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

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

[Out]

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

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Rubi [A]  time = 0.980382, 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*}

Mathematica [A]  time = 0.357787, size = 641, normalized size = 1.32 \[ \frac{210 a^2 c e^4 \left (3 e \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )+2 C \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )\right )+420 a^3 e^6 \left (e (B d-A e)+C \left (-d^2+d e x+e^2 x^2\right )\right )+21 a c^2 e^2 \left (5 e \left (4 A e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+B \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )-6 C \left (-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4-50 d^5 e x+10 d^6+3 d e^5 x^5-2 e^6 x^6\right )\right )-420 (d+e x) \left (a e^2+c d^2\right )^2 \log (d+e x) \left (a e^2 (2 C d-B e)+c d e (6 A e-7 B d)+8 c C d^3\right )+c^3 \left (7 e \left (6 A e \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )+B \left (-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-360 d^6 e x+60 d^7-14 d e^6 x^6+10 e^7 x^7\right )\right )-4 C \left (-420 d^6 e^2 x^2+140 d^5 e^3 x^3-70 d^4 e^4 x^4+42 d^3 e^5 x^5-28 d^2 e^6 x^6-735 d^7 e x+105 d^8+20 d e^7 x^7-15 e^8 x^8\right )\right )}{420 e^9 (d+e x)} \]

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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Maple [A]  time = 0.059, size = 928, normalized size = 1.9 \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)^2,x)

[Out]

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

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Maxima [A]  time = 0.999464, size = 933, normalized size = 1.92 \begin{align*} -\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}} \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)^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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Fricas [A]  time = 1.86145, size = 1987, normalized size = 4.09 \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)^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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Sympy [A]  time = 5.69052, size = 731, normalized size = 1.5 \begin{align*} \frac{C c^{3} x^{7}}{7 e^{2}} - \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{x^{6} \left (- B c^{3} e + 2 C c^{3} d\right )}{6 e^{3}} + \frac{x^{5} \left (A c^{3} e^{2} - 2 B c^{3} d e + 3 C a c^{2} e^{2} + 3 C c^{3} d^{2}\right )}{5 e^{4}} - \frac{x^{4} \left (2 A c^{3} d e^{2} - 3 B a c^{2} e^{3} - 3 B c^{3} d^{2} e + 6 C a c^{2} d e^{2} + 4 C c^{3} d^{3}\right )}{4 e^{5}} + \frac{x^{3} \left (3 A a c^{2} e^{4} + 3 A c^{3} d^{2} e^{2} - 6 B a c^{2} d e^{3} - 4 B c^{3} d^{3} e + 3 C a^{2} c e^{4} + 9 C a c^{2} d^{2} e^{2} + 5 C c^{3} d^{4}\right )}{3 e^{6}} - \frac{x^{2} \left (6 A a c^{2} d e^{4} + 4 A c^{3} d^{3} e^{2} - 3 B a^{2} c e^{5} - 9 B a c^{2} d^{2} e^{3} - 5 B c^{3} d^{4} e + 6 C a^{2} c d e^{4} + 12 C a c^{2} d^{3} e^{2} + 6 C c^{3} d^{5}\right )}{2 e^{7}} + \frac{x \left (3 A a^{2} c e^{6} + 9 A a c^{2} d^{2} e^{4} + 5 A c^{3} d^{4} e^{2} - 6 B a^{2} c d e^{5} - 12 B a c^{2} d^{3} e^{3} - 6 B c^{3} d^{5} e + C a^{3} e^{6} + 9 C a^{2} c d^{2} e^{4} + 15 C a c^{2} d^{4} e^{2} + 7 C c^{3} d^{6}\right )}{e^{8}} - \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}} \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)**2,x)

[Out]

C*c**3*x**7/(7*e**2) - (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) - x**6*(-B*c**3*e + 2*C*c**3*d)/(6*e**3) + x**5*(A*c
**3*e**2 - 2*B*c**3*d*e + 3*C*a*c**2*e**2 + 3*C*c**3*d**2)/(5*e**4) - x**4*(2*A*c**3*d*e**2 - 3*B*a*c**2*e**3
- 3*B*c**3*d**2*e + 6*C*a*c**2*d*e**2 + 4*C*c**3*d**3)/(4*e**5) + x**3*(3*A*a*c**2*e**4 + 3*A*c**3*d**2*e**2 -
6*B*a*c**2*d*e**3 - 4*B*c**3*d**3*e + 3*C*a**2*c*e**4 + 9*C*a*c**2*d**2*e**2 + 5*C*c**3*d**4)/(3*e**6) - x**2
*(6*A*a*c**2*d*e**4 + 4*A*c**3*d**3*e**2 - 3*B*a**2*c*e**5 - 9*B*a*c**2*d**2*e**3 - 5*B*c**3*d**4*e + 6*C*a**2
*c*d*e**4 + 12*C*a*c**2*d**3*e**2 + 6*C*c**3*d**5)/(2*e**7) + x*(3*A*a**2*c*e**6 + 9*A*a*c**2*d**2*e**4 + 5*A*
c**3*d**4*e**2 - 6*B*a**2*c*d*e**5 - 12*B*a*c**2*d**3*e**3 - 6*B*c**3*d**5*e + C*a**3*e**6 + 9*C*a**2*c*d**2*e
**4 + 15*C*a*c**2*d**4*e**2 + 7*C*c**3*d**6)/e**8 - (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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Giac [A]  time = 1.21041, size = 1131, normalized size = 2.33 \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)^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)