3.1319 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx\)

Optimal. Leaf size=290 \[ \frac{x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}-\frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}-\frac{c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac{c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}+\frac{c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac{\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac{c^3 x^6 (B d-A e)}{6 e^2}+\frac{B c^3 x^7}{7 e} \]

[Out]

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

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Rubi [A]  time = 0.816033, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}-\frac{c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}-\frac{c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac{c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}+\frac{c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac{\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac{c^3 x^6 (B d-A e)}{6 e^2}+\frac{B c^3 x^7}{7 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} x^{7}}{7 e} + \frac{c^{3} x^{6} \left (A e - B d\right )}{6 e^{2}} + \frac{c^{2} x^{5} \left (- A c d e + 3 B a e^{2} + B c d^{2}\right )}{5 e^{3}} + \frac{c^{2} x^{4} \left (A e - B d\right ) \left (3 a e^{2} + c d^{2}\right )}{4 e^{4}} + \frac{c x^{3} \left (- 3 A a c d e^{3} - A c^{2} d^{3} e + 3 B a^{2} e^{4} + 3 B a c d^{2} e^{2} + B c^{2} d^{4}\right )}{3 e^{5}} + \frac{c \left (A e - B d\right ) \left (3 a^{2} e^{4} + 3 a c d^{2} e^{2} + c^{2} d^{4}\right ) \int x\, dx}{e^{6}} + \left (- 3 A a^{2} c d e^{5} - 3 A a c^{2} d^{3} e^{3} - A c^{3} d^{5} e + B a^{3} e^{6} + 3 B a^{2} c d^{2} e^{4} + 3 B a c^{2} d^{4} e^{2} + B c^{3} d^{6}\right ) \int \frac{1}{e^{7}}\, dx + \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d),x)

[Out]

B*c**3*x**7/(7*e) + c**3*x**6*(A*e - B*d)/(6*e**2) + c**2*x**5*(-A*c*d*e + 3*B*a
*e**2 + B*c*d**2)/(5*e**3) + c**2*x**4*(A*e - B*d)*(3*a*e**2 + c*d**2)/(4*e**4)
+ c*x**3*(-3*A*a*c*d*e**3 - A*c**2*d**3*e + 3*B*a**2*e**4 + 3*B*a*c*d**2*e**2 +
B*c**2*d**4)/(3*e**5) + c*(A*e - B*d)*(3*a**2*e**4 + 3*a*c*d**2*e**2 + c**2*d**4
)*Integral(x, x)/e**6 + (-3*A*a**2*c*d*e**5 - 3*A*a*c**2*d**3*e**3 - A*c**3*d**5
*e + B*a**3*e**6 + 3*B*a**2*c*d**2*e**4 + 3*B*a*c**2*d**4*e**2 + B*c**3*d**6)*In
tegral(e**(-7), x) + (A*e - B*d)*(a*e**2 + c*d**2)**3*log(d + e*x)/e**8

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Mathematica [A]  time = 0.326537, size = 311, normalized size = 1.07 \[ \frac{e x \left (7 A c e \left (90 a^2 e^4 (e x-2 d)+15 a c e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+B \left (420 a^3 e^6+210 a^2 c e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+21 a c^2 e^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+c^3 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 \left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{420 e^8} \]

Antiderivative was successfully verified.

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

[Out]

(e*x*(7*A*c*e*(90*a^2*e^4*(-2*d + e*x) + 15*a*c*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e
^2*x^2 + 3*e^3*x^3) + c^2*(-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*a^3*e^6 + 210*a^2*c*e^4*(6*d^2 - 3*d*e*
x + 2*e^2*x^2) + 21*a*c^2*e^2*(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*(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))) - 420*(B*d - A*e)*(c*d^2 + a*e
^2)^3*Log[d + e*x])/(420*e^8)

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Maple [A]  time = 0.008, size = 526, normalized size = 1.8 \[{\frac{3\,{a}^{2}Ac{x}^{2}}{2\,e}}-{\frac{{d}^{7}\ln \left ( ex+d \right ) B{c}^{3}}{{e}^{8}}}+{\frac{{d}^{6}\ln \left ( ex+d \right ) A{c}^{3}}{{e}^{7}}}-{\frac{B{c}^{3}{x}^{2}{d}^{5}}{2\,{e}^{6}}}+{\frac{B{c}^{3}{d}^{6}x}{{e}^{7}}}-{\frac{\ln \left ( ex+d \right ) B{a}^{3}d}{{e}^{2}}}+{\frac{3\,aA{c}^{2}{x}^{4}}{4\,e}}+{\frac{B{c}^{3}{x}^{7}}{7\,e}}-{\frac{A{c}^{3}{x}^{3}{d}^{3}}{3\,{e}^{4}}}-{\frac{B{c}^{3}{x}^{6}d}{6\,{e}^{2}}}+{\frac{A{c}^{3}{x}^{4}{d}^{2}}{4\,{e}^{3}}}-{\frac{B{c}^{3}{x}^{4}{d}^{3}}{4\,{e}^{4}}}+{\frac{A{x}^{2}{c}^{3}{d}^{4}}{2\,{e}^{5}}}+{\frac{B{c}^{3}{x}^{5}{d}^{2}}{5\,{e}^{3}}}-{\frac{A{c}^{3}{x}^{5}d}{5\,{e}^{2}}}-{\frac{3\,aB{x}^{2}{c}^{2}{d}^{3}}{2\,{e}^{4}}}-3\,{\frac{Ad{a}^{2}cx}{{e}^{2}}}-3\,{\frac{A{d}^{3}a{c}^{2}x}{{e}^{4}}}-{\frac{aA{c}^{2}{x}^{3}d}{{e}^{2}}}+3\,{\frac{Ba{c}^{2}{d}^{4}x}{{e}^{5}}}+{\frac{3\,aA{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{3}}}-{\frac{3\,{a}^{2}Bc{x}^{2}d}{2\,{e}^{2}}}+{\frac{3\,aB{c}^{2}{x}^{5}}{5\,e}}+{\frac{{a}^{2}Bc{x}^{3}}{e}}+{\frac{B{c}^{3}{x}^{3}{d}^{4}}{3\,{e}^{5}}}-{\frac{3\,aB{c}^{2}{x}^{4}d}{4\,{e}^{2}}}+{\frac{aB{c}^{2}{x}^{3}{d}^{2}}{{e}^{3}}}+3\,{\frac{B{a}^{2}c{d}^{2}x}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}c{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) Aa{c}^{2}{d}^{4}}{{e}^{5}}}-3\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}c{d}^{3}}{{e}^{4}}}-3\,{\frac{\ln \left ( ex+d \right ) Ba{c}^{2}{d}^{5}}{{e}^{6}}}+{\frac{{a}^{3}Bx}{e}}+{\frac{A{c}^{3}{x}^{6}}{6\,e}}-{\frac{A{d}^{5}{c}^{3}x}{{e}^{6}}}+{\frac{\ln \left ( ex+d \right ) A{a}^{3}}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.702597, size = 603, normalized size = 2.08 \[ \frac{60 \, B c^{3} e^{6} x^{7} - 70 \,{\left (B c^{3} d e^{5} - A c^{3} e^{6}\right )} x^{6} + 84 \,{\left (B c^{3} d^{2} e^{4} - A c^{3} d e^{5} + 3 \, B a c^{2} e^{6}\right )} x^{5} - 105 \,{\left (B c^{3} d^{3} e^{3} - A c^{3} d^{2} e^{4} + 3 \, B a c^{2} d e^{5} - 3 \, A a c^{2} e^{6}\right )} x^{4} + 140 \,{\left (B c^{3} d^{4} e^{2} - A c^{3} d^{3} e^{3} + 3 \, B a c^{2} d^{2} e^{4} - 3 \, A a c^{2} d e^{5} + 3 \, B a^{2} c e^{6}\right )} x^{3} - 210 \,{\left (B c^{3} d^{5} e - A c^{3} d^{4} e^{2} + 3 \, B a c^{2} d^{3} e^{3} - 3 \, A a c^{2} d^{2} e^{4} + 3 \, B a^{2} c d e^{5} - 3 \, A a^{2} c e^{6}\right )} x^{2} + 420 \,{\left (B c^{3} d^{6} - A c^{3} d^{5} e + 3 \, B a c^{2} d^{4} e^{2} - 3 \, A a c^{2} d^{3} e^{3} + 3 \, B a^{2} c d^{2} e^{4} - 3 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} x}{420 \, e^{7}} - \frac{{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2729, size = 605, normalized size = 2.09 \[ \frac{60 \, B c^{3} e^{7} x^{7} - 70 \,{\left (B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 84 \,{\left (B c^{3} d^{2} e^{5} - A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} - 105 \,{\left (B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5} + 3 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 140 \,{\left (B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4} + 3 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 210 \,{\left (B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3} + 3 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 420 \,{\left (B c^{3} d^{6} e - A c^{3} d^{5} e^{2} + 3 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} - 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 420 \,{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.75394, size = 394, normalized size = 1.36 \[ \frac{B c^{3} x^{7}}{7 e} - \frac{x^{6} \left (- A c^{3} e + B c^{3} d\right )}{6 e^{2}} + \frac{x^{5} \left (- A c^{3} d e + 3 B a c^{2} e^{2} + B c^{3} d^{2}\right )}{5 e^{3}} - \frac{x^{4} \left (- 3 A a c^{2} e^{3} - A c^{3} d^{2} e + 3 B a c^{2} d e^{2} + B c^{3} d^{3}\right )}{4 e^{4}} + \frac{x^{3} \left (- 3 A a c^{2} d e^{3} - A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 3 B a c^{2} d^{2} e^{2} + B c^{3} d^{4}\right )}{3 e^{5}} - \frac{x^{2} \left (- 3 A a^{2} c e^{5} - 3 A a c^{2} d^{2} e^{3} - A c^{3} d^{4} e + 3 B a^{2} c d e^{4} + 3 B a c^{2} d^{3} e^{2} + B c^{3} d^{5}\right )}{2 e^{6}} + \frac{x \left (- 3 A a^{2} c d e^{5} - 3 A a c^{2} d^{3} e^{3} - A c^{3} d^{5} e + B a^{3} e^{6} + 3 B a^{2} c d^{2} e^{4} + 3 B a c^{2} d^{4} e^{2} + B c^{3} d^{6}\right )}{e^{7}} - \frac{\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285099, size = 620, normalized size = 2.14 \[ -{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} e^{\left (-8\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{420} \,{\left (60 \, B c^{3} x^{7} e^{6} - 70 \, B c^{3} d x^{6} e^{5} + 84 \, B c^{3} d^{2} x^{5} e^{4} - 105 \, B c^{3} d^{3} x^{4} e^{3} + 140 \, B c^{3} d^{4} x^{3} e^{2} - 210 \, B c^{3} d^{5} x^{2} e + 420 \, B c^{3} d^{6} x + 70 \, A c^{3} x^{6} e^{6} - 84 \, A c^{3} d x^{5} e^{5} + 105 \, A c^{3} d^{2} x^{4} e^{4} - 140 \, A c^{3} d^{3} x^{3} e^{3} + 210 \, A c^{3} d^{4} x^{2} e^{2} - 420 \, A c^{3} d^{5} x e + 252 \, B a c^{2} x^{5} e^{6} - 315 \, B a c^{2} d x^{4} e^{5} + 420 \, B a c^{2} d^{2} x^{3} e^{4} - 630 \, B a c^{2} d^{3} x^{2} e^{3} + 1260 \, B a c^{2} d^{4} x e^{2} + 315 \, A a c^{2} x^{4} e^{6} - 420 \, A a c^{2} d x^{3} e^{5} + 630 \, A a c^{2} d^{2} x^{2} e^{4} - 1260 \, A a c^{2} d^{3} x e^{3} + 420 \, B a^{2} c x^{3} e^{6} - 630 \, B a^{2} c d x^{2} e^{5} + 1260 \, B a^{2} c d^{2} x e^{4} + 630 \, A a^{2} c x^{2} e^{6} - 1260 \, A a^{2} c d x e^{5} + 420 \, B a^{3} x e^{6}\right )} e^{\left (-7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d),x, algorithm="giac")

[Out]

-(B*c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^
3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)*e^(-8)*ln(abs(x*e + d)) + 1
/420*(60*B*c^3*x^7*e^6 - 70*B*c^3*d*x^6*e^5 + 84*B*c^3*d^2*x^5*e^4 - 105*B*c^3*d
^3*x^4*e^3 + 140*B*c^3*d^4*x^3*e^2 - 210*B*c^3*d^5*x^2*e + 420*B*c^3*d^6*x + 70*
A*c^3*x^6*e^6 - 84*A*c^3*d*x^5*e^5 + 105*A*c^3*d^2*x^4*e^4 - 140*A*c^3*d^3*x^3*e
^3 + 210*A*c^3*d^4*x^2*e^2 - 420*A*c^3*d^5*x*e + 252*B*a*c^2*x^5*e^6 - 315*B*a*c
^2*d*x^4*e^5 + 420*B*a*c^2*d^2*x^3*e^4 - 630*B*a*c^2*d^3*x^2*e^3 + 1260*B*a*c^2*
d^4*x*e^2 + 315*A*a*c^2*x^4*e^6 - 420*A*a*c^2*d*x^3*e^5 + 630*A*a*c^2*d^2*x^2*e^
4 - 1260*A*a*c^2*d^3*x*e^3 + 420*B*a^2*c*x^3*e^6 - 630*B*a^2*c*d*x^2*e^5 + 1260*
B*a^2*c*d^2*x*e^4 + 630*A*a^2*c*x^2*e^6 - 1260*A*a^2*c*d*x*e^5 + 420*B*a^3*x*e^6
)*e^(-7)