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

Optimal. Leaf size=544 \[ -\frac {(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac {3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac {3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}-\frac {(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac {(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {x \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^7}-\frac {c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac {B c^3 (d+e x)^7}{7 e^8} \]

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Rubi [A]  time = 1.19, antiderivative size = 541, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} -\frac {(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac {3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac {(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac {3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}+\frac {x \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7}-\frac {(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac {c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac {B c^3 (d+e x)^7}{7 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - b*d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e))*x)/e^7 - (3*(c*d^2 - b*d*e +
a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d -
a*e)))*(d + e*x)^2)/(2*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d
^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d +
e*x)^3)/(3*e^8) - ((B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*
(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^4)/(4*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b
^2*e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^5)/(5*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6)/(6*e^8) + (B
*c^3*(d + e*x)^7)/(7*e^8) - ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^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 (a+b x+c x^2\right )^3}{d+e x} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)}{e^7}+\frac {\left (-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^2}{e^7}+\frac {\left (-B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^3}{e^7}+\frac {3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^4}{e^7}+\frac {c^2 (-7 B c d+3 b B e+A c e) (d+e x)^5}{e^7}+\frac {B c^3 (d+e x)^6}{e^7}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right ) x}{e^7}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^2}{2 e^8}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^3}{3 e^8}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^4}{4 e^8}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^5}{5 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^6}{6 e^8}+\frac {B c^3 (d+e x)^7}{7 e^8}-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.46, size = 700, normalized size = 1.29 \begin {gather*} \frac {e x \left (7 A e \left (15 c e^2 \left (6 a^2 e^2 (e x-2 d)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+3 c^2 e \left (5 a e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b \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 )\right )+c^3 \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 (21 c e^2 \left (10 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+10 a b e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^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 )\right )+35 e^3 \left (12 a^3 e^3+18 a^2 b e^2 (e x-2 d)+6 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+21 c^2 e \left (a e \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 )+b \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 )+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 (B d-A e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{420 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(7*A*e*(c^3*(-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) + 10*b*e
^3*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 15*c*e^2*(6*a^2*e^2*(-2*d + e*x)
+ 4*a*b*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 3*c^2*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 + 1
2*e^4*x^4))) + B*(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) + 35*e^3*(12*a^3*e^3 + 18*a^2*b*e^2*(-2*d + e*x) + 6*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) +
b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 21*c*e^2*(10*a^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 10
*a*b*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^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)) + 21*c^2*e*(a*e*(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) + b*(-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)))) - 420*(B*d - A*e)*(c*d^2 +
e*(-(b*d) + a*e))^3*Log[d + e*x])/(420*e^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 843, normalized size = 1.55 \begin {gather*} \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 + {\left (B a + A b\right )} 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 + {\left (B a + A b\right )} c^{2}\right )} d e^{6} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x - 420 \, {\left (B c^{3} d^{7} - A a^{3} e^{7} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^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 + (B*a + 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 + (B*a + A*b)*c^2)*d*e^6 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 140*(B*c^3*d^4*e^3 - (
3*B*b*c^2 + A*c^3)*d^3*e^4 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*
c)*d*e^6 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 - 210*(B*c^3*d^5*e^2 - (3*B*b*c^2 + A*c^3)*d^4
*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + (3*B*a*b^
2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 420*(B*c^3*d^6*e - (3*B*
b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d
^3*e^4 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + (B*a^3
+ 3*A*a^2*b)*e^7)*x - 420*(B*c^3*d^7 - A*a^3*e^7 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d
^5*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3
*e^4 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (B*a^3 + 3*A*a^2*b)*d*e^6)*log(e*x + d))/e^8

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giac [B]  time = 0.20, size = 1130, normalized size = 2.08 \begin {gather*} -{\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 \, B a c^{2} d^{5} e^{2} + 3 \, A b c^{2} d^{5} e^{2} - B b^{3} d^{4} e^{3} - 6 \, B a b c d^{4} e^{3} - 3 \, A b^{2} c d^{4} e^{3} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a b^{2} d^{3} e^{4} + A b^{3} d^{3} e^{4} + 3 \, B a^{2} c d^{3} e^{4} + 6 \, A a b c d^{3} e^{4} - 3 \, B a^{2} b d^{2} e^{5} - 3 \, A a b^{2} d^{2} e^{5} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + 3 \, A a^{2} b d e^{6} - A a^{3} e^{7}\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 \, B a c^{2} x^{5} e^{6} + 252 \, A b c^{2} x^{5} e^{6} - 315 \, B b^{2} c d x^{4} e^{5} - 315 \, B a c^{2} 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 \, B a c^{2} 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 \, B a c^{2} 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 \, B a c^{2} d^{4} x e^{2} + 1260 \, A b c^{2} d^{4} x e^{2} + 105 \, B b^{3} x^{4} e^{6} + 630 \, B a b c x^{4} e^{6} + 315 \, A b^{2} c x^{4} e^{6} + 315 \, A a c^{2} x^{4} e^{6} - 140 \, B b^{3} d x^{3} e^{5} - 840 \, B a b c d x^{3} e^{5} - 420 \, A b^{2} c d x^{3} e^{5} - 420 \, A a c^{2} d x^{3} e^{5} + 210 \, B b^{3} d^{2} x^{2} e^{4} + 1260 \, B a b c d^{2} x^{2} e^{4} + 630 \, A b^{2} c d^{2} x^{2} e^{4} + 630 \, A a c^{2} d^{2} x^{2} e^{4} - 420 \, B b^{3} d^{3} x e^{3} - 2520 \, B a b c d^{3} x e^{3} - 1260 \, A b^{2} c d^{3} x e^{3} - 1260 \, A a c^{2} d^{3} x e^{3} + 420 \, B a b^{2} x^{3} e^{6} + 140 \, A b^{3} x^{3} e^{6} + 420 \, B a^{2} c x^{3} e^{6} + 840 \, A a b c x^{3} e^{6} - 630 \, B a b^{2} d x^{2} e^{5} - 210 \, A b^{3} d x^{2} e^{5} - 630 \, B a^{2} c d x^{2} e^{5} - 1260 \, A a b c d x^{2} e^{5} + 1260 \, B a b^{2} d^{2} x e^{4} + 420 \, A b^{3} d^{2} x e^{4} + 1260 \, B a^{2} c d^{2} x e^{4} + 2520 \, A a b c d^{2} x e^{4} + 630 \, B a^{2} b x^{2} e^{6} + 630 \, A a b^{2} x^{2} e^{6} + 630 \, A a^{2} c x^{2} e^{6} - 1260 \, B a^{2} b d x e^{5} - 1260 \, A a b^{2} d x e^{5} - 1260 \, A a^{2} c d x e^{5} + 420 \, B a^{3} x e^{6} + 1260 \, A a^{2} b x e^{6}\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^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*B*a*c^2*d^5*e^2 + 3*A*b*c^2*d^5*e^2 - B*b^
3*d^4*e^3 - 6*B*a*b*c*d^4*e^3 - 3*A*b^2*c*d^4*e^3 - 3*A*a*c^2*d^4*e^3 + 3*B*a*b^2*d^3*e^4 + A*b^3*d^3*e^4 + 3*
B*a^2*c*d^3*e^4 + 6*A*a*b*c*d^3*e^4 - 3*B*a^2*b*d^2*e^5 - 3*A*a*b^2*d^2*e^5 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6
+ 3*A*a^2*b*d*e^6 - A*a^3*e^7)*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 + 10
5*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*B*a*c^2*x^5*e^6 + 252*A*b*c^2
*x^5*e^6 - 315*B*b^2*c*d*x^4*e^5 - 315*B*a*c^2*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 + 4
20*B*a*c^2*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*B*a*c^2*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*B*a*c^2*d^4*x*e^2 + 1260*A*b*c^2*d^4*x*e^2 + 105*B*b^3*x^4*e^
6 + 630*B*a*b*c*x^4*e^6 + 315*A*b^2*c*x^4*e^6 + 315*A*a*c^2*x^4*e^6 - 140*B*b^3*d*x^3*e^5 - 840*B*a*b*c*d*x^3*
e^5 - 420*A*b^2*c*d*x^3*e^5 - 420*A*a*c^2*d*x^3*e^5 + 210*B*b^3*d^2*x^2*e^4 + 1260*B*a*b*c*d^2*x^2*e^4 + 630*A
*b^2*c*d^2*x^2*e^4 + 630*A*a*c^2*d^2*x^2*e^4 - 420*B*b^3*d^3*x*e^3 - 2520*B*a*b*c*d^3*x*e^3 - 1260*A*b^2*c*d^3
*x*e^3 - 1260*A*a*c^2*d^3*x*e^3 + 420*B*a*b^2*x^3*e^6 + 140*A*b^3*x^3*e^6 + 420*B*a^2*c*x^3*e^6 + 840*A*a*b*c*
x^3*e^6 - 630*B*a*b^2*d*x^2*e^5 - 210*A*b^3*d*x^2*e^5 - 630*B*a^2*c*d*x^2*e^5 - 1260*A*a*b*c*d*x^2*e^5 + 1260*
B*a*b^2*d^2*x*e^4 + 420*A*b^3*d^2*x*e^4 + 1260*B*a^2*c*d^2*x*e^4 + 2520*A*a*b*c*d^2*x*e^4 + 630*B*a^2*b*x^2*e^
6 + 630*A*a*b^2*x^2*e^6 + 630*A*a^2*c*x^2*e^6 - 1260*B*a^2*b*d*x*e^5 - 1260*A*a*b^2*d*x*e^5 - 1260*A*a^2*c*d*x
*e^5 + 420*B*a^3*x*e^6 + 1260*A*a^2*b*x*e^6)*e^(-7)

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maple [B]  time = 0.05, size = 1319, normalized size = 2.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 842, normalized size = 1.55 \begin {gather*} \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 + {\left (B a + A b\right )} 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 + {\left (B a + A b\right )} c^{2}\right )} d e^{5} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{4} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{5} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{2} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{3} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{4} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{5} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{6}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{4} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - A a^{3} e^{7} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^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 + (B*a + 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 + (B*a + A*b)*c^2)*d*e^5 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^6)*x^4 + 140*(B*c^3*d^4*e^2 - (
3*B*b*c^2 + A*c^3)*d^3*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^4 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*
c)*d*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^6)*x^3 - 210*(B*c^3*d^5*e - (3*B*b*c^2 + A*c^3)*d^4*e
^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^4 + (3*B*a*b^2
+ A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^5 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^6)*x^2 + 420*(B*c^3*d^6 - (3*B*b*c^
2 + A*c^3)*d^5*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^3
+ (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^4 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^5 + (B*a^3 + 3*A*
a^2*b)*e^6)*x)/e^7 - (B*c^3*d^7 - A*a^3*e^7 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^
2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4
- 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (B*a^3 + 3*A*a^2*b)*d*e^6)*log(e*x + d)/e^8

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mupad [B]  time = 2.37, size = 968, normalized size = 1.78 \begin {gather*} x^2\,\left (\frac {3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2}{2\,e}-\frac {d\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{e}\right )}{2\,e}\right )+x^5\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{5\,e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}\right )+x^4\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{4\,e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{4\,e}\right )+x\,\left (\frac {B\,a^3+3\,A\,b\,a^2}{e}-\frac {d\,\left (\frac {3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2}{e}-\frac {d\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )+x^3\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{3\,e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e}-\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{3\,e}\right )+x^6\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^3\,d\,e^6+A\,a^3\,e^7+3\,B\,a^2\,b\,d^2\,e^5-3\,A\,a^2\,b\,d\,e^6-3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5-3\,B\,a\,b^2\,d^3\,e^4+3\,A\,a\,b^2\,d^2\,e^5+6\,B\,a\,b\,c\,d^4\,e^3-6\,A\,a\,b\,c\,d^3\,e^4-3\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3+B\,b^3\,d^4\,e^3-A\,b^3\,d^3\,e^4-3\,B\,b^2\,c\,d^5\,e^2+3\,A\,b^2\,c\,d^4\,e^3+3\,B\,b\,c^2\,d^6\,e-3\,A\,b\,c^2\,d^5\,e^2-B\,c^3\,d^7+A\,c^3\,d^6\,e\right )}{e^8}+\frac {B\,c^3\,x^7}{7\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.13, size = 979, normalized size = 1.80 \begin {gather*} \frac {B c^{3} x^{7}}{7 e} + x^{6} \left (\frac {A c^{3}}{6 e} + \frac {B b c^{2}}{2 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \left (\frac {3 A b c^{2}}{5 e} - \frac {A c^{3} d}{5 e^{2}} + \frac {3 B a c^{2}}{5 e} + \frac {3 B b^{2} c}{5 e} - \frac {3 B b c^{2} d}{5 e^{2}} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \left (\frac {3 A a c^{2}}{4 e} + \frac {3 A b^{2} c}{4 e} - \frac {3 A b c^{2} d}{4 e^{2}} + \frac {A c^{3} d^{2}}{4 e^{3}} + \frac {3 B a b c}{2 e} - \frac {3 B a c^{2} d}{4 e^{2}} + \frac {B b^{3}}{4 e} - \frac {3 B b^{2} c d}{4 e^{2}} + \frac {3 B b c^{2} d^{2}}{4 e^{3}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \left (\frac {2 A a b c}{e} - \frac {A a c^{2} d}{e^{2}} + \frac {A b^{3}}{3 e} - \frac {A b^{2} c d}{e^{2}} + \frac {A b c^{2} d^{2}}{e^{3}} - \frac {A c^{3} d^{3}}{3 e^{4}} + \frac {B a^{2} c}{e} + \frac {B a b^{2}}{e} - \frac {2 B a b c d}{e^{2}} + \frac {B a c^{2} d^{2}}{e^{3}} - \frac {B b^{3} d}{3 e^{2}} + \frac {B b^{2} c d^{2}}{e^{3}} - \frac {B b c^{2} d^{3}}{e^{4}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \left (\frac {3 A a^{2} c}{2 e} + \frac {3 A a b^{2}}{2 e} - \frac {3 A a b c d}{e^{2}} + \frac {3 A a c^{2} d^{2}}{2 e^{3}} - \frac {A b^{3} d}{2 e^{2}} + \frac {3 A b^{2} c d^{2}}{2 e^{3}} - \frac {3 A b c^{2} d^{3}}{2 e^{4}} + \frac {A c^{3} d^{4}}{2 e^{5}} + \frac {3 B a^{2} b}{2 e} - \frac {3 B a^{2} c d}{2 e^{2}} - \frac {3 B a b^{2} d}{2 e^{2}} + \frac {3 B a b c d^{2}}{e^{3}} - \frac {3 B a c^{2} d^{3}}{2 e^{4}} + \frac {B b^{3} d^{2}}{2 e^{3}} - \frac {3 B b^{2} c d^{3}}{2 e^{4}} + \frac {3 B b c^{2} d^{4}}{2 e^{5}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (\frac {3 A a^{2} b}{e} - \frac {3 A a^{2} c d}{e^{2}} - \frac {3 A a b^{2} d}{e^{2}} + \frac {6 A a b c d^{2}}{e^{3}} - \frac {3 A a c^{2} d^{3}}{e^{4}} + \frac {A b^{3} d^{2}}{e^{3}} - \frac {3 A b^{2} c d^{3}}{e^{4}} + \frac {3 A b c^{2} d^{4}}{e^{5}} - \frac {A c^{3} d^{5}}{e^{6}} + \frac {B a^{3}}{e} - \frac {3 B a^{2} b d}{e^{2}} + \frac {3 B a^{2} c d^{2}}{e^{3}} + \frac {3 B a b^{2} d^{2}}{e^{3}} - \frac {6 B a b c d^{3}}{e^{4}} + \frac {3 B a c^{2} d^{4}}{e^{5}} - \frac {B b^{3} d^{3}}{e^{4}} + \frac {3 B b^{2} c d^{4}}{e^{5}} - \frac {3 B b c^{2} d^{5}}{e^{6}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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