3.2325 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac {x \left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{e^5}-\frac {x^2 \left (2 A c e (c d-b e)-B \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{2 e^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}-\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6}-\frac {c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac {B c^2 x^4}{4 e^2} \]

[Out]

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

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Rubi [A]  time = 0.49, antiderivative size = 264, 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} \[ -\frac {x^2 \left (2 A c e (c d-b e)-B \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{2 e^4}-\frac {x \left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{e^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6}-\frac {c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac {B c^2 x^4}{4 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.11, size = 250, normalized size = 0.94 \[ \frac {6 e^2 x^2 \left (B \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )+2 A c e (b e-c d)\right )+12 e x \left (A e \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )-2 B (2 c d-b e) \left (e (a e-b d)+c d^2\right )\right )+\frac {12 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}+12 \log (d+e x) \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )+4 c e^3 x^3 (A c e+2 b B e-2 B c d)+3 B c^2 e^4 x^4}{12 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*e*(A*e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e)) - 2*B*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))*x + 6*
e^2*(2*A*c*e*(-(c*d) + b*e) + B*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e)))*x^2 + 4*c*e^3*(-2*B*c*d + 2*b*B*
e + A*c*e)*x^3 + 3*B*c^2*e^4*x^4 + (12*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) + 12*(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))*Log[d + e*x])/(12*e^6)

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fricas [B]  time = 1.10, size = 567, normalized size = 2.12 \[ \frac {3 \, B c^{2} e^{5} x^{5} + 12 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 12 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 12 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} - {\left (5 \, B c^{2} d e^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (5 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} - 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{4} e - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4}\right )} x + 12 \, {\left (5 \, B c^{2} d^{5} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + {\left (5 \, B c^{2} d^{4} e - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/12*(3*B*c^2*e^5*x^5 + 12*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e + 12*(B*b^2 + 2*(B*a + A*b)*c
)*d^3*e^2 - 12*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + 12*(B*a^2 + 2*A*a*b)*d*e^4 - (5*B*c^2*d*e^4 - 4*(2*B*b*c
+ A*c^2)*e^5)*x^4 + 2*(5*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 - 6*
(5*B*c^2*d^3*e^2 - 4*(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 - 12*(4*B*c^2*d^4*e - 3*(2*B*b*c + A*c^2)*d^3*e^2 + 2*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - (2*B*a
*b + A*b^2 + 2*A*a*c)*d*e^4)*x + 12*(5*B*c^2*d^5 - 4*(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 + (B*a^2 + 2*A*a*b)*d*e^4 + (5*B*c^2*d^4*e - 4*(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 + (B*a^2 + 2*A*a*b)*e^5)*x
)*log(e*x + d))/(e^7*x + d*e^6)

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giac [B]  time = 0.17, size = 551, normalized size = 2.06 \[ \frac {1}{12} \, {\left (3 \, B c^{2} - \frac {4 \, {\left (5 \, B c^{2} d e - 2 \, B b c e^{2} - A c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {6 \, {\left (10 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} + B b^{2} e^{4} + 2 \, B a c e^{4} + 2 \, A b c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {12 \, {\left (10 \, B c^{2} d^{3} e^{3} - 12 \, B b c d^{2} e^{4} - 6 \, A c^{2} d^{2} e^{4} + 3 \, B b^{2} d e^{5} + 6 \, B a c d e^{5} + 6 \, A b c d e^{5} - 2 \, B a b e^{6} - A b^{2} e^{6} - 2 \, A a c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-6\right )} - {\left (5 \, B c^{2} d^{4} - 8 \, B b c d^{3} e - 4 \, A c^{2} d^{3} e + 3 \, B b^{2} d^{2} e^{2} + 6 \, B a c d^{2} e^{2} + 6 \, A b c d^{2} e^{2} - 4 \, B a b d e^{3} - 2 \, A b^{2} d e^{3} - 4 \, A a c d e^{3} + B a^{2} e^{4} + 2 \, A a b e^{4}\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {B c^{2} d^{5} e^{4}}{x e + d} - \frac {2 \, B b c d^{4} e^{5}}{x e + d} - \frac {A c^{2} d^{4} e^{5}}{x e + d} + \frac {B b^{2} d^{3} e^{6}}{x e + d} + \frac {2 \, B a c d^{3} e^{6}}{x e + d} + \frac {2 \, A b c d^{3} e^{6}}{x e + d} - \frac {2 \, B a b d^{2} e^{7}}{x e + d} - \frac {A b^{2} d^{2} e^{7}}{x e + d} - \frac {2 \, A a c d^{2} e^{7}}{x e + d} + \frac {B a^{2} d e^{8}}{x e + d} + \frac {2 \, A a b d e^{8}}{x e + d} - \frac {A a^{2} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/12*(3*B*c^2 - 4*(5*B*c^2*d*e - 2*B*b*c*e^2 - A*c^2*e^2)*e^(-1)/(x*e + d) + 6*(10*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)*e^(-2)/(x*e + d)^2 - 12*(10*B*c^2*d^3*e^3 - 12*B*b
*c*d^2*e^4 - 6*A*c^2*d^2*e^4 + 3*B*b^2*d*e^5 + 6*B*a*c*d*e^5 + 6*A*b*c*d*e^5 - 2*B*a*b*e^6 - A*b^2*e^6 - 2*A*a
*c*e^6)*e^(-3)/(x*e + d)^3)*(x*e + d)^4*e^(-6) - (5*B*c^2*d^4 - 8*B*b*c*d^3*e - 4*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 + B*a^2*e^4 + 2*A*a*b*e^
4)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (B*c^2*d^5*e^4/(x*e + d) - 2*B*b*c*d^4*e^5/(x*e + d) - A*c^2*
d^4*e^5/(x*e + d) + B*b^2*d^3*e^6/(x*e + d) + 2*B*a*c*d^3*e^6/(x*e + d) + 2*A*b*c*d^3*e^6/(x*e + d) - 2*B*a*b*
d^2*e^7/(x*e + d) - A*b^2*d^2*e^7/(x*e + d) - 2*A*a*c*d^2*e^7/(x*e + d) + B*a^2*d*e^8/(x*e + d) + 2*A*a*b*d*e^
8/(x*e + d) - A*a^2*e^9/(x*e + d))*e^(-10)

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maple [B]  time = 0.06, size = 609, normalized size = 2.28 \[ \frac {B \,c^{2} x^{4}}{4 e^{2}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}+\frac {2 B b c \,x^{3}}{3 e^{2}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}+\frac {A b c \,x^{2}}{e^{2}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {B a c \,x^{2}}{e^{2}}+\frac {B \,b^{2} x^{2}}{2 e^{2}}-\frac {2 B b c d \,x^{2}}{e^{3}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {A \,a^{2}}{\left (e x +d \right ) e}+\frac {2 A a b d}{\left (e x +d \right ) e^{2}}+\frac {2 A a b \ln \left (e x +d \right )}{e^{2}}-\frac {2 A a c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {4 A a c d \ln \left (e x +d \right )}{e^{3}}+\frac {2 A a c x}{e^{2}}-\frac {A \,b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 A \,b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {A \,b^{2} x}{e^{2}}+\frac {2 A b c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 A b c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 A b c d x}{e^{3}}-\frac {A \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}+\frac {B \,a^{2} d}{\left (e x +d \right ) e^{2}}+\frac {B \,a^{2} \ln \left (e x +d \right )}{e^{2}}-\frac {2 B a b \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {4 B a b d \ln \left (e x +d \right )}{e^{3}}+\frac {2 B a b x}{e^{2}}+\frac {2 B a c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 B a c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 B a c d x}{e^{3}}+\frac {B \,b^{2} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 B \,b^{2} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 B \,b^{2} d x}{e^{3}}-\frac {2 B b c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {8 B b c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {6 B b c \,d^{2} x}{e^{4}}+\frac {B \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {5 B \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}} \]

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)^2,x)

[Out]

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

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maxima [A]  time = 0.61, size = 387, normalized size = 1.45 \[ \frac {B c^{2} d^{5} - A a^{2} e^{5} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (B a^{2} + 2 \, A a b\right )} d e^{4}}{e^{7} x + d e^{6}} + \frac {3 \, B c^{2} e^{3} x^{4} - 4 \, {\left (2 \, B c^{2} d e^{2} - {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{3} + 6 \, {\left (3 \, B c^{2} d^{2} e - 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{3}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{3} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} x}{12 \, e^{5}} + \frac {{\left (5 \, B c^{2} d^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]

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)^2,x, algorithm="maxima")

[Out]

(B*c^2*d^5 - A*a^2*e^5 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*(B*a + A*b)*c)*d^3*e^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)/(e^7*x + d*e^6) + 1/12*(3*B*c^2*e^3*x^4 - 4*(2*B*c^2*d*e^2 - (2*B*b*c
+ A*c^2)*e^3)*x^3 + 6*(3*B*c^2*d^2*e - 2*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*e^3)*x^2 - 12*(4*
B*c^2*d^3 - 3*(2*B*b*c + A*c^2)*d^2*e + 2*(B*b^2 + 2*(B*a + A*b)*c)*d*e^2 - (2*B*a*b + A*b^2 + 2*A*a*c)*e^3)*x
)/e^5 + (5*B*c^2*d^4 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^2 - 2*(2*B*a*b + A*b^2 +
2*A*a*c)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)*log(e*x + d)/e^6

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mupad [B]  time = 0.12, size = 499, normalized size = 1.87 \[ x\,\left (\frac {A\,b^2+2\,B\,a\,b+2\,A\,a\,c}{e^2}-\frac {d^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{e^2}+\frac {B\,c^2\,d^2}{e^4}\right )}{e}\right )+x^3\,\left (\frac {A\,c^2+2\,B\,b\,c}{3\,e^2}-\frac {2\,B\,c^2\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{2\,e^2}+\frac {B\,c^2\,d^2}{2\,e^4}\right )-\frac {-B\,a^2\,d\,e^4+A\,a^2\,e^5+2\,B\,a\,b\,d^2\,e^3-2\,A\,a\,b\,d\,e^4-2\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3-B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3+2\,B\,b\,c\,d^4\,e-2\,A\,b\,c\,d^3\,e^2-B\,c^2\,d^5+A\,c^2\,d^4\,e}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {\ln \left (d+e\,x\right )\,\left (B\,a^2\,e^4-4\,B\,a\,b\,d\,e^3+2\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2-4\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )}{e^6}+\frac {B\,c^2\,x^4}{4\,e^2} \]

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)^2,x)

[Out]

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

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sympy [A]  time = 4.41, size = 442, normalized size = 1.66 \[ \frac {B c^{2} x^{4}}{4 e^{2}} + x^{3} \left (\frac {A c^{2}}{3 e^{2}} + \frac {2 B b c}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}}\right ) + x^{2} \left (\frac {A b c}{e^{2}} - \frac {A c^{2} d}{e^{3}} + \frac {B a c}{e^{2}} + \frac {B b^{2}}{2 e^{2}} - \frac {2 B b c d}{e^{3}} + \frac {3 B c^{2} d^{2}}{2 e^{4}}\right ) + x \left (\frac {2 A a c}{e^{2}} + \frac {A b^{2}}{e^{2}} - \frac {4 A b c d}{e^{3}} + \frac {3 A c^{2} d^{2}}{e^{4}} + \frac {2 B a b}{e^{2}} - \frac {4 B a c d}{e^{3}} - \frac {2 B b^{2} d}{e^{3}} + \frac {6 B b c d^{2}}{e^{4}} - \frac {4 B c^{2} d^{3}}{e^{5}}\right ) + \frac {- A a^{2} e^{5} + 2 A a b d e^{4} - 2 A a c d^{2} e^{3} - A b^{2} d^{2} e^{3} + 2 A b c d^{3} e^{2} - A c^{2} d^{4} e + B a^{2} d e^{4} - 2 B a b d^{2} e^{3} + 2 B a c d^{3} e^{2} + B b^{2} d^{3} e^{2} - 2 B b c d^{4} e + B c^{2} d^{5}}{d e^{6} + e^{7} x} + \frac {\left (a e^{2} - b d e + c d^{2}\right ) \left (2 A b e^{2} - 4 A c d e + B a e^{2} - 3 B b d e + 5 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} \]

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)**2,x)

[Out]

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

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