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

Optimal. Leaf size=490 \[ \frac{c (d+e x)^4 \left (3 a^2 C e^4+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )+5 c^2 d^2 \left (14 C d^2-e (7 B d-3 A e)\right )\right )}{4 e^9}-\frac{c (d+e x)^3 \left (3 a^2 e^4 (4 C d-B e)+6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )+c^2 d^3 \left (56 C d^2-5 e (7 B d-4 A e)\right )\right )}{3 e^9}+\frac{(d+e x)^2 \left (a e^2+c d^2\right ) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 d^2 \left (28 C d^2-3 e (7 B d-5 A e)\right )\right )}{2 e^9}+\frac{c^2 (d+e x)^6 \left (3 a C e^2+c \left (28 C d^2-e (7 B d-A e)\right )\right )}{6 e^9}-\frac{c^2 (d+e x)^5 \left (3 a e^2 (6 C d-B e)+c d \left (56 C d^2-3 e (7 B d-2 A e)\right )\right )}{5 e^9}-\frac{x \left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right )}{e^8}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^9}-\frac{c^3 (d+e x)^7 (8 C d-B e)}{7 e^9}+\frac{c^3 C (d+e x)^8}{8 e^9} \]

[Out]

-(((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)))*x)/e^8) + ((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)))*(d + e*x)^2)/(2*e^
9) - (c*(3*a^2*e^4*(4*C*d - B*e) + c^2*d^3*(56*C*d^2 - 5*e*(7*B*d - 4*A*e)) + 6*a*c*d*e^2*(10*C*d^2 - e*(5*B*d
- 2*A*e)))*(d + e*x)^3)/(3*e^9) + (c*(3*a^2*C*e^4 + 5*c^2*d^2*(14*C*d^2 - e*(7*B*d - 3*A*e)) + 3*a*c*e^2*(15*
C*d^2 - e*(5*B*d - A*e)))*(d + e*x)^4)/(4*e^9) - (c^2*(3*a*e^2*(6*C*d - B*e) + c*d*(56*C*d^2 - 3*e*(7*B*d - 2*
A*e)))*(d + e*x)^5)/(5*e^9) + (c^2*(3*a*C*e^2 + c*(28*C*d^2 - e*(7*B*d - A*e)))*(d + e*x)^6)/(6*e^9) - (c^3*(8
*C*d - B*e)*(d + e*x)^7)/(7*e^9) + (c^3*C*(d + e*x)^8)/(8*e^9) + ((c*d^2 + a*e^2)^3*(C*d^2 - B*d*e + A*e^2)*Lo
g[d + e*x])/e^9

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Rubi [A]  time = 1.09758, antiderivative size = 487, 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} \[ \frac{c (d+e x)^4 \left (3 a^2 C e^4+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )+5 c^2 \left (14 C d^4-d^2 e (7 B d-3 A e)\right )\right )}{4 e^9}-\frac{c (d+e x)^3 \left (3 a^2 e^4 (4 C d-B e)+6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )+c^2 \left (56 C d^5-5 d^3 e (7 B d-4 A e)\right )\right )}{3 e^9}+\frac{(d+e x)^2 \left (a e^2+c d^2\right ) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{2 e^9}+\frac{c^2 (d+e x)^6 \left (3 a C e^2-c e (7 B d-A e)+28 c C d^2\right )}{6 e^9}-\frac{c^2 (d+e x)^5 \left (3 a e^2 (6 C d-B e)-3 c d e (7 B d-2 A e)+56 c C d^3\right )}{5 e^9}-\frac{x \left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^8}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^9}-\frac{c^3 (d+e x)^7 (8 C d-B e)}{7 e^9}+\frac{c^3 C (d+e x)^8}{8 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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))*x)/e^8) + ((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)))*(d + e*x)^2)/(2*e^
9) - (c*(3*a^2*e^4*(4*C*d - B*e) + c^2*(56*C*d^5 - 5*d^3*e*(7*B*d - 4*A*e)) + 6*a*c*d*e^2*(10*C*d^2 - e*(5*B*d
- 2*A*e)))*(d + e*x)^3)/(3*e^9) + (c*(3*a^2*C*e^4 + 5*c^2*(14*C*d^4 - d^2*e*(7*B*d - 3*A*e)) + 3*a*c*e^2*(15*
C*d^2 - e*(5*B*d - A*e)))*(d + e*x)^4)/(4*e^9) - (c^2*(56*c*C*d^3 - 3*c*d*e*(7*B*d - 2*A*e) + 3*a*e^2*(6*C*d -
B*e))*(d + e*x)^5)/(5*e^9) + (c^2*(28*c*C*d^2 + 3*a*C*e^2 - c*e*(7*B*d - A*e))*(d + e*x)^6)/(6*e^9) - (c^3*(8
*C*d - B*e)*(d + e*x)^7)/(7*e^9) + (c^3*C*(d + e*x)^8)/(8*e^9) + ((c*d^2 + a*e^2)^3*(C*d^2 - B*d*e + A*e^2)*Lo
g[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} \, dx &=\int \left (\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}+\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)}+\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 ) (d+e x)}{e^8}+\frac{c \left (-3 a^2 e^4 (4 C d-B e)-c^2 \left (56 C d^5-5 d^3 e (7 B d-4 A e)\right )-6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )\right ) (d+e x)^2}{e^8}+\frac{c \left (3 a^2 C e^4+5 c^2 \left (14 C d^4-d^2 e (7 B d-3 A e)\right )+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )\right ) (d+e x)^3}{e^8}+\frac{c^2 \left (-56 c C d^3+3 c d e (7 B d-2 A e)-3 a e^2 (6 C d-B e)\right ) (d+e x)^4}{e^8}+\frac{c^2 \left (28 c C d^2+3 a C e^2-c e (7 B d-A e)\right ) (d+e x)^5}{e^8}+\frac{c^3 (-8 C d+B e) (d+e x)^6}{e^8}+\frac{c^3 C (d+e x)^7}{e^8}\right ) \, dx\\ &=-\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 ) x}{e^8}+\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 ) (d+e x)^2}{2 e^9}-\frac{c \left (3 a^2 e^4 (4 C d-B e)+c^2 \left (56 C d^5-5 d^3 e (7 B d-4 A e)\right )+6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )\right ) (d+e x)^3}{3 e^9}+\frac{c \left (3 a^2 C e^4+5 c^2 \left (14 C d^4-d^2 e (7 B d-3 A e)\right )+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )\right ) (d+e x)^4}{4 e^9}-\frac{c^2 \left (56 c C d^3-3 c d e (7 B d-2 A e)+3 a e^2 (6 C d-B e)\right ) (d+e x)^5}{5 e^9}+\frac{c^2 \left (28 c C d^2+3 a C e^2-c e (7 B d-A e)\right ) (d+e x)^6}{6 e^9}-\frac{c^3 (8 C d-B e) (d+e x)^7}{7 e^9}+\frac{c^3 C (d+e x)^8}{8 e^9}+\frac{\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.504503, size = 498, normalized size = 1.02 \[ \frac{x \left (210 a^2 c e^4 \left (2 e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+C \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+420 a^3 e^6 (2 B e-2 C d+C e x)+42 a c^2 e^2 \left (e \left (5 A e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+B \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+C \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+c^3 \left (2 e \left (7 A e \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )+B \left (140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-210 d^5 e x+420 d^6-70 d e^5 x^5+60 e^6 x^6\right )\right )+C \left (-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5+420 d^6 e x-840 d^7-120 d e^6 x^6+105 e^7 x^7\right )\right )\right )}{840 e^8}+\frac{\left (a e^2+c d^2\right )^3 \log (d+e x) \left (e (A e-B d)+C d^2\right )}{e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(420*a^3*e^6*(-2*C*d + 2*B*e + C*e*x) + 210*a^2*c*e^4*(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))) + 42*a*c^2*e^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))) + c^3*(C*(-840*d^7 + 420*d^6*e*x -
280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + 2*e*(7*
A*e*(-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) + B*(420*d^6 - 210*d^
5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)))))/(840*e^8) + ((c*d^
2 + a*e^2)^3*(C*d^2 + e*(-(B*d) + A*e))*Log[d + e*x])/e^9

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Maple [A]  time = 0.054, size = 880, normalized size = 1.8 \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),x)

[Out]

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

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Maxima [A]  time = 1.0654, size = 907, normalized size = 1.85 \begin{align*} \frac{105 \, C c^{3} e^{7} x^{8} - 120 \,{\left (C c^{3} d e^{6} - B c^{3} e^{7}\right )} x^{7} + 140 \,{\left (C c^{3} d^{2} e^{5} - B c^{3} d e^{6} +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} - 168 \,{\left (C c^{3} d^{3} e^{4} - B c^{3} d^{2} e^{5} - 3 \, B a c^{2} e^{7} +{\left (3 \, C a c^{2} + A c^{3}\right )} d e^{6}\right )} x^{5} + 210 \,{\left (C c^{3} d^{4} e^{3} - B c^{3} d^{3} e^{4} - 3 \, B a c^{2} d e^{6} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{5} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{7}\right )} x^{4} - 280 \,{\left (C c^{3} d^{5} e^{2} - B c^{3} d^{4} e^{3} - 3 \, B a c^{2} d^{2} e^{5} - 3 \, B a^{2} c e^{7} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d e^{6}\right )} x^{3} + 420 \,{\left (C c^{3} d^{6} e - B c^{3} d^{5} e^{2} - 3 \, B a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d e^{6} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{5} +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{7}\right )} x^{2} - 840 \,{\left (C c^{3} d^{7} - B c^{3} d^{6} e - 3 \, B a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{2} e^{5} - B a^{3} e^{7} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{4} +{\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{6}\right )} x}{840 \, e^{8}} + \frac{{\left (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}\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),x, algorithm="maxima")

[Out]

1/840*(105*C*c^3*e^7*x^8 - 120*(C*c^3*d*e^6 - B*c^3*e^7)*x^7 + 140*(C*c^3*d^2*e^5 - B*c^3*d*e^6 + (3*C*a*c^2 +
A*c^3)*e^7)*x^6 - 168*(C*c^3*d^3*e^4 - B*c^3*d^2*e^5 - 3*B*a*c^2*e^7 + (3*C*a*c^2 + A*c^3)*d*e^6)*x^5 + 210*(
C*c^3*d^4*e^3 - B*c^3*d^3*e^4 - 3*B*a*c^2*d*e^6 + (3*C*a*c^2 + A*c^3)*d^2*e^5 + 3*(C*a^2*c + A*a*c^2)*e^7)*x^4
- 280*(C*c^3*d^5*e^2 - B*c^3*d^4*e^3 - 3*B*a*c^2*d^2*e^5 - 3*B*a^2*c*e^7 + (3*C*a*c^2 + A*c^3)*d^3*e^4 + 3*(C
*a^2*c + A*a*c^2)*d*e^6)*x^3 + 420*(C*c^3*d^6*e - B*c^3*d^5*e^2 - 3*B*a*c^2*d^3*e^4 - 3*B*a^2*c*d*e^6 + (3*C*a
*c^2 + A*c^3)*d^4*e^3 + 3*(C*a^2*c + A*a*c^2)*d^2*e^5 + (C*a^3 + 3*A*a^2*c)*e^7)*x^2 - 840*(C*c^3*d^7 - B*c^3*
d^6*e - 3*B*a*c^2*d^4*e^3 - 3*B*a^2*c*d^2*e^5 - B*a^3*e^7 + (3*C*a*c^2 + A*c^3)*d^5*e^2 + 3*(C*a^2*c + A*a*c^2
)*d^3*e^4 + (C*a^3 + 3*A*a^2*c)*d*e^6)*x)/e^8 + (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)*log(e*x + d)/e^9

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Fricas [A]  time = 1.72348, size = 1385, normalized size = 2.83 \begin{align*} \frac{105 \, C c^{3} e^{8} x^{8} - 120 \,{\left (C c^{3} d e^{7} - B c^{3} e^{8}\right )} x^{7} + 140 \,{\left (C c^{3} d^{2} e^{6} - B c^{3} d e^{7} +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{8}\right )} x^{6} - 168 \,{\left (C c^{3} d^{3} e^{5} - B c^{3} d^{2} e^{6} - 3 \, B a c^{2} e^{8} +{\left (3 \, C a c^{2} + A c^{3}\right )} d e^{7}\right )} x^{5} + 210 \,{\left (C c^{3} d^{4} e^{4} - B c^{3} d^{3} e^{5} - 3 \, B a c^{2} d e^{7} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{6} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{8}\right )} x^{4} - 280 \,{\left (C c^{3} d^{5} e^{3} - B c^{3} d^{4} e^{4} - 3 \, B a c^{2} d^{2} e^{6} - 3 \, B a^{2} c e^{8} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{5} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d e^{7}\right )} x^{3} + 420 \,{\left (C c^{3} d^{6} e^{2} - B c^{3} d^{5} e^{3} - 3 \, B a c^{2} d^{3} e^{5} - 3 \, B a^{2} c d e^{7} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{4} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{6} +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{8}\right )} x^{2} - 840 \,{\left (C c^{3} d^{7} e - B c^{3} d^{6} e^{2} - 3 \, B a c^{2} d^{4} e^{4} - 3 \, B a^{2} c d^{2} e^{6} - B a^{3} e^{8} +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 3 \,{\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 + 840 \,{\left (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}\right )} \log \left (e x + d\right )}{840 \, 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),x, algorithm="fricas")

[Out]

1/840*(105*C*c^3*e^8*x^8 - 120*(C*c^3*d*e^7 - B*c^3*e^8)*x^7 + 140*(C*c^3*d^2*e^6 - B*c^3*d*e^7 + (3*C*a*c^2 +
A*c^3)*e^8)*x^6 - 168*(C*c^3*d^3*e^5 - B*c^3*d^2*e^6 - 3*B*a*c^2*e^8 + (3*C*a*c^2 + A*c^3)*d*e^7)*x^5 + 210*(
C*c^3*d^4*e^4 - B*c^3*d^3*e^5 - 3*B*a*c^2*d*e^7 + (3*C*a*c^2 + A*c^3)*d^2*e^6 + 3*(C*a^2*c + A*a*c^2)*e^8)*x^4
- 280*(C*c^3*d^5*e^3 - B*c^3*d^4*e^4 - 3*B*a*c^2*d^2*e^6 - 3*B*a^2*c*e^8 + (3*C*a*c^2 + A*c^3)*d^3*e^5 + 3*(C
*a^2*c + A*a*c^2)*d*e^7)*x^3 + 420*(C*c^3*d^6*e^2 - B*c^3*d^5*e^3 - 3*B*a*c^2*d^3*e^5 - 3*B*a^2*c*d*e^7 + (3*C
*a*c^2 + A*c^3)*d^4*e^4 + 3*(C*a^2*c + A*a*c^2)*d^2*e^6 + (C*a^3 + 3*A*a^2*c)*e^8)*x^2 - 840*(C*c^3*d^7*e - B*
c^3*d^6*e^2 - 3*B*a*c^2*d^4*e^4 - 3*B*a^2*c*d^2*e^6 - B*a^3*e^8 + (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 + (C*a^3 + 3*A*a^2*c)*d*e^7)*x + 840*(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)*log(e*x + d))/e^9

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Sympy [A]  time = 2.32097, size = 658, normalized size = 1.34 \begin{align*} \frac{C c^{3} x^{8}}{8 e} - \frac{x^{7} \left (- B c^{3} e + C c^{3} d\right )}{7 e^{2}} + \frac{x^{6} \left (A c^{3} e^{2} - B c^{3} d e + 3 C a c^{2} e^{2} + C c^{3} d^{2}\right )}{6 e^{3}} - \frac{x^{5} \left (A c^{3} d e^{2} - 3 B a c^{2} e^{3} - B c^{3} d^{2} e + 3 C a c^{2} d e^{2} + C c^{3} d^{3}\right )}{5 e^{4}} + \frac{x^{4} \left (3 A a c^{2} e^{4} + A c^{3} d^{2} e^{2} - 3 B a c^{2} d e^{3} - B c^{3} d^{3} e + 3 C a^{2} c e^{4} + 3 C a c^{2} d^{2} e^{2} + C c^{3} d^{4}\right )}{4 e^{5}} - \frac{x^{3} \left (3 A a c^{2} d e^{4} + A c^{3} d^{3} e^{2} - 3 B a^{2} c e^{5} - 3 B a c^{2} d^{2} e^{3} - B c^{3} d^{4} e + 3 C a^{2} c d e^{4} + 3 C a c^{2} d^{3} e^{2} + C c^{3} d^{5}\right )}{3 e^{6}} + \frac{x^{2} \left (3 A a^{2} c e^{6} + 3 A a c^{2} d^{2} e^{4} + A c^{3} d^{4} e^{2} - 3 B a^{2} c d e^{5} - 3 B a c^{2} d^{3} e^{3} - B c^{3} d^{5} e + C a^{3} e^{6} + 3 C a^{2} c d^{2} e^{4} + 3 C a c^{2} d^{4} e^{2} + C c^{3} d^{6}\right )}{2 e^{7}} - \frac{x \left (3 A a^{2} c d e^{6} + 3 A a c^{2} d^{3} e^{4} + A c^{3} d^{5} e^{2} - B a^{3} e^{7} - 3 B a^{2} c d^{2} e^{5} - 3 B a c^{2} d^{4} e^{3} - B c^{3} d^{6} e + C a^{3} d e^{6} + 3 C a^{2} c d^{3} e^{4} + 3 C a c^{2} d^{5} e^{2} + C c^{3} d^{7}\right )}{e^{8}} + \frac{\left (a e^{2} + c d^{2}\right )^{3} \left (A e^{2} - B d e + C d^{2}\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),x)

[Out]

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

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Giac [A]  time = 1.14769, size = 1031, normalized size = 2.1 \begin{align*}{\left (C c^{3} d^{8} - B c^{3} d^{7} e + 3 \, C a c^{2} d^{6} e^{2} + A c^{3} d^{6} e^{2} - 3 \, B a c^{2} d^{5} e^{3} + 3 \, C a^{2} c d^{4} e^{4} + 3 \, A a c^{2} d^{4} e^{4} - 3 \, B a^{2} c d^{3} e^{5} + C a^{3} d^{2} e^{6} + 3 \, A a^{2} c d^{2} e^{6} - B a^{3} d e^{7} + A a^{3} e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{840} \,{\left (105 \, C c^{3} x^{8} e^{7} - 120 \, C c^{3} d x^{7} e^{6} + 140 \, C c^{3} d^{2} x^{6} e^{5} - 168 \, C c^{3} d^{3} x^{5} e^{4} + 210 \, C c^{3} d^{4} x^{4} e^{3} - 280 \, C c^{3} d^{5} x^{3} e^{2} + 420 \, C c^{3} d^{6} x^{2} e - 840 \, C c^{3} d^{7} x + 120 \, B c^{3} x^{7} e^{7} - 140 \, B c^{3} d x^{6} e^{6} + 168 \, B c^{3} d^{2} x^{5} e^{5} - 210 \, B c^{3} d^{3} x^{4} e^{4} + 280 \, B c^{3} d^{4} x^{3} e^{3} - 420 \, B c^{3} d^{5} x^{2} e^{2} + 840 \, B c^{3} d^{6} x e + 420 \, C a c^{2} x^{6} e^{7} + 140 \, A c^{3} x^{6} e^{7} - 504 \, C a c^{2} d x^{5} e^{6} - 168 \, A c^{3} d x^{5} e^{6} + 630 \, C a c^{2} d^{2} x^{4} e^{5} + 210 \, A c^{3} d^{2} x^{4} e^{5} - 840 \, C a c^{2} d^{3} x^{3} e^{4} - 280 \, A c^{3} d^{3} x^{3} e^{4} + 1260 \, C a c^{2} d^{4} x^{2} e^{3} + 420 \, A c^{3} d^{4} x^{2} e^{3} - 2520 \, C a c^{2} d^{5} x e^{2} - 840 \, A c^{3} d^{5} x e^{2} + 504 \, B a c^{2} x^{5} e^{7} - 630 \, B a c^{2} d x^{4} e^{6} + 840 \, B a c^{2} d^{2} x^{3} e^{5} - 1260 \, B a c^{2} d^{3} x^{2} e^{4} + 2520 \, B a c^{2} d^{4} x e^{3} + 630 \, C a^{2} c x^{4} e^{7} + 630 \, A a c^{2} x^{4} e^{7} - 840 \, C a^{2} c d x^{3} e^{6} - 840 \, A a c^{2} d x^{3} e^{6} + 1260 \, C a^{2} c d^{2} x^{2} e^{5} + 1260 \, A a c^{2} d^{2} x^{2} e^{5} - 2520 \, C a^{2} c d^{3} x e^{4} - 2520 \, A a c^{2} d^{3} x e^{4} + 840 \, B a^{2} c x^{3} e^{7} - 1260 \, B a^{2} c d x^{2} e^{6} + 2520 \, B a^{2} c d^{2} x e^{5} + 420 \, C a^{3} x^{2} e^{7} + 1260 \, A a^{2} c x^{2} e^{7} - 840 \, C a^{3} d x e^{6} - 2520 \, A a^{2} c d x e^{6} + 840 \, B a^{3} x e^{7}\right )} e^{\left (-8\right )} \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),x, algorithm="giac")

[Out]

(C*c^3*d^8 - B*c^3*d^7*e + 3*C*a*c^2*d^6*e^2 + A*c^3*d^6*e^2 - 3*B*a*c^2*d^5*e^3 + 3*C*a^2*c*d^4*e^4 + 3*A*a*c
^2*d^4*e^4 - 3*B*a^2*c*d^3*e^5 + C*a^3*d^2*e^6 + 3*A*a^2*c*d^2*e^6 - B*a^3*d*e^7 + A*a^3*e^8)*e^(-9)*log(abs(x
*e + d)) + 1/840*(105*C*c^3*x^8*e^7 - 120*C*c^3*d*x^7*e^6 + 140*C*c^3*d^2*x^6*e^5 - 168*C*c^3*d^3*x^5*e^4 + 21
0*C*c^3*d^4*x^4*e^3 - 280*C*c^3*d^5*x^3*e^2 + 420*C*c^3*d^6*x^2*e - 840*C*c^3*d^7*x + 120*B*c^3*x^7*e^7 - 140*
B*c^3*d*x^6*e^6 + 168*B*c^3*d^2*x^5*e^5 - 210*B*c^3*d^3*x^4*e^4 + 280*B*c^3*d^4*x^3*e^3 - 420*B*c^3*d^5*x^2*e^
2 + 840*B*c^3*d^6*x*e + 420*C*a*c^2*x^6*e^7 + 140*A*c^3*x^6*e^7 - 504*C*a*c^2*d*x^5*e^6 - 168*A*c^3*d*x^5*e^6
+ 630*C*a*c^2*d^2*x^4*e^5 + 210*A*c^3*d^2*x^4*e^5 - 840*C*a*c^2*d^3*x^3*e^4 - 280*A*c^3*d^3*x^3*e^4 + 1260*C*a
*c^2*d^4*x^2*e^3 + 420*A*c^3*d^4*x^2*e^3 - 2520*C*a*c^2*d^5*x*e^2 - 840*A*c^3*d^5*x*e^2 + 504*B*a*c^2*x^5*e^7
- 630*B*a*c^2*d*x^4*e^6 + 840*B*a*c^2*d^2*x^3*e^5 - 1260*B*a*c^2*d^3*x^2*e^4 + 2520*B*a*c^2*d^4*x*e^3 + 630*C*
a^2*c*x^4*e^7 + 630*A*a*c^2*x^4*e^7 - 840*C*a^2*c*d*x^3*e^6 - 840*A*a*c^2*d*x^3*e^6 + 1260*C*a^2*c*d^2*x^2*e^5
+ 1260*A*a*c^2*d^2*x^2*e^5 - 2520*C*a^2*c*d^3*x*e^4 - 2520*A*a*c^2*d^3*x*e^4 + 840*B*a^2*c*x^3*e^7 - 1260*B*a
^2*c*d*x^2*e^6 + 2520*B*a^2*c*d^2*x*e^5 + 420*C*a^3*x^2*e^7 + 1260*A*a^2*c*x^2*e^7 - 840*C*a^3*d*x*e^6 - 2520*
A*a^2*c*d*x*e^6 + 840*B*a^3*x*e^7)*e^(-8)