3} e + 3 C a^{2} c e^{4} + 18 C a c^{2} d^{2} e^{2} + 15 C c^{3} d^{4}\right )}{2 e^{7}} - \frac{x \left (9 A a c^{2} d e^{4} + 10 A c^{3} d^{3} e^{2} - 3 B a^{2} c e^{5} - 18 B a c^{2} d^{2} e^{3} - 15 B c^{3} d^{4} e + 9 C a^{2} c d e^{4} + 30 C a c^{2} d^{3} e^{2} + 21 C c^{3} d^{5}\right )}{e^{8}} + \frac{\left (a e^{2} + c d^{2}\right ) \left (3 A a c e^{4} + 15 A c^{2} d^{2} e^{2} - 9 B a c d e^{3} - 21 B c^{2} d^{3} e + C a^{2} e^{4} + 17 C a c d^{2} e^{2} + 28 C c^{2} d^{4}\right ) \log{\left (d + e x \right )}}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*c**3*x**6/(6*e**3) + (-A*a**3*e**8 + 9*A*a**2*c*d**2*e**6 + 21*A*a*c**2*d**4*e**4 + 11*A*c**3*d**6*e**2 - B*
a**3*d*e**7 - 15*B*a**2*c*d**3*e**5 - 27*B*a*c**2*d**5*e**3 - 13*B*c**3*d**7*e + 3*C*a**3*d**2*e**6 + 21*C*a**
2*c*d**4*e**4 + 33*C*a*c**2*d**6*e**2 + 15*C*c**3*d**8 + x*(12*A*a**2*c*d*e**7 + 24*A*a*c**2*d**3*e**5 + 12*A*
c**3*d**5*e**3 - 2*B*a**3*e**8 - 18*B*a**2*c*d**2*e**6 - 30*B*a*c**2*d**4*e**4 - 14*B*c**3*d**6*e**2 + 4*C*a**
3*d*e**7 + 24*C*a**2*c*d**3*e**5 + 36*C*a*c**2*d**5*e**3 + 16*C*c**3*d**7*e))/(2*d**2*e**9 + 4*d*e**10*x + 2*e
**11*x**2) - x**5*(-B*c**3*e + 3*C*c**3*d)/(5*e**4) + x**4*(A*c**3*e**2 - 3*B*c**3*d*e + 3*C*a*c**2*e**2 + 6*C
*c**3*d**2)/(4*e**5) - x**3*(3*A*c**3*d*e**2 - 3*B*a*c**2*e**3 - 6*B*c**3*d**2*e + 9*C*a*c**2*d*e**2 + 10*C*c*
*3*d**3)/(3*e**6) + x**2*(3*A*a*c**2*e**4 + 6*A*c**3*d**2*e**2 - 9*B*a*c**2*d*e**3 - 10*B*c**3*d**3*e + 3*C*a*
*2*c*e**4 + 18*C*a*c**2*d**2*e**2 + 15*C*c**3*d**4)/(2*e**7) - x*(9*A*a*c**2*d*e**4 + 10*A*c**3*d**3*e**2 - 3*
B*a**2*c*e**5 - 18*B*a*c**2*d**2*e**3 - 15*B*c**3*d**4*e + 9*C*a**2*c*d*e**4 + 30*C*a*c**2*d**3*e**2 + 21*C*c*
*3*d**5)/e**8 + (a*e**2 + c*d**2)*(3*A*a*c*e**4 + 15*A*c**2*d**2*e**2 - 9*B*a*c*d*e**3 - 21*B*c**2*d**3*e + C*
a**2*e**4 + 17*C*a*c*d**2*e**2 + 28*C*c**2*d**4)*log(d + e*x)/e**9

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Giac [A]  time = 1.15108, size = 981, normalized size = 2.11 \begin{align*}{\left (28 \, C c^{3} d^{6} - 21 \, B c^{3} d^{5} e + 45 \, C a c^{2} d^{4} e^{2} + 15 \, A c^{3} d^{4} e^{2} - 30 \, B a c^{2} d^{3} e^{3} + 18 \, C a^{2} c d^{2} e^{4} + 18 \, A a c^{2} d^{2} e^{4} - 9 \, B a^{2} c d e^{5} + C a^{3} e^{6} + 3 \, A a^{2} c e^{6}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, C c^{3} x^{6} e^{15} - 36 \, C c^{3} d x^{5} e^{14} + 90 \, C c^{3} d^{2} x^{4} e^{13} - 200 \, C c^{3} d^{3} x^{3} e^{12} + 450 \, C c^{3} d^{4} x^{2} e^{11} - 1260 \, C c^{3} d^{5} x e^{10} + 12 \, B c^{3} x^{5} e^{15} - 45 \, B c^{3} d x^{4} e^{14} + 120 \, B c^{3} d^{2} x^{3} e^{13} - 300 \, B c^{3} d^{3} x^{2} e^{12} + 900 \, B c^{3} d^{4} x e^{11} + 45 \, C a c^{2} x^{4} e^{15} + 15 \, A c^{3} x^{4} e^{15} - 180 \, C a c^{2} d x^{3} e^{14} - 60 \, A c^{3} d x^{3} e^{14} + 540 \, C a c^{2} d^{2} x^{2} e^{13} + 180 \, A c^{3} d^{2} x^{2} e^{13} - 1800 \, C a c^{2} d^{3} x e^{12} - 600 \, A c^{3} d^{3} x e^{12} + 60 \, B a c^{2} x^{3} e^{15} - 270 \, B a c^{2} d x^{2} e^{14} + 1080 \, B a c^{2} d^{2} x e^{13} + 90 \, C a^{2} c x^{2} e^{15} + 90 \, A a c^{2} x^{2} e^{15} - 540 \, C a^{2} c d x e^{14} - 540 \, A a c^{2} d x e^{14} + 180 \, B a^{2} c x e^{15}\right )} e^{\left (-18\right )} + \frac{{\left (15 \, C c^{3} d^{8} - 13 \, B c^{3} d^{7} e + 33 \, C a c^{2} d^{6} e^{2} + 11 \, A c^{3} d^{6} e^{2} - 27 \, B a c^{2} d^{5} e^{3} + 21 \, C a^{2} c d^{4} e^{4} + 21 \, A a c^{2} d^{4} e^{4} - 15 \, B a^{2} c d^{3} e^{5} + 3 \, C a^{3} d^{2} e^{6} + 9 \, A a^{2} c d^{2} e^{6} - B a^{3} d e^{7} - A a^{3} e^{8} + 2 \,{\left (8 \, C c^{3} d^{7} e - 7 \, B c^{3} d^{6} e^{2} + 18 \, C a c^{2} d^{5} e^{3} + 6 \, A c^{3} d^{5} e^{3} - 15 \, B a c^{2} d^{4} e^{4} + 12 \, C a^{2} c d^{3} e^{5} + 12 \, A a c^{2} d^{3} e^{5} - 9 \, B a^{2} c d^{2} e^{6} + 2 \, C a^{3} d e^{7} + 6 \, A a^{2} c d e^{7} - B a^{3} e^{8}\right )} x\right )} e^{\left (-9\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)^3*(C*x^2+B*x+A)/(e*x+d)^3,x, algorithm="giac")

[Out]

(28*C*c^3*d^6 - 21*B*c^3*d^5*e + 45*C*a*c^2*d^4*e^2 + 15*A*c^3*d^4*e^2 - 30*B*a*c^2*d^3*e^3 + 18*C*a^2*c*d^2*e
^4 + 18*A*a*c^2*d^2*e^4 - 9*B*a^2*c*d*e^5 + C*a^3*e^6 + 3*A*a^2*c*e^6)*e^(-9)*log(abs(x*e + d)) + 1/60*(10*C*c
^3*x^6*e^15 - 36*C*c^3*d*x^5*e^14 + 90*C*c^3*d^2*x^4*e^13 - 200*C*c^3*d^3*x^3*e^12 + 450*C*c^3*d^4*x^2*e^11 -
1260*C*c^3*d^5*x*e^10 + 12*B*c^3*x^5*e^15 - 45*B*c^3*d*x^4*e^14 + 120*B*c^3*d^2*x^3*e^13 - 300*B*c^3*d^3*x^2*e
^12 + 900*B*c^3*d^4*x*e^11 + 45*C*a*c^2*x^4*e^15 + 15*A*c^3*x^4*e^15 - 180*C*a*c^2*d*x^3*e^14 - 60*A*c^3*d*x^3
*e^14 + 540*C*a*c^2*d^2*x^2*e^13 + 180*A*c^3*d^2*x^2*e^13 - 1800*C*a*c^2*d^3*x*e^12 - 600*A*c^3*d^3*x*e^12 + 6
0*B*a*c^2*x^3*e^15 - 270*B*a*c^2*d*x^2*e^14 + 1080*B*a*c^2*d^2*x*e^13 + 90*C*a^2*c*x^2*e^15 + 90*A*a*c^2*x^2*e
^15 - 540*C*a^2*c*d*x*e^14 - 540*A*a*c^2*d*x*e^14 + 180*B*a^2*c*x*e^15)*e^(-18) + 1/2*(15*C*c^3*d^8 - 13*B*c^3
*d^7*e + 33*C*a*c^2*d^6*e^2 + 11*A*c^3*d^6*e^2 - 27*B*a*c^2*d^5*e^3 + 21*C*a^2*c*d^4*e^4 + 21*A*a*c^2*d^4*e^4
- 15*B*a^2*c*d^3*e^5 + 3*C*a^3*d^2*e^6 + 9*A*a^2*c*d^2*e^6 - B*a^3*d*e^7 - A*a^3*e^8 + 2*(8*C*c^3*d^7*e - 7*B*
c^3*d^6*e^2 + 18*C*a*c^2*d^5*e^3 + 6*A*c^3*d^5*e^3 - 15*B*a*c^2*d^4*e^4 + 12*C*a^2*c*d^3*e^5 + 12*A*a*c^2*d^3*
e^5 - 9*B*a^2*c*d^2*e^6 + 2*C*a^3*d*e^7 + 6*A*a^2*c*d*e^7 - B*a^3*e^8)*x)*e^(-9)/(x*e + d)^2