3.22.7 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=555 \[ \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{6 e^8 (d+e x)^6}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8 (d+e x)^7}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^8 (d+e x)^5}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{8 e^8 (d+e x)^8}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{9 e^8 (d+e x)^9}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2} \]

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Rubi [A]  time = 0.74, antiderivative size = 553, normalized size of antiderivative = 1.00, 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 {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 )}{6 e^8 (d+e x)^6}+\frac {3 c \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 )}{4 e^8 (d+e x)^4}+\frac {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 )}{5 e^8 (d+e x)^5}+\frac {3 \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 )}{7 e^8 (d+e x)^7}-\frac {\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 )}{8 e^8 (d+e x)^8}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{9 e^8 (d+e x)^9}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^10,x]

[Out]

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

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)^{10}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^{10}}+\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 (d+e x)^9}+\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 )}{e^7 (d+e x)^8}+\frac {-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 )}{e^7 (d+e x)^7}+\frac {-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 )}{e^7 (d+e x)^6}+\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 )}{e^7 (d+e x)^5}+\frac {c^2 (-7 B c d+3 b B e+A c e)}{e^7 (d+e x)^4}+\frac {B c^3}{e^7 (d+e x)^3}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{9 e^8 (d+e x)^9}-\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 )}{8 e^8 (d+e x)^8}+\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 )}{7 e^8 (d+e x)^7}+\frac {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 )}{6 e^8 (d+e x)^6}+\frac {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 )}{5 e^8 (d+e x)^5}+\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 )}{4 e^8 (d+e x)^4}+\frac {c^2 (7 B c d-3 b B e-A c e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 852, normalized size = 1.54 \begin {gather*} -\frac {A e \left (10 \left (d^6+9 e x d^5+36 e^2 x^2 d^4+84 e^3 x^3 d^3+126 e^4 x^4 d^2+126 e^5 x^5 d+84 e^6 x^6\right ) c^3+3 e \left (4 a e \left (d^4+9 e x d^3+36 e^2 x^2 d^2+84 e^3 x^3 d+126 e^4 x^4\right )+5 b \left (d^5+9 e x d^4+36 e^2 x^2 d^3+84 e^3 x^3 d^2+126 e^4 x^4 d+126 e^5 x^5\right )\right ) c^2+6 e^2 \left (2 \left (d^4+9 e x d^3+36 e^2 x^2 d^2+84 e^3 x^3 d+126 e^4 x^4\right ) b^2+5 a e \left (d^3+9 e x d^2+36 e^2 x^2 d+84 e^3 x^3\right ) b+5 a^2 e^2 \left (d^2+9 e x d+36 e^2 x^2\right )\right ) c+5 e^3 \left (\left (d^3+9 e x d^2+36 e^2 x^2 d+84 e^3 x^3\right ) b^3+6 a e \left (d^2+9 e x d+36 e^2 x^2\right ) b^2+21 a^2 e^2 (d+9 e x) b+56 a^3 e^3\right )\right )+B \left (35 \left (d^7+9 e x d^6+36 e^2 x^2 d^5+84 e^3 x^3 d^4+126 e^4 x^4 d^3+126 e^5 x^5 d^2+84 e^6 x^6 d+36 e^7 x^7\right ) c^3+15 e \left (a e \left (d^5+9 e x d^4+36 e^2 x^2 d^3+84 e^3 x^3 d^2+126 e^4 x^4 d+126 e^5 x^5\right )+2 b \left (d^6+9 e x d^5+36 e^2 x^2 d^4+84 e^3 x^3 d^3+126 e^4 x^4 d^2+126 e^5 x^5 d+84 e^6 x^6\right )\right ) c^2+3 e^2 \left (5 \left (d^5+9 e x d^4+36 e^2 x^2 d^3+84 e^3 x^3 d^2+126 e^4 x^4 d+126 e^5 x^5\right ) b^2+8 a e \left (d^4+9 e x d^3+36 e^2 x^2 d^2+84 e^3 x^3 d+126 e^4 x^4\right ) b+5 a^2 e^2 \left (d^3+9 e x d^2+36 e^2 x^2 d+84 e^3 x^3\right )\right ) c+e^3 \left (4 \left (d^4+9 e x d^3+36 e^2 x^2 d^2+84 e^3 x^3 d+126 e^4 x^4\right ) b^3+15 a e \left (d^3+9 e x d^2+36 e^2 x^2 d+84 e^3 x^3\right ) b^2+30 a^2 e^2 \left (d^2+9 e x d+36 e^2 x^2\right ) b+35 a^3 e^3 (d+9 e x)\right )\right )}{2520 e^8 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^10,x]

[Out]

-1/2520*(A*e*(10*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84
*e^6*x^6) + 5*e^3*(56*a^3*e^3 + 21*a^2*b*e^2*(d + 9*e*x) + 6*a*b^2*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + b^3*(d^3 +
9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 6*c*e^2*(5*a^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b*e*(d^3 + 9*
d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 2*b^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4))
+ 3*c^2*e*(4*a*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b*(d^5 + 9*d^4*e*x + 36*d
^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5))) + B*(35*c^3*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 8
4*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7) + e^3*(35*a^3*e^3*(d + 9*e*x) +
30*a^2*b*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 15*a*b^2*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*b^3*
(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + 3*c*e^2*(5*a^2*e^2*(d^3 + 9*d^2*e*x + 36*d*
e^2*x^2 + 84*e^3*x^3) + 8*a*b*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b^2*(d^5 +
9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 15*c^2*e*(a*e*(d^5 + 9*d^4*e*x
+ 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*b*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*
d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))))/(e^8*(d + e*x)^9)

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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)^{10}} \, 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)^10,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^10, x]

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fricas [A]  time = 0.39, size = 945, normalized size = 1.70 \begin {gather*} -\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 280 \, A a^{3} e^{7} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + 4 \, {\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} + 5 \, {\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} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, {\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} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + 4 \, {\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} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + 5 \, {\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} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} + 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} + 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} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + 4 \, {\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} + 5 \, {\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} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\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)^10,x, algorithm="fricas")

[Out]

-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 280*A*a^3*e^7 + 10*(3*B*b*c^2 + A*c^3)*d^6*e + 15*(B*b^2*c + (B*a
+ A*b)*c^2)*d^5*e^2 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2
+ 2*A*a*b)*c)*d^3*e^4 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 35*(B*a^3 + 3*A*a^2*b)*d*e^6 + 420*(7*B*c^
3*d*e^6 + 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 630*(7*B*c^3*d^2*e^5 + 2*(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 + 126*(35*B*c^3*d^3*e^4 + 10*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15*(B*b^2*c + (B*a + A*b)*
c^2)*d*e^6 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 84*(35*B*c^3*d^4*e^3 + 10*(3*B*b*c^2 + A
*c^3)*d^3*e^4 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 +
5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*(3*B*b*c^2 + A*c^3)*d^4*e^
3 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + 5*(3*B*a*
b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 9*(35*B*c^3*d^6*e + 1
0*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A
*b^2)*c)*d^3*e^4 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*
e^6 + 35*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 + 84*d^3*e^14*x^6 + 126*d^4*e^
13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*e^8)

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giac [B]  time = 0.21, size = 1129, normalized size = 2.03 \begin {gather*} -\frac {{\left (1260 \, B c^{3} x^{7} e^{7} + 2940 \, B c^{3} d x^{6} e^{6} + 4410 \, B c^{3} d^{2} x^{5} e^{5} + 4410 \, B c^{3} d^{3} x^{4} e^{4} + 2940 \, B c^{3} d^{4} x^{3} e^{3} + 1260 \, B c^{3} d^{5} x^{2} e^{2} + 315 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 2520 \, B b c^{2} x^{6} e^{7} + 840 \, A c^{3} x^{6} e^{7} + 3780 \, B b c^{2} d x^{5} e^{6} + 1260 \, A c^{3} d x^{5} e^{6} + 3780 \, B b c^{2} d^{2} x^{4} e^{5} + 1260 \, A c^{3} d^{2} x^{4} e^{5} + 2520 \, B b c^{2} d^{3} x^{3} e^{4} + 840 \, A c^{3} d^{3} x^{3} e^{4} + 1080 \, B b c^{2} d^{4} x^{2} e^{3} + 360 \, A c^{3} d^{4} x^{2} e^{3} + 270 \, B b c^{2} d^{5} x e^{2} + 90 \, A c^{3} d^{5} x e^{2} + 30 \, B b c^{2} d^{6} e + 10 \, A c^{3} d^{6} e + 1890 \, B b^{2} c x^{5} e^{7} + 1890 \, B a c^{2} x^{5} e^{7} + 1890 \, A b c^{2} x^{5} e^{7} + 1890 \, B b^{2} c d x^{4} e^{6} + 1890 \, B a c^{2} d x^{4} e^{6} + 1890 \, A b c^{2} d x^{4} e^{6} + 1260 \, B b^{2} c d^{2} x^{3} e^{5} + 1260 \, B a c^{2} d^{2} x^{3} e^{5} + 1260 \, A b c^{2} d^{2} x^{3} e^{5} + 540 \, B b^{2} c d^{3} x^{2} e^{4} + 540 \, B a c^{2} d^{3} x^{2} e^{4} + 540 \, A b c^{2} d^{3} x^{2} e^{4} + 135 \, B b^{2} c d^{4} x e^{3} + 135 \, B a c^{2} d^{4} x e^{3} + 135 \, A b c^{2} d^{4} x e^{3} + 15 \, B b^{2} c d^{5} e^{2} + 15 \, B a c^{2} d^{5} e^{2} + 15 \, A b c^{2} d^{5} e^{2} + 504 \, B b^{3} x^{4} e^{7} + 3024 \, B a b c x^{4} e^{7} + 1512 \, A b^{2} c x^{4} e^{7} + 1512 \, A a c^{2} x^{4} e^{7} + 336 \, B b^{3} d x^{3} e^{6} + 2016 \, B a b c d x^{3} e^{6} + 1008 \, A b^{2} c d x^{3} e^{6} + 1008 \, A a c^{2} d x^{3} e^{6} + 144 \, B b^{3} d^{2} x^{2} e^{5} + 864 \, B a b c d^{2} x^{2} e^{5} + 432 \, A b^{2} c d^{2} x^{2} e^{5} + 432 \, A a c^{2} d^{2} x^{2} e^{5} + 36 \, B b^{3} d^{3} x e^{4} + 216 \, B a b c d^{3} x e^{4} + 108 \, A b^{2} c d^{3} x e^{4} + 108 \, A a c^{2} d^{3} x e^{4} + 4 \, B b^{3} d^{4} e^{3} + 24 \, B a b c d^{4} e^{3} + 12 \, A b^{2} c d^{4} e^{3} + 12 \, A a c^{2} d^{4} e^{3} + 1260 \, B a b^{2} x^{3} e^{7} + 420 \, A b^{3} x^{3} e^{7} + 1260 \, B a^{2} c x^{3} e^{7} + 2520 \, A a b c x^{3} e^{7} + 540 \, B a b^{2} d x^{2} e^{6} + 180 \, A b^{3} d x^{2} e^{6} + 540 \, B a^{2} c d x^{2} e^{6} + 1080 \, A a b c d x^{2} e^{6} + 135 \, B a b^{2} d^{2} x e^{5} + 45 \, A b^{3} d^{2} x e^{5} + 135 \, B a^{2} c d^{2} x e^{5} + 270 \, A a b c d^{2} x e^{5} + 15 \, B a b^{2} d^{3} e^{4} + 5 \, A b^{3} d^{3} e^{4} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a b c d^{3} e^{4} + 1080 \, B a^{2} b x^{2} e^{7} + 1080 \, A a b^{2} x^{2} e^{7} + 1080 \, A a^{2} c x^{2} e^{7} + 270 \, B a^{2} b d x e^{6} + 270 \, A a b^{2} d x e^{6} + 270 \, A a^{2} c d x e^{6} + 30 \, B a^{2} b d^{2} e^{5} + 30 \, A a b^{2} d^{2} e^{5} + 30 \, A a^{2} c d^{2} e^{5} + 315 \, B a^{3} x e^{7} + 945 \, A a^{2} b x e^{7} + 35 \, B a^{3} d e^{6} + 105 \, A a^{2} b d e^{6} + 280 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{2520 \, {\left (x e + d\right )}^{9}} \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)^10,x, algorithm="giac")

[Out]

-1/2520*(1260*B*c^3*x^7*e^7 + 2940*B*c^3*d*x^6*e^6 + 4410*B*c^3*d^2*x^5*e^5 + 4410*B*c^3*d^3*x^4*e^4 + 2940*B*
c^3*d^4*x^3*e^3 + 1260*B*c^3*d^5*x^2*e^2 + 315*B*c^3*d^6*x*e + 35*B*c^3*d^7 + 2520*B*b*c^2*x^6*e^7 + 840*A*c^3
*x^6*e^7 + 3780*B*b*c^2*d*x^5*e^6 + 1260*A*c^3*d*x^5*e^6 + 3780*B*b*c^2*d^2*x^4*e^5 + 1260*A*c^3*d^2*x^4*e^5 +
2520*B*b*c^2*d^3*x^3*e^4 + 840*A*c^3*d^3*x^3*e^4 + 1080*B*b*c^2*d^4*x^2*e^3 + 360*A*c^3*d^4*x^2*e^3 + 270*B*b
*c^2*d^5*x*e^2 + 90*A*c^3*d^5*x*e^2 + 30*B*b*c^2*d^6*e + 10*A*c^3*d^6*e + 1890*B*b^2*c*x^5*e^7 + 1890*B*a*c^2*
x^5*e^7 + 1890*A*b*c^2*x^5*e^7 + 1890*B*b^2*c*d*x^4*e^6 + 1890*B*a*c^2*d*x^4*e^6 + 1890*A*b*c^2*d*x^4*e^6 + 12
60*B*b^2*c*d^2*x^3*e^5 + 1260*B*a*c^2*d^2*x^3*e^5 + 1260*A*b*c^2*d^2*x^3*e^5 + 540*B*b^2*c*d^3*x^2*e^4 + 540*B
*a*c^2*d^3*x^2*e^4 + 540*A*b*c^2*d^3*x^2*e^4 + 135*B*b^2*c*d^4*x*e^3 + 135*B*a*c^2*d^4*x*e^3 + 135*A*b*c^2*d^4
*x*e^3 + 15*B*b^2*c*d^5*e^2 + 15*B*a*c^2*d^5*e^2 + 15*A*b*c^2*d^5*e^2 + 504*B*b^3*x^4*e^7 + 3024*B*a*b*c*x^4*e
^7 + 1512*A*b^2*c*x^4*e^7 + 1512*A*a*c^2*x^4*e^7 + 336*B*b^3*d*x^3*e^6 + 2016*B*a*b*c*d*x^3*e^6 + 1008*A*b^2*c
*d*x^3*e^6 + 1008*A*a*c^2*d*x^3*e^6 + 144*B*b^3*d^2*x^2*e^5 + 864*B*a*b*c*d^2*x^2*e^5 + 432*A*b^2*c*d^2*x^2*e^
5 + 432*A*a*c^2*d^2*x^2*e^5 + 36*B*b^3*d^3*x*e^4 + 216*B*a*b*c*d^3*x*e^4 + 108*A*b^2*c*d^3*x*e^4 + 108*A*a*c^2
*d^3*x*e^4 + 4*B*b^3*d^4*e^3 + 24*B*a*b*c*d^4*e^3 + 12*A*b^2*c*d^4*e^3 + 12*A*a*c^2*d^4*e^3 + 1260*B*a*b^2*x^3
*e^7 + 420*A*b^3*x^3*e^7 + 1260*B*a^2*c*x^3*e^7 + 2520*A*a*b*c*x^3*e^7 + 540*B*a*b^2*d*x^2*e^6 + 180*A*b^3*d*x
^2*e^6 + 540*B*a^2*c*d*x^2*e^6 + 1080*A*a*b*c*d*x^2*e^6 + 135*B*a*b^2*d^2*x*e^5 + 45*A*b^3*d^2*x*e^5 + 135*B*a
^2*c*d^2*x*e^5 + 270*A*a*b*c*d^2*x*e^5 + 15*B*a*b^2*d^3*e^4 + 5*A*b^3*d^3*e^4 + 15*B*a^2*c*d^3*e^4 + 30*A*a*b*
c*d^3*e^4 + 1080*B*a^2*b*x^2*e^7 + 1080*A*a*b^2*x^2*e^7 + 1080*A*a^2*c*x^2*e^7 + 270*B*a^2*b*d*x*e^6 + 270*A*a
*b^2*d*x*e^6 + 270*A*a^2*c*d*x*e^6 + 30*B*a^2*b*d^2*e^5 + 30*A*a*b^2*d^2*e^5 + 30*A*a^2*c*d^2*e^5 + 315*B*a^3*
x*e^7 + 945*A*a^2*b*x*e^7 + 35*B*a^3*d*e^6 + 105*A*a^2*b*d*e^6 + 280*A*a^3*e^7)*e^(-8)/(x*e + d)^9

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maple [A]  time = 0.05, size = 1067, normalized size = 1.92 \begin {gather*} -\frac {B \,c^{3}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {\left (A c e +3 B b e -7 B c d \right ) c^{2}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {3 \left (A b c \,e^{2}-2 A \,c^{2} d e +a B c \,e^{2}+b^{2} B \,e^{2}-6 B b c d e +7 B \,c^{2} d^{2}\right ) c}{4 \left (e x +d \right )^{4} e^{8}}-\frac {3 A \,a^{2} b \,e^{6}-6 A \,a^{2} c d \,e^{5}-6 A d a \,b^{2} e^{5}+18 A \,d^{2} a b c \,e^{4}-12 A \,d^{3} a \,c^{2} e^{3}+3 A \,d^{2} b^{3} e^{4}-12 A \,d^{3} b^{2} c \,e^{3}+15 A b \,c^{2} d^{4} e^{2}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}-6 B d \,a^{2} b \,e^{5}+9 B \,a^{2} c \,d^{2} e^{4}+9 B \,d^{2} a \,b^{2} e^{4}-24 B \,d^{3} a b c \,e^{3}+15 B \,d^{4} a \,c^{2} e^{2}-4 B \,b^{3} d^{3} e^{3}+15 B \,d^{4} b^{2} c \,e^{2}-18 B \,d^{5} b \,c^{2} e +7 B \,d^{6} c^{3}}{8 \left (e x +d \right )^{8} e^{8}}-\frac {6 A a b c \,e^{4}-12 A d a \,c^{2} e^{3}+A \,b^{3} e^{4}-12 A d \,b^{2} c \,e^{3}+30 A b \,c^{2} d^{2} e^{2}-20 A \,c^{3} d^{3} e +3 B \,a^{2} c \,e^{4}+3 B a \,b^{2} e^{4}-24 B d a b c \,e^{3}+30 B \,d^{2} a \,c^{2} e^{2}-4 B \,b^{3} d \,e^{3}+30 B \,d^{2} b^{2} c \,e^{2}-60 B \,d^{3} b \,c^{2} e +35 B \,c^{3} d^{4}}{6 \left (e x +d \right )^{6} e^{8}}-\frac {3 A \,a^{2} c \,e^{5}+3 A a \,b^{2} e^{5}-18 A d a b c \,e^{4}+18 A a \,c^{2} d^{2} e^{3}-3 A d \,b^{3} e^{4}+18 A \,b^{2} c \,d^{2} e^{3}-30 A b \,c^{2} d^{3} e^{2}+15 A \,c^{3} d^{4} e +3 B \,a^{2} b \,e^{5}-9 B \,a^{2} c d \,e^{4}-9 B d a \,b^{2} e^{4}+36 B \,d^{2} a b c \,e^{3}-30 B a \,c^{2} d^{3} e^{2}+6 B \,d^{2} b^{3} e^{3}-30 B \,d^{3} b^{2} c \,e^{2}+45 B \,d^{4} b \,c^{2} e -21 B \,d^{5} c^{3}}{7 \left (e x +d \right )^{7} e^{8}}-\frac {A \,a^{3} e^{7}-3 A d \,a^{2} b \,e^{6}+3 A \,d^{2} a^{2} c \,e^{5}+3 A \,d^{2} a \,b^{2} e^{5}-6 A \,d^{3} a b c \,e^{4}+3 A a \,c^{2} d^{4} e^{3}-A \,b^{3} d^{3} e^{4}+3 A \,d^{4} b^{2} c \,e^{3}-3 A b \,c^{2} d^{5} e^{2}+A \,d^{6} c^{3} e -B \,a^{3} d \,e^{6}+3 B \,d^{2} a^{2} b \,e^{5}-3 B \,d^{3} a^{2} c \,e^{4}-3 B a \,b^{2} d^{3} e^{4}+6 B \,d^{4} a b c \,e^{3}-3 B a \,c^{2} d^{5} e^{2}+B \,d^{4} b^{3} e^{3}-3 B \,d^{5} b^{2} c \,e^{2}+3 B \,d^{6} b \,c^{2} e -B \,d^{7} c^{3}}{9 \left (e x +d \right )^{9} e^{8}}-\frac {3 A a \,c^{2} e^{3}+3 A \,b^{2} c \,e^{3}-15 A b \,c^{2} d \,e^{2}+15 A \,c^{3} d^{2} e +6 a b B c \,e^{3}-15 B d a \,c^{2} e^{2}+b^{3} B \,e^{3}-15 B d \,b^{2} c \,e^{2}+45 B b \,c^{2} d^{2} e -35 B \,d^{3} c^{3}}{5 \left (e x +d \right )^{5} e^{8}} \end {gather*}

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)^10,x)

[Out]

-1/8*(3*A*a^2*b*e^6-6*A*a^2*c*d*e^5-6*A*a*b^2*d*e^5+18*A*a*b*c*d^2*e^4-12*A*a*c^2*d^3*e^3+3*A*b^3*d^2*e^4-12*A
*b^2*c*d^3*e^3+15*A*b*c^2*d^4*e^2-6*A*c^3*d^5*e+B*a^3*e^6-6*B*a^2*b*d*e^5+9*B*a^2*c*d^2*e^4+9*B*a*b^2*d^2*e^4-
24*B*a*b*c*d^3*e^3+15*B*a*c^2*d^4*e^2-4*B*b^3*d^3*e^3+15*B*b^2*c*d^4*e^2-18*B*b*c^2*d^5*e+7*B*c^3*d^6)/e^8/(e*
x+d)^8-1/2/(e*x+d)^2*B*c^3/e^8-3/4*c*(A*b*c*e^2-2*A*c^2*d*e+B*a*c*e^2+B*b^2*e^2-6*B*b*c*d*e+7*B*c^2*d^2)/e^8/(
e*x+d)^4-1/6*(6*A*a*b*c*e^4-12*A*a*c^2*d*e^3+A*b^3*e^4-12*A*b^2*c*d*e^3+30*A*b*c^2*d^2*e^2-20*A*c^3*d^3*e+3*B*
a^2*c*e^4+3*B*a*b^2*e^4-24*B*a*b*c*d*e^3+30*B*a*c^2*d^2*e^2-4*B*b^3*d*e^3+30*B*b^2*c*d^2*e^2-60*B*b*c^2*d^3*e+
35*B*c^3*d^4)/e^8/(e*x+d)^6-1/7*(3*A*a^2*c*e^5+3*A*a*b^2*e^5-18*A*a*b*c*d*e^4+18*A*a*c^2*d^2*e^3-3*A*b^3*d*e^4
+18*A*b^2*c*d^2*e^3-30*A*b*c^2*d^3*e^2+15*A*c^3*d^4*e+3*B*a^2*b*e^5-9*B*a^2*c*d*e^4-9*B*a*b^2*d*e^4+36*B*a*b*c
*d^2*e^3-30*B*a*c^2*d^3*e^2+6*B*b^3*d^2*e^3-30*B*b^2*c*d^3*e^2+45*B*b*c^2*d^4*e-21*B*c^3*d^5)/e^8/(e*x+d)^7-1/
9*(A*a^3*e^7-3*A*a^2*b*d*e^6+3*A*a^2*c*d^2*e^5+3*A*a*b^2*d^2*e^5-6*A*a*b*c*d^3*e^4+3*A*a*c^2*d^4*e^3-A*b^3*d^3
*e^4+3*A*b^2*c*d^4*e^3-3*A*b*c^2*d^5*e^2+A*c^3*d^6*e-B*a^3*d*e^6+3*B*a^2*b*d^2*e^5-3*B*a^2*c*d^3*e^4-3*B*a*b^2
*d^3*e^4+6*B*a*b*c*d^4*e^3-3*B*a*c^2*d^5*e^2+B*b^3*d^4*e^3-3*B*b^2*c*d^5*e^2+3*B*b*c^2*d^6*e-B*c^3*d^7)/e^8/(e
*x+d)^9-1/5*(3*A*a*c^2*e^3+3*A*b^2*c*e^3-15*A*b*c^2*d*e^2+15*A*c^3*d^2*e+6*B*a*b*c*e^3-15*B*a*c^2*d*e^2+B*b^3*
e^3-15*B*b^2*c*d*e^2+45*B*b*c^2*d^2*e-35*B*c^3*d^3)/e^8/(e*x+d)^5-1/3*c^2*(A*c*e+3*B*b*e-7*B*c*d)/e^8/(e*x+d)^
3

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maxima [A]  time = 1.05, size = 945, normalized size = 1.70 \begin {gather*} -\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 280 \, A a^{3} e^{7} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + 4 \, {\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} + 5 \, {\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} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, {\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} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + 4 \, {\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} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + 5 \, {\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} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} + 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} + 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} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + 4 \, {\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} + 5 \, {\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} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\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)^10,x, algorithm="maxima")

[Out]

-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 280*A*a^3*e^7 + 10*(3*B*b*c^2 + A*c^3)*d^6*e + 15*(B*b^2*c + (B*a
+ A*b)*c^2)*d^5*e^2 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2
+ 2*A*a*b)*c)*d^3*e^4 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 35*(B*a^3 + 3*A*a^2*b)*d*e^6 + 420*(7*B*c^
3*d*e^6 + 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 630*(7*B*c^3*d^2*e^5 + 2*(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 + 126*(35*B*c^3*d^3*e^4 + 10*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15*(B*b^2*c + (B*a + A*b)*
c^2)*d*e^6 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 84*(35*B*c^3*d^4*e^3 + 10*(3*B*b*c^2 + A
*c^3)*d^3*e^4 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 +
5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*(3*B*b*c^2 + A*c^3)*d^4*e^
3 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + 5*(3*B*a*
b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 9*(35*B*c^3*d^6*e + 1
0*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A
*b^2)*c)*d^3*e^4 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*
e^6 + 35*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 + 84*d^3*e^14*x^6 + 126*d^4*e^
13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*e^8)

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mupad [B]  time = 2.57, size = 1142, normalized size = 2.06 \begin {gather*} -\frac {\frac {35\,B\,a^3\,d\,e^6+280\,A\,a^3\,e^7+30\,B\,a^2\,b\,d^2\,e^5+105\,A\,a^2\,b\,d\,e^6+15\,B\,a^2\,c\,d^3\,e^4+30\,A\,a^2\,c\,d^2\,e^5+15\,B\,a\,b^2\,d^3\,e^4+30\,A\,a\,b^2\,d^2\,e^5+24\,B\,a\,b\,c\,d^4\,e^3+30\,A\,a\,b\,c\,d^3\,e^4+15\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3+4\,B\,b^3\,d^4\,e^3+5\,A\,b^3\,d^3\,e^4+15\,B\,b^2\,c\,d^5\,e^2+12\,A\,b^2\,c\,d^4\,e^3+30\,B\,b\,c^2\,d^6\,e+15\,A\,b\,c^2\,d^5\,e^2+35\,B\,c^3\,d^7+10\,A\,c^3\,d^6\,e}{2520\,e^8}+\frac {x^4\,\left (4\,B\,b^3\,e^3+15\,B\,b^2\,c\,d\,e^2+12\,A\,b^2\,c\,e^3+30\,B\,b\,c^2\,d^2\,e+15\,A\,b\,c^2\,d\,e^2+24\,B\,a\,b\,c\,e^3+35\,B\,c^3\,d^3+10\,A\,c^3\,d^2\,e+15\,B\,a\,c^2\,d\,e^2+12\,A\,a\,c^2\,e^3\right )}{20\,e^4}+\frac {x\,\left (35\,B\,a^3\,e^6+30\,B\,a^2\,b\,d\,e^5+105\,A\,a^2\,b\,e^6+15\,B\,a^2\,c\,d^2\,e^4+30\,A\,a^2\,c\,d\,e^5+15\,B\,a\,b^2\,d^2\,e^4+30\,A\,a\,b^2\,d\,e^5+24\,B\,a\,b\,c\,d^3\,e^3+30\,A\,a\,b\,c\,d^2\,e^4+15\,B\,a\,c^2\,d^4\,e^2+12\,A\,a\,c^2\,d^3\,e^3+4\,B\,b^3\,d^3\,e^3+5\,A\,b^3\,d^2\,e^4+15\,B\,b^2\,c\,d^4\,e^2+12\,A\,b^2\,c\,d^3\,e^3+30\,B\,b\,c^2\,d^5\,e+15\,A\,b\,c^2\,d^4\,e^2+35\,B\,c^3\,d^6+10\,A\,c^3\,d^5\,e\right )}{280\,e^7}+\frac {x^2\,\left (30\,B\,a^2\,b\,e^5+15\,B\,a^2\,c\,d\,e^4+30\,A\,a^2\,c\,e^5+15\,B\,a\,b^2\,d\,e^4+30\,A\,a\,b^2\,e^5+24\,B\,a\,b\,c\,d^2\,e^3+30\,A\,a\,b\,c\,d\,e^4+15\,B\,a\,c^2\,d^3\,e^2+12\,A\,a\,c^2\,d^2\,e^3+4\,B\,b^3\,d^2\,e^3+5\,A\,b^3\,d\,e^4+15\,B\,b^2\,c\,d^3\,e^2+12\,A\,b^2\,c\,d^2\,e^3+30\,B\,b\,c^2\,d^4\,e+15\,A\,b\,c^2\,d^3\,e^2+35\,B\,c^3\,d^5+10\,A\,c^3\,d^4\,e\right )}{70\,e^6}+\frac {x^5\,\left (3\,B\,b^2\,c\,e^2+6\,B\,b\,c^2\,d\,e+3\,A\,b\,c^2\,e^2+7\,B\,c^3\,d^2+2\,A\,c^3\,d\,e+3\,B\,a\,c^2\,e^2\right )}{4\,e^3}+\frac {x^3\,\left (15\,B\,a^2\,c\,e^4+15\,B\,a\,b^2\,e^4+24\,B\,a\,b\,c\,d\,e^3+30\,A\,a\,b\,c\,e^4+15\,B\,a\,c^2\,d^2\,e^2+12\,A\,a\,c^2\,d\,e^3+4\,B\,b^3\,d\,e^3+5\,A\,b^3\,e^4+15\,B\,b^2\,c\,d^2\,e^2+12\,A\,b^2\,c\,d\,e^3+30\,B\,b\,c^2\,d^3\,e+15\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4+10\,A\,c^3\,d^3\,e\right )}{30\,e^5}+\frac {c^2\,x^6\,\left (2\,A\,c\,e+6\,B\,b\,e+7\,B\,c\,d\right )}{6\,e^2}+\frac {B\,c^3\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \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)^10,x)

[Out]

-((280*A*a^3*e^7 + 35*B*c^3*d^7 + 35*B*a^3*d*e^6 + 10*A*c^3*d^6*e + 5*A*b^3*d^3*e^4 + 4*B*b^3*d^4*e^3 + 30*A*a
*b^2*d^2*e^5 + 12*A*a*c^2*d^4*e^3 + 30*A*a^2*c*d^2*e^5 + 15*B*a*b^2*d^3*e^4 + 30*B*a^2*b*d^2*e^5 + 15*A*b*c^2*
d^5*e^2 + 12*A*b^2*c*d^4*e^3 + 15*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^4 + 15*B*b^2*c*d^5*e^2 + 105*A*a^2*b*d*e^
6 + 30*B*b*c^2*d^6*e + 30*A*a*b*c*d^3*e^4 + 24*B*a*b*c*d^4*e^3)/(2520*e^8) + (x^4*(4*B*b^3*e^3 + 35*B*c^3*d^3
+ 12*A*a*c^2*e^3 + 12*A*b^2*c*e^3 + 10*A*c^3*d^2*e + 24*B*a*b*c*e^3 + 15*A*b*c^2*d*e^2 + 15*B*a*c^2*d*e^2 + 30
*B*b*c^2*d^2*e + 15*B*b^2*c*d*e^2))/(20*e^4) + (x*(35*B*a^3*e^6 + 35*B*c^3*d^6 + 105*A*a^2*b*e^6 + 10*A*c^3*d^
5*e + 5*A*b^3*d^2*e^4 + 4*B*b^3*d^3*e^3 + 12*A*a*c^2*d^3*e^3 + 15*B*a*b^2*d^2*e^4 + 15*A*b*c^2*d^4*e^2 + 12*A*
b^2*c*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 15*B*a^2*c*d^2*e^4 + 15*B*b^2*c*d^4*e^2 + 30*A*a*b^2*d*e^5 + 30*A*a^2*c*d
*e^5 + 30*B*a^2*b*d*e^5 + 30*B*b*c^2*d^5*e + 30*A*a*b*c*d^2*e^4 + 24*B*a*b*c*d^3*e^3))/(280*e^7) + (x^2*(35*B*
c^3*d^5 + 30*A*a*b^2*e^5 + 30*A*a^2*c*e^5 + 30*B*a^2*b*e^5 + 5*A*b^3*d*e^4 + 10*A*c^3*d^4*e + 4*B*b^3*d^2*e^3
+ 12*A*a*c^2*d^2*e^3 + 15*A*b*c^2*d^3*e^2 + 12*A*b^2*c*d^2*e^3 + 15*B*a*c^2*d^3*e^2 + 15*B*b^2*c*d^3*e^2 + 15*
B*a*b^2*d*e^4 + 15*B*a^2*c*d*e^4 + 30*B*b*c^2*d^4*e + 24*B*a*b*c*d^2*e^3 + 30*A*a*b*c*d*e^4))/(70*e^6) + (x^5*
(7*B*c^3*d^2 + 2*A*c^3*d*e + 3*A*b*c^2*e^2 + 3*B*a*c^2*e^2 + 3*B*b^2*c*e^2 + 6*B*b*c^2*d*e))/(4*e^3) + (x^3*(5
*A*b^3*e^4 + 35*B*c^3*d^4 + 15*B*a*b^2*e^4 + 15*B*a^2*c*e^4 + 10*A*c^3*d^3*e + 4*B*b^3*d*e^3 + 15*A*b*c^2*d^2*
e^2 + 15*B*a*c^2*d^2*e^2 + 15*B*b^2*c*d^2*e^2 + 30*A*a*b*c*e^4 + 12*A*a*c^2*d*e^3 + 12*A*b^2*c*d*e^3 + 30*B*b*
c^2*d^3*e + 24*B*a*b*c*d*e^3))/(30*e^5) + (c^2*x^6*(2*A*c*e + 6*B*b*e + 7*B*c*d))/(6*e^2) + (B*c^3*x^7)/(2*e))
/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e
^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**10,x)

[Out]

Timed out

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