3.21.86 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=286 \[ -\frac {\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^6 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{2 e^6 (d+e x)^2}-\frac {c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac {B c^2 x^2}{2 e^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.41, antiderivative size = 284, 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 {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 )}{e^6 (d+e x)}-\frac {\log (d+e x) \left (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 )\right )}{e^6}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{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 )}{2 e^6 (d+e x)^2}-\frac {c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac {B c^2 x^2}{2 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^4,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 263, normalized size = 0.92 \begin {gather*} \frac {6 \log (d+e x) \left (B \left (2 c e (a e-4 b d)+b^2 e^2+10 c^2 d^2\right )+2 A c e (b e-2 c d)\right )-\frac {6 \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )}{d+e x}+\frac {2 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-\frac {3 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{(d+e x)^2}+6 c e x (A c e+2 b B e-4 B c d)+3 B c^2 e^2 x^2}{6 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^4,x]

[Out]

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

________________________________________________________________________________________

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)^4} \, 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)^4,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^4, x]

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 618, normalized size = 2.16 \begin {gather*} \frac {3 \, B c^{2} e^{5} x^{5} + 47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \, {\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} - {\left (B a^{2} + 2 \, A a b\right )} d e^{4} - 3 \, {\left (5 \, B c^{2} d e^{4} - 2 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} - 9 \, {\left (7 \, B c^{2} d^{2} e^{3} - 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} - 3 \, {\left (3 \, B c^{2} d^{3} e^{2} + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 6 \, {\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} + 3 \, {\left (27 \, B c^{2} d^{4} e - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \, {\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} - {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x + 6 \, {\left (10 \, B c^{2} d^{5} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + {\left (10 \, 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} + 3 \, {\left (10 \, B c^{2} d^{3} e^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 3 \, {\left (10 \, B c^{2} d^{4} e - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/6*(3*B*c^2*e^5*x^5 + 47*B*c^2*d^5 - 2*A*a^2*e^5 - 26*(2*B*b*c + A*c^2)*d^4*e + 11*(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 - (B*a^2 + 2*A*a*b)*d*e^4 - 3*(5*B*c^2*d*e^4 - 2*(2*B*b*c + A*
c^2)*e^5)*x^4 - 9*(7*B*c^2*d^2*e^3 - 2*(2*B*b*c + A*c^2)*d*e^4)*x^3 - 3*(3*B*c^2*d^3*e^2 + 6*(2*B*b*c + A*c^2)
*d^2*e^3 - 6*(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 + 3*(27*B*c^2*d^4*e - 18
*(2*B*b*c + A*c^2)*d^3*e^2 + 9*(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 - (B*a^
2 + 2*A*a*b)*e^5)*x + 6*(10*B*c^2*d^5 - 4*(2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + (10*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 + 3*(10*B*c^2*d^3*e^2 - 4*(2*B*b*
c + A*c^2)*d^2*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d*e^4)*x^2 + 3*(10*B*c^2*d^4*e - 4*(2*B*b*c + A*c^2)*d^3*e^2 +
(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 424, normalized size = 1.48 \begin {gather*} {\left (10 \, B c^{2} d^{2} - 8 \, B b c d e - 4 \, A c^{2} d e + B b^{2} e^{2} + 2 \, B a c e^{2} + 2 \, A b c e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B c^{2} x^{2} e^{4} - 8 \, B c^{2} d x e^{3} + 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac {{\left (47 \, B c^{2} d^{5} - 52 \, B b c d^{4} e - 26 \, A c^{2} d^{4} e + 11 \, B b^{2} d^{3} e^{2} + 22 \, B a c d^{3} e^{2} + 22 \, 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} - B a^{2} d e^{4} - 2 \, A a b d e^{4} - 2 \, A a^{2} e^{5} + 6 \, {\left (10 \, B c^{2} d^{3} e^{2} - 12 \, B b c d^{2} e^{3} - 6 \, A c^{2} d^{2} e^{3} + 3 \, B b^{2} d e^{4} + 6 \, B a c d e^{4} + 6 \, A b c d e^{4} - 2 \, B a b e^{5} - A b^{2} e^{5} - 2 \, A a c e^{5}\right )} x^{2} + 3 \, {\left (35 \, B c^{2} d^{4} e - 40 \, B b c d^{3} e^{2} - 20 \, A c^{2} d^{3} e^{2} + 9 \, B b^{2} d^{2} e^{3} + 18 \, B a c d^{2} e^{3} + 18 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} - B a^{2} e^{5} - 2 \, A a b e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \, {\left (x e + d\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

(10*B*c^2*d^2 - 8*B*b*c*d*e - 4*A*c^2*d*e + B*b^2*e^2 + 2*B*a*c*e^2 + 2*A*b*c*e^2)*e^(-6)*log(abs(x*e + d)) +
1/2*(B*c^2*x^2*e^4 - 8*B*c^2*d*x*e^3 + 4*B*b*c*x*e^4 + 2*A*c^2*x*e^4)*e^(-8) + 1/6*(47*B*c^2*d^5 - 52*B*b*c*d^
4*e - 26*A*c^2*d^4*e + 11*B*b^2*d^3*e^2 + 22*B*a*c*d^3*e^2 + 22*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 - B*a^2*d*e^4 - 2*A*a*b*d*e^4 - 2*A*a^2*e^5 + 6*(10*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 - 6
*A*c^2*d^2*e^3 + 3*B*b^2*d*e^4 + 6*B*a*c*d*e^4 + 6*A*b*c*d*e^4 - 2*B*a*b*e^5 - A*b^2*e^5 - 2*A*a*c*e^5)*x^2 +
3*(35*B*c^2*d^4*e - 40*B*b*c*d^3*e^2 - 20*A*c^2*d^3*e^2 + 9*B*b^2*d^2*e^3 + 18*B*a*c*d^2*e^3 + 18*A*b*c*d^2*e^
3 - 4*B*a*b*d*e^4 - 2*A*b^2*d*e^4 - 4*A*a*c*d*e^4 - B*a^2*e^5 - 2*A*a*b*e^5)*x)*e^(-6)/(x*e + d)^3

________________________________________________________________________________________

maple [B]  time = 0.06, size = 690, normalized size = 2.41 \begin {gather*} -\frac {A \,a^{2}}{3 \left (e x +d \right )^{3} e}+\frac {2 A a b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 A a c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A \,b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 A b c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,a^{2} d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 B a b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 B a c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}+\frac {B \,b^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {2 B b c \,d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,c^{2} d^{5}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A a b}{\left (e x +d \right )^{2} e^{2}}+\frac {2 A a c d}{\left (e x +d \right )^{2} e^{3}}+\frac {A \,b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 A b c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {2 A \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {B \,a^{2}}{2 \left (e x +d \right )^{2} e^{2}}+\frac {2 B a b d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 B a c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}-\frac {3 B \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {4 B b c \,d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {5 B \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {B \,c^{2} x^{2}}{2 e^{4}}-\frac {2 A a c}{\left (e x +d \right ) e^{3}}-\frac {A \,b^{2}}{\left (e x +d \right ) e^{3}}+\frac {6 A b c d}{\left (e x +d \right ) e^{4}}+\frac {2 A b c \ln \left (e x +d \right )}{e^{4}}-\frac {6 A \,c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {A \,c^{2} x}{e^{4}}-\frac {2 B a b}{\left (e x +d \right ) e^{3}}+\frac {6 B a c d}{\left (e x +d \right ) e^{4}}+\frac {2 B a c \ln \left (e x +d \right )}{e^{4}}+\frac {3 B \,b^{2} d}{\left (e x +d \right ) e^{4}}+\frac {B \,b^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {12 B b c \,d^{2}}{\left (e x +d \right ) e^{5}}-\frac {8 B b c d \ln \left (e x +d \right )}{e^{5}}+\frac {2 B b c x}{e^{4}}+\frac {10 B \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {10 B \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d x}{e^{5}} \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)^4,x)

[Out]

-2/3/e^5/(e*x+d)^3*B*b*c*d^4+2/(e*x+d)^2*A*a*c*d/e^3-3/(e*x+d)^2*B*a*c*d^2/e^4+1/3/(e*x+d)^3*B*b^2*d^3/e^4-1/(
e*x+d)^2*A*a*b/e^2+1/(e*x+d)^2*A*b^2*d/e^3+A*c^2/e^4*x-1/2/(e*x+d)^2*B*a^2/e^2-1/3/(e*x+d)^3*A*a^2/e+6/(e*x+d)
*B*a*c*d/e^4-1/(e*x+d)*A*b^2/e^3+2/(e*x+d)^2*B*a*b*d/e^3-2/3/(e*x+d)^3*B*a*b*d^2/e^3+2/3/(e*x+d)^3*A*a*b*d/e^2
+1/3/(e*x+d)^3*B*c^2*d^5/e^6-4*A*c^2*d/e^5*ln(e*x+d)+2*B*a*c/e^4*ln(e*x+d)+10*B*c^2*d^2/e^6*ln(e*x+d)+4/e^5/(e
*x+d)^2*B*b*c*d^3-1/3/(e*x+d)^3*A*b^2*d^2/e^3-2/3/(e*x+d)^3*A*a*c*d^2/e^3+2/3/(e*x+d)^3*B*a*c*d^3/e^4-2/(e*x+d
)*B*a*b/e^3+3/(e*x+d)*B*b^2*d/e^4-6/(e*x+d)*A*c^2*d^2/e^5+10/(e*x+d)*B*c^2*d^3/e^6-3/2/(e*x+d)^2*B*b^2*d^2/e^4
-3/e^4/(e*x+d)^2*A*b*c*d^2-12/e^5/(e*x+d)*B*b*c*d^2+6/e^4/(e*x+d)*A*b*c*d-2/(e*x+d)*A*a*c/e^3+2/3/e^4/(e*x+d)^
3*A*d^3*b*c-8/e^5*ln(e*x+d)*B*b*c*d-4*B*c^2*d/e^5*x+2/(e*x+d)^2*A*c^2*d^3/e^5-5/2/(e*x+d)^2*B*c^2*d^4/e^6-1/3/
(e*x+d)^3*A*c^2*d^4/e^5+1/3/(e*x+d)^3*B*a^2*d/e^2+2*c/e^4*B*b*x+2/e^4*ln(e*x+d)*A*b*c+1/2*B*c^2/e^4*x^2+B*b^2/
e^4*ln(e*x+d)

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 409, normalized size = 1.43 \begin {gather*} \frac {47 \, B c^{2} d^{5} - 2 \, A a^{2} e^{5} - 26 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \, {\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} - {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 6 \, {\left (10 \, B c^{2} d^{3} e^{2} - 6 \, {\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} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B c^{2} d^{4} e - 20 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \, {\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} - {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac {B c^{2} e x^{2} - 2 \, {\left (4 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} x}{2 \, e^{5}} + \frac {{\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} \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)^4,x, algorithm="maxima")

[Out]

1/6*(47*B*c^2*d^5 - 2*A*a^2*e^5 - 26*(2*B*b*c + A*c^2)*d^4*e + 11*(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 - (B*a^2 + 2*A*a*b)*d*e^4 + 6*(10*B*c^2*d^3*e^2 - 6*(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*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 + 3*(35*B*c^2*d^4*e - 20*(2*B*b*c + A
*c^2)*d^3*e^2 + 9*(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 - (B*a^2 + 2*A*a*b)*
e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) + 1/2*(B*c^2*e*x^2 - 2*(4*B*c^2*d - (2*B*b*c + A*c^2)*
e)*x)/e^5 + (10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*log(e*x + d)/e^6

________________________________________________________________________________________

mupad [B]  time = 2.37, size = 465, normalized size = 1.63 \begin {gather*} x\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^4}-\frac {4\,B\,c^2\,d}{e^5}\right )-\frac {x^2\,\left (-3\,B\,b^2\,d\,e^3+A\,b^2\,e^4+12\,B\,b\,c\,d^2\,e^2-6\,A\,b\,c\,d\,e^3+2\,B\,a\,b\,e^4-10\,B\,c^2\,d^3\,e+6\,A\,c^2\,d^2\,e^2-6\,B\,a\,c\,d\,e^3+2\,A\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{2}+2\,B\,a\,b\,d\,e^3+A\,a\,b\,e^4-9\,B\,a\,c\,d^2\,e^2+2\,A\,a\,c\,d\,e^3-\frac {9\,B\,b^2\,d^2\,e^2}{2}+A\,b^2\,d\,e^3+20\,B\,b\,c\,d^3\,e-9\,A\,b\,c\,d^2\,e^2-\frac {35\,B\,c^2\,d^4}{2}+10\,A\,c^2\,d^3\,e\right )+\frac {B\,a^2\,d\,e^4+2\,A\,a^2\,e^5+4\,B\,a\,b\,d^2\,e^3+2\,A\,a\,b\,d\,e^4-22\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3-11\,B\,b^2\,d^3\,e^2+2\,A\,b^2\,d^2\,e^3+52\,B\,b\,c\,d^4\,e-22\,A\,b\,c\,d^3\,e^2-47\,B\,c^2\,d^5+26\,A\,c^2\,d^4\,e}{6\,e}}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (B\,b^2\,e^2-8\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2+10\,B\,c^2\,d^2-4\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{e^6}+\frac {B\,c^2\,x^2}{2\,e^4} \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)^4,x)

[Out]

x*((A*c^2 + 2*B*b*c)/e^4 - (4*B*c^2*d)/e^5) - (x^2*(A*b^2*e^4 + 2*A*a*c*e^4 + 2*B*a*b*e^4 - 3*B*b^2*d*e^3 - 10
*B*c^2*d^3*e + 6*A*c^2*d^2*e^2 - 6*A*b*c*d*e^3 - 6*B*a*c*d*e^3 + 12*B*b*c*d^2*e^2) + x*((B*a^2*e^4)/2 - (35*B*
c^2*d^4)/2 + A*a*b*e^4 + A*b^2*d*e^3 + 10*A*c^2*d^3*e - (9*B*b^2*d^2*e^2)/2 + 2*A*a*c*d*e^3 + 2*B*a*b*d*e^3 +
20*B*b*c*d^3*e - 9*A*b*c*d^2*e^2 - 9*B*a*c*d^2*e^2) + (2*A*a^2*e^5 - 47*B*c^2*d^5 + B*a^2*d*e^4 + 26*A*c^2*d^4
*e + 2*A*b^2*d^2*e^3 - 11*B*b^2*d^3*e^2 + 2*A*a*b*d*e^4 + 52*B*b*c*d^4*e + 4*A*a*c*d^2*e^3 + 4*B*a*b*d^2*e^3 -
22*A*b*c*d^3*e^2 - 22*B*a*c*d^3*e^2)/(6*e))/(d^3*e^5 + e^8*x^3 + 3*d^2*e^6*x + 3*d*e^7*x^2) + (log(d + e*x)*(
B*b^2*e^2 + 10*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))/e^6 + (B*c^2*x^2)/(2*e^4)

________________________________________________________________________________________

sympy [A]  time = 152.98, size = 547, normalized size = 1.91 \begin {gather*} \frac {B c^{2} x^{2}}{2 e^{4}} + x \left (\frac {A c^{2}}{e^{4}} + \frac {2 B b c}{e^{4}} - \frac {4 B c^{2} d}{e^{5}}\right ) + \frac {- 2 A a^{2} e^{5} - 2 A a b d e^{4} - 4 A a c d^{2} e^{3} - 2 A b^{2} d^{2} e^{3} + 22 A b c d^{3} e^{2} - 26 A c^{2} d^{4} e - B a^{2} d e^{4} - 4 B a b d^{2} e^{3} + 22 B a c d^{3} e^{2} + 11 B b^{2} d^{3} e^{2} - 52 B b c d^{4} e + 47 B c^{2} d^{5} + x^{2} \left (- 12 A a c e^{5} - 6 A b^{2} e^{5} + 36 A b c d e^{4} - 36 A c^{2} d^{2} e^{3} - 12 B a b e^{5} + 36 B a c d e^{4} + 18 B b^{2} d e^{4} - 72 B b c d^{2} e^{3} + 60 B c^{2} d^{3} e^{2}\right ) + x \left (- 6 A a b e^{5} - 12 A a c d e^{4} - 6 A b^{2} d e^{4} + 54 A b c d^{2} e^{3} - 60 A c^{2} d^{3} e^{2} - 3 B a^{2} e^{5} - 12 B a b d e^{4} + 54 B a c d^{2} e^{3} + 27 B b^{2} d^{2} e^{3} - 120 B b c d^{3} e^{2} + 105 B c^{2} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} + \frac {\left (2 A b c e^{2} - 4 A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right ) \log {\left (d + e x \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)**4,x)

[Out]

B*c**2*x**2/(2*e**4) + x*(A*c**2/e**4 + 2*B*b*c/e**4 - 4*B*c**2*d/e**5) + (-2*A*a**2*e**5 - 2*A*a*b*d*e**4 - 4
*A*a*c*d**2*e**3 - 2*A*b**2*d**2*e**3 + 22*A*b*c*d**3*e**2 - 26*A*c**2*d**4*e - B*a**2*d*e**4 - 4*B*a*b*d**2*e
**3 + 22*B*a*c*d**3*e**2 + 11*B*b**2*d**3*e**2 - 52*B*b*c*d**4*e + 47*B*c**2*d**5 + x**2*(-12*A*a*c*e**5 - 6*A
*b**2*e**5 + 36*A*b*c*d*e**4 - 36*A*c**2*d**2*e**3 - 12*B*a*b*e**5 + 36*B*a*c*d*e**4 + 18*B*b**2*d*e**4 - 72*B
*b*c*d**2*e**3 + 60*B*c**2*d**3*e**2) + x*(-6*A*a*b*e**5 - 12*A*a*c*d*e**4 - 6*A*b**2*d*e**4 + 54*A*b*c*d**2*e
**3 - 60*A*c**2*d**3*e**2 - 3*B*a**2*e**5 - 12*B*a*b*d*e**4 + 54*B*a*c*d**2*e**3 + 27*B*b**2*d**2*e**3 - 120*B
*b*c*d**3*e**2 + 105*B*c**2*d**4*e))/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + (2*A*b*c*
e**2 - 4*A*c**2*d*e + 2*B*a*c*e**2 + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)*log(d + e*x)/e**6

________________________________________________________________________________________