3.21.88 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=297 \[ \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{4 e^6 (d+e x)^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6 (d+e x)^5}+\frac {c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \]

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Rubi [A]  time = 0.34, antiderivative size = 295, 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 {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6 (d+e x)^5}+\frac {c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^6*(d + e*x)^5) - ((c*d^2 - b*d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d
- a*e) - 2*A*e*(2*c*d - b*e)))/(4*e^6*(d + e*x)^4) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d
- a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(3*e^6*(d + e*x)^3) + (2*A*c*e*(2*c*d - b*e) - B*(1
0*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(2*e^6*(d + e*x)^2) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(e^6*(d + e*
x)) + (B*c^2*Log[d + e*x])/e^6

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

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Mathematica [A]  time = 0.21, size = 386, normalized size = 1.30 \begin {gather*} \frac {-2 A e \left (e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-e^2 \left (3 a^2 e^2 (d+5 e x)+4 a b e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-6 c e \left (a e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 b \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

(-2*A*e*(6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5
*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2)) + c*e*(2*a*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 1
0*d*e^2*x^2 + 10*e^3*x^3))) + B*(c^2*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4
) - e^2*(3*a^2*e^2*(d + 5*e*x) + 4*a*b*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2
+ 10*e^3*x^3)) - 6*c*e*(a*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*b*(d^4 + 5*d^3*e*x + 10*d^2*e^2*
x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))) + 60*B*c^2*(d + e*x)^5*Log[d + e*x])/(60*e^6*(d + e*x)^5)

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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 )^2}{(d+e x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6, x]

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fricas [A]  time = 0.39, size = 505, normalized size = 1.70 \begin {gather*} \frac {137 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 60 \, {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \, {\left (30 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} - {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 10 \, {\left (110 \, B c^{2} d^{3} e^{2} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x + 60 \, {\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/60*(137*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B
*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - 3*(B*a^2 + 2*A*a*b)*d*e^4 + 60*(5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 +
30*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 + 10*(110*B*c^2*d^3*e^2
- 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 +
5*(125*B*c^2*d^4*e - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2
+ 2*A*a*c)*d*e^4 - 3*(B*a^2 + 2*A*a*b)*e^5)*x + 60*(B*c^2*e^5*x^5 + 5*B*c^2*d*e^4*x^4 + 10*B*c^2*d^2*e^3*x^3 +
10*B*c^2*d^3*e^2*x^2 + 5*B*c^2*d^4*e*x + B*c^2*d^5)*log(e*x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 +
10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)

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giac [A]  time = 0.19, size = 426, normalized size = 1.43 \begin {gather*} B c^{2} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (60 \, {\left (5 \, B c^{2} d e^{3} - 2 \, B b c e^{4} - A c^{2} e^{4}\right )} x^{4} + 30 \, {\left (30 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} - B b^{2} e^{4} - 2 \, B a c e^{4} - 2 \, A b c e^{4}\right )} x^{3} + 10 \, {\left (110 \, B c^{2} d^{3} e - 24 \, B b c d^{2} e^{2} - 12 \, A c^{2} d^{2} e^{2} - 3 \, B b^{2} d e^{3} - 6 \, B a c d e^{3} - 6 \, A b c d e^{3} - 4 \, B a b e^{4} - 2 \, A b^{2} e^{4} - 4 \, A a c e^{4}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} - 24 \, B b c d^{3} e - 12 \, A c^{2} d^{3} e - 3 \, B b^{2} d^{2} e^{2} - 6 \, B a c d^{2} e^{2} - 6 \, A b c d^{2} e^{2} - 4 \, B a b d e^{3} - 2 \, A b^{2} d e^{3} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4} - 6 \, A a b e^{4}\right )} x + {\left (137 \, B c^{2} d^{5} - 24 \, B b c d^{4} e - 12 \, A c^{2} d^{4} e - 3 \, B b^{2} d^{3} e^{2} - 6 \, B a c d^{3} e^{2} - 6 \, A b c d^{3} e^{2} - 4 \, B a b d^{2} e^{3} - 2 \, A b^{2} d^{2} e^{3} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 6 \, A a b d e^{4} - 12 \, A a^{2} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

B*c^2*e^(-6)*log(abs(x*e + d)) + 1/60*(60*(5*B*c^2*d*e^3 - 2*B*b*c*e^4 - A*c^2*e^4)*x^4 + 30*(30*B*c^2*d^2*e^2
- 8*B*b*c*d*e^3 - 4*A*c^2*d*e^3 - B*b^2*e^4 - 2*B*a*c*e^4 - 2*A*b*c*e^4)*x^3 + 10*(110*B*c^2*d^3*e - 24*B*b*c
*d^2*e^2 - 12*A*c^2*d^2*e^2 - 3*B*b^2*d*e^3 - 6*B*a*c*d*e^3 - 6*A*b*c*d*e^3 - 4*B*a*b*e^4 - 2*A*b^2*e^4 - 4*A*
a*c*e^4)*x^2 + 5*(125*B*c^2*d^4 - 24*B*b*c*d^3*e - 12*A*c^2*d^3*e - 3*B*b^2*d^2*e^2 - 6*B*a*c*d^2*e^2 - 6*A*b*
c*d^2*e^2 - 4*B*a*b*d*e^3 - 2*A*b^2*d*e^3 - 4*A*a*c*d*e^3 - 3*B*a^2*e^4 - 6*A*a*b*e^4)*x + (137*B*c^2*d^5 - 24
*B*b*c*d^4*e - 12*A*c^2*d^4*e - 3*B*b^2*d^3*e^2 - 6*B*a*c*d^3*e^2 - 6*A*b*c*d^3*e^2 - 4*B*a*b*d^2*e^3 - 2*A*b^
2*d^2*e^3 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 6*A*a*b*d*e^4 - 12*A*a^2*e^5)*e^(-1))*e^(-5)/(x*e + d)^5

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maple [B]  time = 0.06, size = 715, normalized size = 2.41 \begin {gather*} -\frac {A \,a^{2}}{5 \left (e x +d \right )^{5} e}+\frac {2 A a b d}{5 \left (e x +d \right )^{5} e^{2}}-\frac {2 A a c \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}-\frac {A \,b^{2} d^{2}}{5 \left (e x +d \right )^{5} e^{3}}+\frac {2 A b c \,d^{3}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {A \,c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {B \,a^{2} d}{5 \left (e x +d \right )^{5} e^{2}}-\frac {2 B a b \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}+\frac {2 B a c \,d^{3}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {B \,b^{2} d^{3}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {2 B b c \,d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {B \,c^{2} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {A a b}{2 \left (e x +d \right )^{4} e^{2}}+\frac {A a c d}{\left (e x +d \right )^{4} e^{3}}+\frac {A \,b^{2} d}{2 \left (e x +d \right )^{4} e^{3}}-\frac {3 A b c \,d^{2}}{2 \left (e x +d \right )^{4} e^{4}}+\frac {A \,c^{2} d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {B \,a^{2}}{4 \left (e x +d \right )^{4} e^{2}}+\frac {B a b d}{\left (e x +d \right )^{4} e^{3}}-\frac {3 B a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {3 B \,b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{4}}+\frac {2 B b c \,d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {5 B \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 A a c}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A \,b^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 A b c d}{\left (e x +d \right )^{3} e^{4}}-\frac {2 A \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{5}}-\frac {2 B a b}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 B a c d}{\left (e x +d \right )^{3} e^{4}}+\frac {B \,b^{2} d}{\left (e x +d \right )^{3} e^{4}}-\frac {4 B b c \,d^{2}}{\left (e x +d \right )^{3} e^{5}}+\frac {10 B \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A b c}{\left (e x +d \right )^{2} e^{4}}+\frac {2 A \,c^{2} d}{\left (e x +d \right )^{2} e^{5}}-\frac {B a c}{\left (e x +d \right )^{2} e^{4}}-\frac {B \,b^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {4 B b c d}{\left (e x +d \right )^{2} e^{5}}-\frac {5 B \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{6}}-\frac {A \,c^{2}}{\left (e x +d \right ) e^{5}}-\frac {2 B b c}{\left (e x +d \right ) e^{5}}+\frac {5 B \,c^{2} d}{\left (e x +d \right ) e^{6}}+\frac {B \,c^{2} \ln \left (e x +d \right )}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^6,x)

[Out]

-1/3/e^3/(e*x+d)^3*A*b^2-1/5/e/(e*x+d)^5*A*a^2-1/4/e^2/(e*x+d)^4*B*a^2-c^2/e^5/(e*x+d)*A+1/e^3/(e*x+d)^4*A*a*c
*d+1/e^3/(e*x+d)^4*B*d*a*b+4/e^5/(e*x+d)^2*B*b*c*d-3/2/e^4/(e*x+d)^4*A*b*c*d^2-2/5/e^3/(e*x+d)^5*B*d^2*a*b+2/5
/e^2/(e*x+d)^5*A*a*b*d-3/2/e^4/(e*x+d)^4*B*a*c*d^2+2/e^5/(e*x+d)^4*B*d^3*b*c-2/5/e^3/(e*x+d)^5*A*d^2*a*c+2/5/e
^4/(e*x+d)^5*A*b*c*d^3+2/e^4/(e*x+d)^3*B*a*c*d-4/e^5/(e*x+d)^3*B*b*c*d^2+2/5/e^4/(e*x+d)^5*B*d^3*a*c-2/5/e^5/(
e*x+d)^5*B*d^4*b*c+2/e^4/(e*x+d)^3*A*b*c*d+5*c^2/e^6/(e*x+d)*B*d-1/e^4/(e*x+d)^2*A*b*c+2/e^5/(e*x+d)^2*A*c^2*d
-1/e^4/(e*x+d)^2*a*B*c-2*c/e^5/(e*x+d)*B*b+10/3/e^6/(e*x+d)^3*B*c^2*d^3+1/e^4/(e*x+d)^3*B*b^2*d+1/5/e^2/(e*x+d
)^5*B*a^2*d+1/e^5/(e*x+d)^4*A*c^2*d^3-5/e^6/(e*x+d)^2*B*d^2*c^2+1/5/e^6/(e*x+d)^5*B*c^2*d^5-2/3/e^3/(e*x+d)^3*
A*a*c+1/2/e^3/(e*x+d)^4*A*d*b^2+1/5/e^4/(e*x+d)^5*B*d^3*b^2-2/e^5/(e*x+d)^3*A*c^2*d^2-1/5/e^3/(e*x+d)^5*A*d^2*
b^2-1/5/e^5/(e*x+d)^5*A*c^2*d^4-1/2/e^2/(e*x+d)^4*A*a*b-3/4/e^4/(e*x+d)^4*B*b^2*d^2-5/4/e^6/(e*x+d)^4*B*d^4*c^
2-2/3/e^3/(e*x+d)^3*B*a*b+B*c^2/e^6*ln(e*x+d)-1/2/(e*x+d)^2*B*b^2/e^4

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maxima [A]  time = 0.69, size = 438, normalized size = 1.47 \begin {gather*} \frac {137 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 60 \, {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \, {\left (30 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} - {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 10 \, {\left (110 \, B c^{2} d^{3} e^{2} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac {B c^{2} \log \left (e x + d\right )}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

1/60*(137*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B
*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - 3*(B*a^2 + 2*A*a*b)*d*e^4 + 60*(5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 +
30*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 + 10*(110*B*c^2*d^3*e^2
- 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 +
5*(125*B*c^2*d^4*e - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2
+ 2*A*a*c)*d*e^4 - 3*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*
d^4*e^7*x + d^5*e^6) + B*c^2*log(e*x + d)/e^6

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mupad [B]  time = 2.50, size = 483, normalized size = 1.63 \begin {gather*} \frac {B\,c^2\,\ln \left (d+e\,x\right )}{e^6}-\frac {\frac {3\,B\,a^2\,d\,e^4+12\,A\,a^2\,e^5+4\,B\,a\,b\,d^2\,e^3+6\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3+3\,B\,b^2\,d^3\,e^2+2\,A\,b^2\,d^2\,e^3+24\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2-137\,B\,c^2\,d^5+12\,A\,c^2\,d^4\,e}{60\,e^6}+\frac {x^3\,\left (B\,b^2\,e^2+8\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2-30\,B\,c^2\,d^2+4\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{2\,e^3}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+4\,B\,a\,b\,e^3-110\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e+6\,B\,a\,c\,d\,e^2+4\,A\,a\,c\,e^3\right )}{6\,e^4}+\frac {x\,\left (3\,B\,a^2\,e^4+4\,B\,a\,b\,d\,e^3+6\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2+4\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2+2\,A\,b^2\,d\,e^3+24\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2-125\,B\,c^2\,d^4+12\,A\,c^2\,d^3\,e\right )}{12\,e^5}+\frac {c\,x^4\,\left (A\,c\,e+2\,B\,b\,e-5\,B\,c\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6,x)

[Out]

(B*c^2*log(d + e*x))/e^6 - ((12*A*a^2*e^5 - 137*B*c^2*d^5 + 3*B*a^2*d*e^4 + 12*A*c^2*d^4*e + 2*A*b^2*d^2*e^3 +
3*B*b^2*d^3*e^2 + 6*A*a*b*d*e^4 + 24*B*b*c*d^4*e + 4*A*a*c*d^2*e^3 + 4*B*a*b*d^2*e^3 + 6*A*b*c*d^3*e^2 + 6*B*
a*c*d^3*e^2)/(60*e^6) + (x^3*(B*b^2*e^2 - 30*B*c^2*d^2 + 2*A*b*c*e^2 + 2*B*a*c*e^2 + 4*A*c^2*d*e + 8*B*b*c*d*e
))/(2*e^3) + (x^2*(2*A*b^2*e^3 - 110*B*c^2*d^3 + 4*A*a*c*e^3 + 4*B*a*b*e^3 + 12*A*c^2*d^2*e + 3*B*b^2*d*e^2 +
6*A*b*c*d*e^2 + 6*B*a*c*d*e^2 + 24*B*b*c*d^2*e))/(6*e^4) + (x*(3*B*a^2*e^4 - 125*B*c^2*d^4 + 6*A*a*b*e^4 + 2*A
*b^2*d*e^3 + 12*A*c^2*d^3*e + 3*B*b^2*d^2*e^2 + 4*A*a*c*d*e^3 + 4*B*a*b*d*e^3 + 24*B*b*c*d^3*e + 6*A*b*c*d^2*e
^2 + 6*B*a*c*d^2*e^2))/(12*e^5) + (c*x^4*(A*c*e + 2*B*b*e - 5*B*c*d))/e^2)/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d
^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*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)**2/(e*x+d)**6,x)

[Out]

Timed out

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