3.1133 \(\int \frac{(A+B x) (b x+c x^2)^3}{d+e x} \, dx\)

Optimal. Leaf size=257 \[ -\frac{c x^5 \left (A c e (c d-3 b e)-B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{5 e^3}-\frac{x^4 \left (B (c d-b e)^3-A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{4 e^4}-\frac{c^2 x^6 (-A c e-3 b B e+B c d)}{6 e^2}+\frac{d^2 x (B d-A e) (c d-b e)^3}{e^7}-\frac{d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8}+\frac{x^3 (B d-A e) (c d-b e)^3}{3 e^5}-\frac{d x^2 (B d-A e) (c d-b e)^3}{2 e^6}+\frac{B c^3 x^7}{7 e} \]

[Out]

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

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Rubi [A]  time = 0.448647, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{c x^5 \left (A c e (c d-3 b e)-B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{5 e^3}-\frac{x^4 \left (B (c d-b e)^3-A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{4 e^4}-\frac{c^2 x^6 (-A c e-3 b B e+B c d)}{6 e^2}+\frac{d^2 x (B d-A e) (c d-b e)^3}{e^7}-\frac{d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8}+\frac{x^3 (B d-A e) (c d-b e)^3}{3 e^5}-\frac{d x^2 (B d-A e) (c d-b e)^3}{2 e^6}+\frac{B c^3 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.109854, size = 248, normalized size = 0.96 \[ \frac{84 c e^5 x^5 \left (A c e (3 b e-c d)+B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )+105 e^4 x^4 \left (A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-B (c d-b e)^3\right )+70 c^2 e^6 x^6 (A c e+3 b B e-B c d)+420 d^2 e x (B d-A e) (c d-b e)^3-420 d^3 (B d-A e) (c d-b e)^3 \log (d+e x)+140 e^3 x^3 (A e-B d) (b e-c d)^3-210 d e^2 x^2 (B d-A e) (c d-b e)^3+60 B c^3 e^7 x^7}{420 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(420*d^2*e*(B*d - A*e)*(c*d - b*e)^3*x - 210*d*e^2*(B*d - A*e)*(c*d - b*e)^3*x^2 + 140*e^3*(-(B*d) + A*e)*(-(c
*d) + b*e)^3*x^3 + 105*e^4*(-(B*(c*d - b*e)^3) + A*c*e*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2))*x^4 + 84*c*e^5*(A*c*
e*(-(c*d) + 3*b*e) + B*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2))*x^5 + 70*c^2*e^6*(-(B*c*d) + 3*b*B*e + A*c*e)*x^6 +
60*B*c^3*e^7*x^7 - 420*d^3*(B*d - A*e)*(c*d - b*e)^3*Log[d + e*x])/(420*e^8)

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Maple [B]  time = 0.007, size = 708, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.06785, size = 716, normalized size = 2.79 \begin{align*} \frac{60 \, B c^{3} e^{6} x^{7} - 70 \,{\left (B c^{3} d e^{5} -{\left (3 \, B b c^{2} + A c^{3}\right )} e^{6}\right )} x^{6} + 84 \,{\left (B c^{3} d^{2} e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} d e^{5} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{6}\right )} x^{5} - 105 \,{\left (B c^{3} d^{3} e^{3} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{4} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d e^{5} -{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{6}\right )} x^{4} + 140 \,{\left (B c^{3} d^{4} e^{2} + A b^{3} e^{6} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{3} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{4} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{5}\right )} x^{3} - 210 \,{\left (B c^{3} d^{5} e + A b^{3} d e^{5} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{3} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{4}\right )} x^{2} + 420 \,{\left (B c^{3} d^{6} + A b^{3} d^{2} e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{2} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{3}\right )} x}{420 \, e^{7}} - \frac{{\left (B c^{3} d^{7} + A b^{3} d^{3} e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.48554, size = 1079, normalized size = 4.2 \begin{align*} \frac{60 \, B c^{3} e^{7} x^{7} - 70 \,{\left (B c^{3} d e^{6} -{\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 84 \,{\left (B c^{3} d^{2} e^{5} -{\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{7}\right )} x^{5} - 105 \,{\left (B c^{3} d^{3} e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d e^{6} -{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{7}\right )} x^{4} + 140 \,{\left (B c^{3} d^{4} e^{3} + A b^{3} e^{7} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} - 210 \,{\left (B c^{3} d^{5} e^{2} + A b^{3} d e^{6} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 420 \,{\left (B c^{3} d^{6} e + A b^{3} d^{2} e^{5} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x - 420 \,{\left (B c^{3} d^{7} + A b^{3} d^{3} e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(60*B*c^3*e^7*x^7 - 70*(B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 84*(B*c^3*d^2*e^5 - (3*B*b*c^2 + A*
c^3)*d*e^6 + 3*(B*b^2*c + A*b*c^2)*e^7)*x^5 - 105*(B*c^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^2*e^5 + 3*(B*b^2*c +
A*b*c^2)*d*e^6 - (B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 140*(B*c^3*d^4*e^3 + A*b^3*e^7 - (3*B*b*c^2 + A*c^3)*d^3*e^4 +
3*(B*b^2*c + A*b*c^2)*d^2*e^5 - (B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 - 210*(B*c^3*d^5*e^2 + A*b^3*d*e^6 - (3*B*b*c^
2 + A*c^3)*d^4*e^3 + 3*(B*b^2*c + A*b*c^2)*d^3*e^4 - (B*b^3 + 3*A*b^2*c)*d^2*e^5)*x^2 + 420*(B*c^3*d^6*e + A*b
^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4*e^3 - (B*b^3 + 3*A*b^2*c)*d^3*e^4)*x - 42
0*(B*c^3*d^7 + A*b^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + A*b*c^2)*d^5*e^2 - (B*b^3 + 3*A*b^2*c)
*d^4*e^3)*log(e*x + d))/e^8

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Sympy [B]  time = 2.12315, size = 541, normalized size = 2.11 \begin{align*} \frac{B c^{3} x^{7}}{7 e} + \frac{d^{3} \left (- A e + B d\right ) \left (b e - c d\right )^{3} \log{\left (d + e x \right )}}{e^{8}} + \frac{x^{6} \left (A c^{3} e + 3 B b c^{2} e - B c^{3} d\right )}{6 e^{2}} + \frac{x^{5} \left (3 A b c^{2} e^{2} - A c^{3} d e + 3 B b^{2} c e^{2} - 3 B b c^{2} d e + B c^{3} d^{2}\right )}{5 e^{3}} + \frac{x^{4} \left (3 A b^{2} c e^{3} - 3 A b c^{2} d e^{2} + A c^{3} d^{2} e + B b^{3} e^{3} - 3 B b^{2} c d e^{2} + 3 B b c^{2} d^{2} e - B c^{3} d^{3}\right )}{4 e^{4}} - \frac{x^{3} \left (- A b^{3} e^{4} + 3 A b^{2} c d e^{3} - 3 A b c^{2} d^{2} e^{2} + A c^{3} d^{3} e + B b^{3} d e^{3} - 3 B b^{2} c d^{2} e^{2} + 3 B b c^{2} d^{3} e - B c^{3} d^{4}\right )}{3 e^{5}} + \frac{x^{2} \left (- A b^{3} d e^{4} + 3 A b^{2} c d^{2} e^{3} - 3 A b c^{2} d^{3} e^{2} + A c^{3} d^{4} e + B b^{3} d^{2} e^{3} - 3 B b^{2} c d^{3} e^{2} + 3 B b c^{2} d^{4} e - B c^{3} d^{5}\right )}{2 e^{6}} - \frac{x \left (- A b^{3} d^{2} e^{4} + 3 A b^{2} c d^{3} e^{3} - 3 A b c^{2} d^{4} e^{2} + A c^{3} d^{5} e + B b^{3} d^{3} e^{3} - 3 B b^{2} c d^{4} e^{2} + 3 B b c^{2} d^{5} e - B c^{3} d^{6}\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.25814, size = 849, normalized size = 3.3 \begin{align*} -{\left (B c^{3} d^{7} - 3 \, B b c^{2} d^{6} e - A c^{3} d^{6} e + 3 \, B b^{2} c d^{5} e^{2} + 3 \, A b c^{2} d^{5} e^{2} - B b^{3} d^{4} e^{3} - 3 \, A b^{2} c d^{4} e^{3} + A b^{3} d^{3} e^{4}\right )} e^{\left (-8\right )} \log \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 + 210 \, B b c^{2} x^{6} e^{6} + 70 \, A c^{3} x^{6} e^{6} - 252 \, B b c^{2} d x^{5} e^{5} - 84 \, A c^{3} d x^{5} e^{5} + 315 \, B b c^{2} d^{2} x^{4} e^{4} + 105 \, A c^{3} d^{2} x^{4} e^{4} - 420 \, B b c^{2} d^{3} x^{3} e^{3} - 140 \, A c^{3} d^{3} x^{3} e^{3} + 630 \, B b c^{2} d^{4} x^{2} e^{2} + 210 \, A c^{3} d^{4} x^{2} e^{2} - 1260 \, B b c^{2} d^{5} x e - 420 \, A c^{3} d^{5} x e + 252 \, B b^{2} c x^{5} e^{6} + 252 \, A b c^{2} x^{5} e^{6} - 315 \, B b^{2} c d x^{4} e^{5} - 315 \, A b c^{2} d x^{4} e^{5} + 420 \, B b^{2} c d^{2} x^{3} e^{4} + 420 \, A b c^{2} d^{2} x^{3} e^{4} - 630 \, B b^{2} c d^{3} x^{2} e^{3} - 630 \, A b c^{2} d^{3} x^{2} e^{3} + 1260 \, B b^{2} c d^{4} x e^{2} + 1260 \, A b c^{2} d^{4} x e^{2} + 105 \, B b^{3} x^{4} e^{6} + 315 \, A b^{2} c x^{4} e^{6} - 140 \, B b^{3} d x^{3} e^{5} - 420 \, A b^{2} c d x^{3} e^{5} + 210 \, B b^{3} d^{2} x^{2} e^{4} + 630 \, A b^{2} c d^{2} x^{2} e^{4} - 420 \, B b^{3} d^{3} x e^{3} - 1260 \, A b^{2} c d^{3} x e^{3} + 140 \, A b^{3} x^{3} e^{6} - 210 \, A b^{3} d x^{2} e^{5} + 420 \, A b^{3} d^{2} x e^{4}\right )} e^{\left (-7\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*c^3*d^7 - 3*B*b*c^2*d^6*e - A*c^3*d^6*e + 3*B*b^2*c*d^5*e^2 + 3*A*b*c^2*d^5*e^2 - B*b^3*d^4*e^3 - 3*A*b^2*
c*d^4*e^3 + A*b^3*d^3*e^4)*e^(-8)*log(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 + 210*B*b*
c^2*x^6*e^6 + 70*A*c^3*x^6*e^6 - 252*B*b*c^2*d*x^5*e^5 - 84*A*c^3*d*x^5*e^5 + 315*B*b*c^2*d^2*x^4*e^4 + 105*A*
c^3*d^2*x^4*e^4 - 420*B*b*c^2*d^3*x^3*e^3 - 140*A*c^3*d^3*x^3*e^3 + 630*B*b*c^2*d^4*x^2*e^2 + 210*A*c^3*d^4*x^
2*e^2 - 1260*B*b*c^2*d^5*x*e - 420*A*c^3*d^5*x*e + 252*B*b^2*c*x^5*e^6 + 252*A*b*c^2*x^5*e^6 - 315*B*b^2*c*d*x
^4*e^5 - 315*A*b*c^2*d*x^4*e^5 + 420*B*b^2*c*d^2*x^3*e^4 + 420*A*b*c^2*d^2*x^3*e^4 - 630*B*b^2*c*d^3*x^2*e^3 -
630*A*b*c^2*d^3*x^2*e^3 + 1260*B*b^2*c*d^4*x*e^2 + 1260*A*b*c^2*d^4*x*e^2 + 105*B*b^3*x^4*e^6 + 315*A*b^2*c*x
^4*e^6 - 140*B*b^3*d*x^3*e^5 - 420*A*b^2*c*d*x^3*e^5 + 210*B*b^3*d^2*x^2*e^4 + 630*A*b^2*c*d^2*x^2*e^4 - 420*B
*b^3*d^3*x*e^3 - 1260*A*b^2*c*d^3*x*e^3 + 140*A*b^3*x^3*e^6 - 210*A*b^3*d*x^2*e^5 + 420*A*b^3*d^2*x*e^4)*e^(-7
)