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

Optimal. Leaf size=486 \[ \frac {c x^3 \left (3 a^2 C e^4+3 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 d^2 \left (5 C d^2-e (4 B d-3 A e)\right )\right )}{3 e^6}-\frac {c x^2 \left (3 a^2 e^4 (2 C d-B e)+3 a c d e^2 \left (4 C d^2-e (3 B d-2 A e)\right )+c^2 d^3 \left (6 C d^2-e (5 B d-4 A e)\right )\right )}{2 e^7}+\frac {x \left (a^3 C e^6+3 a^2 c e^4 \left (3 C d^2-e (2 B d-A e)\right )+3 a c^2 d^2 e^2 \left (5 C d^2-e (4 B d-3 A e)\right )+c^3 d^4 \left (7 C d^2-e (6 B d-5 A e)\right )\right )}{e^8}-\frac {c^2 x^4 \left (3 a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{4 e^5}+\frac {c^2 x^5 \left (3 a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right )}{5 e^4}-\frac {\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{e^9 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right )}{e^9}-\frac {c^3 x^6 (2 C d-B e)}{6 e^3}+\frac {c^3 C x^7}{7 e^2} \]

[Out]

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

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Rubi [A]  time = 0.98, antiderivative size = 483, 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} \[ \frac {c x^3 \left (3 a^2 C e^4+3 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )\right )}{3 e^6}-\frac {c x^2 \left (3 a^2 e^4 (2 C d-B e)+3 a c d e^2 \left (4 C d^2-e (3 B d-2 A e)\right )+c^2 \left (6 C d^5-d^3 e (5 B d-4 A e)\right )\right )}{2 e^7}+\frac {x \left (3 a^2 c e^4 \left (3 C d^2-e (2 B d-A e)\right )+a^3 C e^6+3 a c^2 d^2 e^2 \left (5 C d^2-e (4 B d-3 A e)\right )+c^3 \left (7 C d^6-d^4 e (6 B d-5 A e)\right )\right )}{e^8}+\frac {c^2 x^5 \left (3 a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{5 e^4}-\frac {c^2 x^4 \left (3 a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{4 e^5}-\frac {\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{e^9 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2 \log (d+e x) \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^9}-\frac {c^3 x^6 (2 C d-B e)}{6 e^3}+\frac {c^3 C x^7}{7 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.40, size = 641, normalized size = 1.32 \[ \frac {420 a^3 e^6 \left (e (B d-A e)+C \left (-d^2+d e x+e^2 x^2\right )\right )+210 a^2 c e^4 \left (3 e \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+2 C \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+21 a c^2 e^2 \left (5 e \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )-6 C \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )-420 (d+e x) \left (a e^2+c d^2\right )^2 \log (d+e x) \left (a e^2 (2 C d-B e)+c d e (6 A e-7 B d)+8 c C d^3\right )+c^3 \left (7 e \left (6 A e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+B \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )-4 C \left (105 d^8-735 d^7 e x-420 d^6 e^2 x^2+140 d^5 e^3 x^3-70 d^4 e^4 x^4+42 d^3 e^5 x^5-28 d^2 e^6 x^6+20 d e^7 x^7-15 e^8 x^8\right )\right )}{420 e^9 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(420*a^3*e^6*(e*(B*d - A*e) + C*(-d^2 + d*e*x + e^2*x^2)) + 210*a^2*c*e^4*(2*C*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2
*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 3*e*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3
*x^3))) + 21*a*c^2*e^2*(-6*C*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*
x^5 - 2*e^6*x^6) + 5*e*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^
4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5))) + c^3*(-4*C*(105*d^8 - 735*d^7*e*x - 420*
d^6*e^2*x^2 + 140*d^5*e^3*x^3 - 70*d^4*e^4*x^4 + 42*d^3*e^5*x^5 - 28*d^2*e^6*x^6 + 20*d*e^7*x^7 - 15*e^8*x^8)
+ 7*e*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6
) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^
6 + 10*e^7*x^7))) - 420*(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)
*Log[d + e*x])/(420*e^9*(d + e*x))

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fricas [A]  time = 0.93, size = 932, normalized size = 1.92 \[ \frac {60 \, C c^{3} e^{8} x^{8} - 420 \, C c^{3} d^{8} + 420 \, B c^{3} d^{7} e + 1260 \, B a c^{2} d^{5} e^{3} + 1260 \, B a^{2} c d^{3} e^{5} + 420 \, B a^{3} d e^{7} - 420 \, A a^{3} e^{8} - 420 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} - 1260 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} - 420 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} - 10 \, {\left (8 \, C c^{3} d e^{7} - 7 \, B c^{3} e^{8}\right )} x^{7} + 14 \, {\left (8 \, C c^{3} d^{2} e^{6} - 7 \, B c^{3} d e^{7} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} e^{8}\right )} x^{6} - 21 \, {\left (8 \, C c^{3} d^{3} e^{5} - 7 \, B c^{3} d^{2} e^{6} - 15 \, B a c^{2} e^{8} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d e^{7}\right )} x^{5} + 35 \, {\left (8 \, C c^{3} d^{4} e^{4} - 7 \, B c^{3} d^{3} e^{5} - 15 \, B a c^{2} d e^{7} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{6} + 12 \, {\left (C a^{2} c + A a c^{2}\right )} e^{8}\right )} x^{4} - 70 \, {\left (8 \, C c^{3} d^{5} e^{3} - 7 \, B c^{3} d^{4} e^{4} - 15 \, B a c^{2} d^{2} e^{6} - 9 \, B a^{2} c e^{8} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{5} + 12 \, {\left (C a^{2} c + A a c^{2}\right )} d e^{7}\right )} x^{3} + 210 \, {\left (8 \, C c^{3} d^{6} e^{2} - 7 \, B c^{3} d^{5} e^{3} - 15 \, B a c^{2} d^{3} e^{5} - 9 \, B a^{2} c d e^{7} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{4} + 12 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{6} + 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{8}\right )} x^{2} + 420 \, {\left (7 \, C c^{3} d^{7} e - 6 \, B c^{3} d^{6} e^{2} - 12 \, B a c^{2} d^{4} e^{4} - 6 \, B a^{2} c d^{2} e^{6} + 5 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 9 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x - 420 \, {\left (8 \, C c^{3} d^{8} - 7 \, B c^{3} d^{7} e - 15 \, B a c^{2} d^{5} e^{3} - 9 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 12 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} + {\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\right )} \log \left (e x + d\right )}{420 \, {\left (e^{10} x + d e^{9}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/420*(60*C*c^3*e^8*x^8 - 420*C*c^3*d^8 + 420*B*c^3*d^7*e + 1260*B*a*c^2*d^5*e^3 + 1260*B*a^2*c*d^3*e^5 + 420*
B*a^3*d*e^7 - 420*A*a^3*e^8 - 420*(3*C*a*c^2 + A*c^3)*d^6*e^2 - 1260*(C*a^2*c + A*a*c^2)*d^4*e^4 - 420*(C*a^3
+ 3*A*a^2*c)*d^2*e^6 - 10*(8*C*c^3*d*e^7 - 7*B*c^3*e^8)*x^7 + 14*(8*C*c^3*d^2*e^6 - 7*B*c^3*d*e^7 + 6*(3*C*a*c
^2 + A*c^3)*e^8)*x^6 - 21*(8*C*c^3*d^3*e^5 - 7*B*c^3*d^2*e^6 - 15*B*a*c^2*e^8 + 6*(3*C*a*c^2 + A*c^3)*d*e^7)*x
^5 + 35*(8*C*c^3*d^4*e^4 - 7*B*c^3*d^3*e^5 - 15*B*a*c^2*d*e^7 + 6*(3*C*a*c^2 + A*c^3)*d^2*e^6 + 12*(C*a^2*c +
A*a*c^2)*e^8)*x^4 - 70*(8*C*c^3*d^5*e^3 - 7*B*c^3*d^4*e^4 - 15*B*a*c^2*d^2*e^6 - 9*B*a^2*c*e^8 + 6*(3*C*a*c^2
+ A*c^3)*d^3*e^5 + 12*(C*a^2*c + A*a*c^2)*d*e^7)*x^3 + 210*(8*C*c^3*d^6*e^2 - 7*B*c^3*d^5*e^3 - 15*B*a*c^2*d^3
*e^5 - 9*B*a^2*c*d*e^7 + 6*(3*C*a*c^2 + A*c^3)*d^4*e^4 + 12*(C*a^2*c + A*a*c^2)*d^2*e^6 + 2*(C*a^3 + 3*A*a^2*c
)*e^8)*x^2 + 420*(7*C*c^3*d^7*e - 6*B*c^3*d^6*e^2 - 12*B*a*c^2*d^4*e^4 - 6*B*a^2*c*d^2*e^6 + 5*(3*C*a*c^2 + A*
c^3)*d^5*e^3 + 9*(C*a^2*c + A*a*c^2)*d^3*e^5 + (C*a^3 + 3*A*a^2*c)*d*e^7)*x - 420*(8*C*c^3*d^8 - 7*B*c^3*d^7*e
- 15*B*a*c^2*d^5*e^3 - 9*B*a^2*c*d^3*e^5 - B*a^3*d*e^7 + 6*(3*C*a*c^2 + A*c^3)*d^6*e^2 + 12*(C*a^2*c + A*a*c^
2)*d^4*e^4 + 2*(C*a^3 + 3*A*a^2*c)*d^2*e^6 + (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)*log(e*x + d))/(e^10*x + d*e^9)

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giac [A]  time = 0.20, size = 838, normalized size = 1.72 \[ \frac {1}{420} \, {\left (60 \, C c^{3} - \frac {70 \, {\left (8 \, C c^{3} d e - B c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {84 \, {\left (28 \, C c^{3} d^{2} e^{2} - 7 \, B c^{3} d e^{3} + 3 \, C a c^{2} e^{4} + A c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {105 \, {\left (56 \, C c^{3} d^{3} e^{3} - 21 \, B c^{3} d^{2} e^{4} + 18 \, C a c^{2} d e^{5} + 6 \, A c^{3} d e^{5} - 3 \, B a c^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {140 \, {\left (70 \, C c^{3} d^{4} e^{4} - 35 \, B c^{3} d^{3} e^{5} + 45 \, C a c^{2} d^{2} e^{6} + 15 \, A c^{3} d^{2} e^{6} - 15 \, B a c^{2} d e^{7} + 3 \, C a^{2} c e^{8} + 3 \, A a c^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {210 \, {\left (56 \, C c^{3} d^{5} e^{5} - 35 \, B c^{3} d^{4} e^{6} + 60 \, C a c^{2} d^{3} e^{7} + 20 \, A c^{3} d^{3} e^{7} - 30 \, B a c^{2} d^{2} e^{8} + 12 \, C a^{2} c d e^{9} + 12 \, A a c^{2} d e^{9} - 3 \, B a^{2} c e^{10}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} + \frac {420 \, {\left (28 \, C c^{3} d^{6} e^{6} - 21 \, B c^{3} d^{5} e^{7} + 45 \, C a c^{2} d^{4} e^{8} + 15 \, A c^{3} d^{4} e^{8} - 30 \, B a c^{2} d^{3} e^{9} + 18 \, C a^{2} c d^{2} e^{10} + 18 \, A a c^{2} d^{2} e^{10} - 9 \, B a^{2} c d e^{11} + C a^{3} e^{12} + 3 \, A a^{2} c e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}}\right )} {\left (x e + d\right )}^{7} e^{\left (-9\right )} + {\left (8 \, C c^{3} d^{7} - 7 \, B c^{3} d^{6} e + 18 \, C a c^{2} d^{5} e^{2} + 6 \, A c^{3} d^{5} e^{2} - 15 \, B a c^{2} d^{4} e^{3} + 12 \, C a^{2} c d^{3} e^{4} + 12 \, A a c^{2} d^{3} e^{4} - 9 \, B a^{2} c d^{2} e^{5} + 2 \, C a^{3} d e^{6} + 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {C c^{3} d^{8} e^{7}}{x e + d} - \frac {B c^{3} d^{7} e^{8}}{x e + d} + \frac {3 \, C a c^{2} d^{6} e^{9}}{x e + d} + \frac {A c^{3} d^{6} e^{9}}{x e + d} - \frac {3 \, B a c^{2} d^{5} e^{10}}{x e + d} + \frac {3 \, C a^{2} c d^{4} e^{11}}{x e + d} + \frac {3 \, A a c^{2} d^{4} e^{11}}{x e + d} - \frac {3 \, B a^{2} c d^{3} e^{12}}{x e + d} + \frac {C a^{3} d^{2} e^{13}}{x e + d} + \frac {3 \, A a^{2} c d^{2} e^{13}}{x e + d} - \frac {B a^{3} d e^{14}}{x e + d} + \frac {A a^{3} e^{15}}{x e + d}\right )} e^{\left (-16\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/420*(60*C*c^3 - 70*(8*C*c^3*d*e - B*c^3*e^2)*e^(-1)/(x*e + d) + 84*(28*C*c^3*d^2*e^2 - 7*B*c^3*d*e^3 + 3*C*a
*c^2*e^4 + A*c^3*e^4)*e^(-2)/(x*e + d)^2 - 105*(56*C*c^3*d^3*e^3 - 21*B*c^3*d^2*e^4 + 18*C*a*c^2*d*e^5 + 6*A*c
^3*d*e^5 - 3*B*a*c^2*e^6)*e^(-3)/(x*e + d)^3 + 140*(70*C*c^3*d^4*e^4 - 35*B*c^3*d^3*e^5 + 45*C*a*c^2*d^2*e^6 +
15*A*c^3*d^2*e^6 - 15*B*a*c^2*d*e^7 + 3*C*a^2*c*e^8 + 3*A*a*c^2*e^8)*e^(-4)/(x*e + d)^4 - 210*(56*C*c^3*d^5*e
^5 - 35*B*c^3*d^4*e^6 + 60*C*a*c^2*d^3*e^7 + 20*A*c^3*d^3*e^7 - 30*B*a*c^2*d^2*e^8 + 12*C*a^2*c*d*e^9 + 12*A*a
*c^2*d*e^9 - 3*B*a^2*c*e^10)*e^(-5)/(x*e + d)^5 + 420*(28*C*c^3*d^6*e^6 - 21*B*c^3*d^5*e^7 + 45*C*a*c^2*d^4*e^
8 + 15*A*c^3*d^4*e^8 - 30*B*a*c^2*d^3*e^9 + 18*C*a^2*c*d^2*e^10 + 18*A*a*c^2*d^2*e^10 - 9*B*a^2*c*d*e^11 + C*a
^3*e^12 + 3*A*a^2*c*e^12)*e^(-6)/(x*e + d)^6)*(x*e + d)^7*e^(-9) + (8*C*c^3*d^7 - 7*B*c^3*d^6*e + 18*C*a*c^2*d
^5*e^2 + 6*A*c^3*d^5*e^2 - 15*B*a*c^2*d^4*e^3 + 12*C*a^2*c*d^3*e^4 + 12*A*a*c^2*d^3*e^4 - 9*B*a^2*c*d^2*e^5 +
2*C*a^3*d*e^6 + 6*A*a^2*c*d*e^6 - B*a^3*e^7)*e^(-9)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (C*c^3*d^8*e^7/(x*e
+ d) - B*c^3*d^7*e^8/(x*e + d) + 3*C*a*c^2*d^6*e^9/(x*e + d) + A*c^3*d^6*e^9/(x*e + d) - 3*B*a*c^2*d^5*e^10/(
x*e + d) + 3*C*a^2*c*d^4*e^11/(x*e + d) + 3*A*a*c^2*d^4*e^11/(x*e + d) - 3*B*a^2*c*d^3*e^12/(x*e + d) + C*a^3*
d^2*e^13/(x*e + d) + 3*A*a^2*c*d^2*e^13/(x*e + d) - B*a^3*d*e^14/(x*e + d) + A*a^3*e^15/(x*e + d))*e^(-16)

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maple [A]  time = 0.02, size = 928, normalized size = 1.91 \[ \frac {C \,c^{3} x^{7}}{7 e^{2}}+\frac {B \,c^{3} x^{6}}{6 e^{2}}-\frac {C \,c^{3} d \,x^{6}}{3 e^{3}}+\frac {A \,c^{3} x^{5}}{5 e^{2}}-\frac {2 B \,c^{3} d \,x^{5}}{5 e^{3}}+\frac {3 C a \,c^{2} x^{5}}{5 e^{2}}+\frac {3 C \,c^{3} d^{2} x^{5}}{5 e^{4}}-\frac {A \,c^{3} d \,x^{4}}{2 e^{3}}+\frac {3 B a \,c^{2} x^{4}}{4 e^{2}}+\frac {3 B \,c^{3} d^{2} x^{4}}{4 e^{4}}-\frac {3 C a \,c^{2} d \,x^{4}}{2 e^{3}}-\frac {C \,c^{3} d^{3} x^{4}}{e^{5}}+\frac {A a \,c^{2} x^{3}}{e^{2}}+\frac {A \,c^{3} d^{2} x^{3}}{e^{4}}-\frac {2 B a \,c^{2} d \,x^{3}}{e^{3}}-\frac {4 B \,c^{3} d^{3} x^{3}}{3 e^{5}}+\frac {C \,a^{2} c \,x^{3}}{e^{2}}+\frac {3 C a \,c^{2} d^{2} x^{3}}{e^{4}}+\frac {5 C \,c^{3} d^{4} x^{3}}{3 e^{6}}-\frac {3 A a \,c^{2} d \,x^{2}}{e^{3}}-\frac {2 A \,c^{3} d^{3} x^{2}}{e^{5}}+\frac {3 B \,a^{2} c \,x^{2}}{2 e^{2}}+\frac {9 B a \,c^{2} d^{2} x^{2}}{2 e^{4}}+\frac {5 B \,c^{3} d^{4} x^{2}}{2 e^{6}}-\frac {3 C \,a^{2} c d \,x^{2}}{e^{3}}-\frac {6 C a \,c^{2} d^{3} x^{2}}{e^{5}}-\frac {3 C \,c^{3} d^{5} x^{2}}{e^{7}}-\frac {A \,a^{3}}{\left (e x +d \right ) e}-\frac {3 A \,a^{2} c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {6 A \,a^{2} c d \ln \left (e x +d \right )}{e^{3}}+\frac {3 A \,a^{2} c x}{e^{2}}-\frac {3 A a \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {12 A a \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {9 A a \,c^{2} d^{2} x}{e^{4}}-\frac {A \,c^{3} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {6 A \,c^{3} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {5 A \,c^{3} d^{4} x}{e^{6}}+\frac {B \,a^{3} d}{\left (e x +d \right ) e^{2}}+\frac {B \,a^{3} \ln \left (e x +d \right )}{e^{2}}+\frac {3 B \,a^{2} c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {9 B \,a^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {6 B \,a^{2} c d x}{e^{3}}+\frac {3 B a \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {15 B a \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {12 B a \,c^{2} d^{3} x}{e^{5}}+\frac {B \,c^{3} d^{7}}{\left (e x +d \right ) e^{8}}+\frac {7 B \,c^{3} d^{6} \ln \left (e x +d \right )}{e^{8}}-\frac {6 B \,c^{3} d^{5} x}{e^{7}}-\frac {C \,a^{3} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 C \,a^{3} d \ln \left (e x +d \right )}{e^{3}}+\frac {C \,a^{3} x}{e^{2}}-\frac {3 C \,a^{2} c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {12 C \,a^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {9 C \,a^{2} c \,d^{2} x}{e^{4}}-\frac {3 C a \,c^{2} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {18 C a \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {15 C a \,c^{2} d^{4} x}{e^{6}}-\frac {C \,c^{3} d^{8}}{\left (e x +d \right ) e^{9}}-\frac {8 C \,c^{3} d^{7} \ln \left (e x +d \right )}{e^{9}}+\frac {7 C \,c^{3} d^{6} x}{e^{8}} \]

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)^2,x)

[Out]

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

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maxima [A]  time = 0.49, size = 691, normalized size = 1.42 \[ -\frac {C c^{3} d^{8} - B c^{3} d^{7} e - 3 \, B a c^{2} d^{5} e^{3} - 3 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} + A a^{3} e^{8} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6}}{e^{10} x + d e^{9}} + \frac {60 \, C c^{3} e^{6} x^{7} - 70 \, {\left (2 \, C c^{3} d e^{5} - B c^{3} e^{6}\right )} x^{6} + 84 \, {\left (3 \, C c^{3} d^{2} e^{4} - 2 \, B c^{3} d e^{5} + {\left (3 \, C a c^{2} + A c^{3}\right )} e^{6}\right )} x^{5} - 105 \, {\left (4 \, C c^{3} d^{3} e^{3} - 3 \, B c^{3} d^{2} e^{4} - 3 \, B a c^{2} e^{6} + 2 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d e^{5}\right )} x^{4} + 140 \, {\left (5 \, C c^{3} d^{4} e^{2} - 4 \, B c^{3} d^{3} e^{3} - 6 \, B a c^{2} d e^{5} + 3 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{4} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} e^{6}\right )} x^{3} - 210 \, {\left (6 \, C c^{3} d^{5} e - 5 \, B c^{3} d^{4} e^{2} - 9 \, B a c^{2} d^{2} e^{4} - 3 \, B a^{2} c e^{6} + 4 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{3} + 6 \, {\left (C a^{2} c + A a c^{2}\right )} d e^{5}\right )} x^{2} + 420 \, {\left (7 \, C c^{3} d^{6} - 6 \, B c^{3} d^{5} e - 12 \, B a c^{2} d^{3} e^{3} - 6 \, B a^{2} c d e^{5} + 5 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{2} + 9 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{6}\right )} x}{420 \, e^{8}} - \frac {{\left (8 \, C c^{3} d^{7} - 7 \, B c^{3} d^{6} e - 15 \, B a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{2} e^{5} - B a^{3} e^{7} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{2} + 12 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{4} + 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \]

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)^2,x, algorithm="maxima")

[Out]

-(C*c^3*d^8 - B*c^3*d^7*e - 3*B*a*c^2*d^5*e^3 - 3*B*a^2*c*d^3*e^5 - B*a^3*d*e^7 + A*a^3*e^8 + (3*C*a*c^2 + A*c
^3)*d^6*e^2 + 3*(C*a^2*c + A*a*c^2)*d^4*e^4 + (C*a^3 + 3*A*a^2*c)*d^2*e^6)/(e^10*x + d*e^9) + 1/420*(60*C*c^3*
e^6*x^7 - 70*(2*C*c^3*d*e^5 - B*c^3*e^6)*x^6 + 84*(3*C*c^3*d^2*e^4 - 2*B*c^3*d*e^5 + (3*C*a*c^2 + A*c^3)*e^6)*
x^5 - 105*(4*C*c^3*d^3*e^3 - 3*B*c^3*d^2*e^4 - 3*B*a*c^2*e^6 + 2*(3*C*a*c^2 + A*c^3)*d*e^5)*x^4 + 140*(5*C*c^3
*d^4*e^2 - 4*B*c^3*d^3*e^3 - 6*B*a*c^2*d*e^5 + 3*(3*C*a*c^2 + A*c^3)*d^2*e^4 + 3*(C*a^2*c + A*a*c^2)*e^6)*x^3
- 210*(6*C*c^3*d^5*e - 5*B*c^3*d^4*e^2 - 9*B*a*c^2*d^2*e^4 - 3*B*a^2*c*e^6 + 4*(3*C*a*c^2 + A*c^3)*d^3*e^3 + 6
*(C*a^2*c + A*a*c^2)*d*e^5)*x^2 + 420*(7*C*c^3*d^6 - 6*B*c^3*d^5*e - 12*B*a*c^2*d^3*e^3 - 6*B*a^2*c*d*e^5 + 5*
(3*C*a*c^2 + A*c^3)*d^4*e^2 + 9*(C*a^2*c + A*a*c^2)*d^2*e^4 + (C*a^3 + 3*A*a^2*c)*e^6)*x)/e^8 - (8*C*c^3*d^7 -
7*B*c^3*d^6*e - 15*B*a*c^2*d^4*e^3 - 9*B*a^2*c*d^2*e^5 - B*a^3*e^7 + 6*(3*C*a*c^2 + A*c^3)*d^5*e^2 + 12*(C*a^
2*c + A*a*c^2)*d^3*e^4 + 2*(C*a^3 + 3*A*a^2*c)*d*e^6)*log(e*x + d)/e^9

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mupad [B]  time = 3.99, size = 1511, normalized size = 3.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((C*a^3 + 3*A*a^2*c)/e^2 + (2*d*((2*d*((d^2*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e
^2 + (C*c^3*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (
C*c^3*d^2)/e^4))/e - (d^2*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e^2 + (3*B*a*c^2)/e^2))/e + (3*a*c*(A*c + C*a))/e^2
))/e + (d^2*((2*d*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/e - (
d^2*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e^2 + (3*B*a*c^2)/e^2))/e^2 - (3*B*a^2*c)/e^2))/e - (d^2*((d^2*((2*d*((B*
c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((B*c^3)/e
^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/e - (d^2*((B*c^3)/e^2 - (2*C*c^3*d)/e^3
))/e^2 + (3*B*a*c^2)/e^2))/e + (3*a*c*(A*c + C*a))/e^2))/e^2) + x^4*((d*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))
/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/(2*e) - (d^2*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/(4*e^2) + (3*B*
a*c^2)/(4*e^2)) - x^2*((d*((d^2*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^
2)/e^4))/e^2 - (2*d*((2*d*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4
))/e - (d^2*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e^2 + (3*B*a*c^2)/e^2))/e + (3*a*c*(A*c + C*a))/e^2))/e + (d^2*((
2*d*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/e - (d^2*((B*c^3)/e
^2 - (2*C*c^3*d)/e^3))/e^2 + (3*B*a*c^2)/e^2))/(2*e^2) - (3*B*a^2*c)/(2*e^2)) + x^6*((B*c^3)/(6*e^2) - (C*c^3*
d)/(3*e^3)) - x^5*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/(5*e) - (A*c^3 + 3*C*a*c^2)/(5*e^2) + (C*c^3*d^2)/(5*
e^4)) + x^3*((d^2*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/(3*e^
2) - (2*d*((2*d*((2*d*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e - (A*c^3 + 3*C*a*c^2)/e^2 + (C*c^3*d^2)/e^4))/e - (d^
2*((B*c^3)/e^2 - (2*C*c^3*d)/e^3))/e^2 + (3*B*a*c^2)/e^2))/(3*e) + (a*c*(A*c + C*a))/e^2) - (A*a^3*e^8 + C*c^3
*d^8 - B*a^3*d*e^7 - B*c^3*d^7*e + A*c^3*d^6*e^2 + C*a^3*d^2*e^6 + 3*A*a*c^2*d^4*e^4 + 3*A*a^2*c*d^2*e^6 - 3*B
*a*c^2*d^5*e^3 - 3*B*a^2*c*d^3*e^5 + 3*C*a*c^2*d^6*e^2 + 3*C*a^2*c*d^4*e^4)/(e*(d*e^8 + e^9*x)) - (log(d + e*x
)*(8*C*c^3*d^7 - B*a^3*e^7 + 2*C*a^3*d*e^6 - 7*B*c^3*d^6*e + 6*A*c^3*d^5*e^2 + 12*A*a*c^2*d^3*e^4 - 15*B*a*c^2
*d^4*e^3 - 9*B*a^2*c*d^2*e^5 + 18*C*a*c^2*d^5*e^2 + 12*C*a^2*c*d^3*e^4 + 6*A*a^2*c*d*e^6))/e^9 + (C*c^3*x^7)/(
7*e^2)

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sympy [A]  time = 4.95, size = 748, normalized size = 1.54 \[ \frac {C c^{3} x^{7}}{7 e^{2}} + x^{6} \left (\frac {B c^{3}}{6 e^{2}} - \frac {C c^{3} d}{3 e^{3}}\right ) + x^{5} \left (\frac {A c^{3}}{5 e^{2}} - \frac {2 B c^{3} d}{5 e^{3}} + \frac {3 C a c^{2}}{5 e^{2}} + \frac {3 C c^{3} d^{2}}{5 e^{4}}\right ) + x^{4} \left (- \frac {A c^{3} d}{2 e^{3}} + \frac {3 B a c^{2}}{4 e^{2}} + \frac {3 B c^{3} d^{2}}{4 e^{4}} - \frac {3 C a c^{2} d}{2 e^{3}} - \frac {C c^{3} d^{3}}{e^{5}}\right ) + x^{3} \left (\frac {A a c^{2}}{e^{2}} + \frac {A c^{3} d^{2}}{e^{4}} - \frac {2 B a c^{2} d}{e^{3}} - \frac {4 B c^{3} d^{3}}{3 e^{5}} + \frac {C a^{2} c}{e^{2}} + \frac {3 C a c^{2} d^{2}}{e^{4}} + \frac {5 C c^{3} d^{4}}{3 e^{6}}\right ) + x^{2} \left (- \frac {3 A a c^{2} d}{e^{3}} - \frac {2 A c^{3} d^{3}}{e^{5}} + \frac {3 B a^{2} c}{2 e^{2}} + \frac {9 B a c^{2} d^{2}}{2 e^{4}} + \frac {5 B c^{3} d^{4}}{2 e^{6}} - \frac {3 C a^{2} c d}{e^{3}} - \frac {6 C a c^{2} d^{3}}{e^{5}} - \frac {3 C c^{3} d^{5}}{e^{7}}\right ) + x \left (\frac {3 A a^{2} c}{e^{2}} + \frac {9 A a c^{2} d^{2}}{e^{4}} + \frac {5 A c^{3} d^{4}}{e^{6}} - \frac {6 B a^{2} c d}{e^{3}} - \frac {12 B a c^{2} d^{3}}{e^{5}} - \frac {6 B c^{3} d^{5}}{e^{7}} + \frac {C a^{3}}{e^{2}} + \frac {9 C a^{2} c d^{2}}{e^{4}} + \frac {15 C a c^{2} d^{4}}{e^{6}} + \frac {7 C c^{3} d^{6}}{e^{8}}\right ) + \frac {- A a^{3} e^{8} - 3 A a^{2} c d^{2} e^{6} - 3 A a c^{2} d^{4} e^{4} - A c^{3} d^{6} e^{2} + B a^{3} d e^{7} + 3 B a^{2} c d^{3} e^{5} + 3 B a c^{2} d^{5} e^{3} + B c^{3} d^{7} e - C a^{3} d^{2} e^{6} - 3 C a^{2} c d^{4} e^{4} - 3 C a c^{2} d^{6} e^{2} - C c^{3} d^{8}}{d e^{9} + e^{10} x} - \frac {\left (a e^{2} + c d^{2}\right )^{2} \left (6 A c d e^{2} - B a e^{3} - 7 B c d^{2} e + 2 C a d e^{2} + 8 C c d^{3}\right ) \log {\left (d + e x \right )}}{e^{9}} \]

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)**2,x)

[Out]

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

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