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

Optimal. Leaf size=186 \[ \frac {2 e^2 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac {2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x)}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{2 b^6 (a+b x)^2}-\frac {(A b-a B) (b d-a e)^4}{3 b^6 (a+b x)^3}+\frac {e^3 x (-4 a B e+A b e+4 b B d)}{b^5}+\frac {B e^4 x^2}{2 b^4} \]

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Rubi [A]  time = 0.23, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \begin {gather*} \frac {e^3 x (-4 a B e+A b e+4 b B d)}{b^5}+\frac {2 e^2 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac {2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x)}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{2 b^6 (a+b x)^2}-\frac {(A b-a B) (b d-a e)^4}{3 b^6 (a+b x)^3}+\frac {B e^4 x^2}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.19, size = 362, normalized size = 1.95 \begin {gather*} \frac {-2 A b \left (13 a^4 e^4+a^3 b e^3 (27 e x-22 d)+3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a b^3 e \left (2 d^3+18 d^2 e x-36 d e^2 x^2-9 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )\right )+B \left (47 a^5 e^4+a^4 b e^3 (81 e x-104 d)-3 a^3 b^2 e^2 \left (-22 d^2+72 d e x+3 e^2 x^2\right )-a^2 b^3 e \left (8 d^3-162 d^2 e x+72 d e^2 x^2+63 e^3 x^3\right )-a b^4 \left (d^4+24 d^3 e x-108 d^2 e^2 x^2-72 d e^3 x^3+15 e^4 x^4\right )+3 b^5 x \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )\right )+12 e^2 (a+b x)^3 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{6 b^6 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b*(13*a^4*e^4 + a^3*b*e^3*(-22*d + 27*e*x) + 3*a^2*b^2*e^2*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*b^3*e*(2*d
^3 + 18*d^2*e*x - 36*d*e^2*x^2 - 9*e^3*x^3) + b^4*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4*x^4)) + B*(47*a^5*
e^4 + a^4*b*e^3*(-104*d + 81*e*x) - 3*a^3*b^2*e^2*(-22*d^2 + 72*d*e*x + 3*e^2*x^2) - a^2*b^3*e*(8*d^3 - 162*d^
2*e*x + 72*d*e^2*x^2 + 63*e^3*x^3) + 3*b^5*x*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) - a*b^4*(d^4 + 24*d^3*
e*x - 108*d^2*e^2*x^2 - 72*d*e^3*x^3 + 15*e^4*x^4)) + 12*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a + b*
x)^3*Log[a + b*x])/(6*b^6*(a + b*x)^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 671, normalized size = 3.61 \begin {gather*} \frac {3 \, B b^{5} e^{4} x^{5} - {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} - 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e + 6 \, {\left (11 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (26 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} d e^{3} + {\left (47 \, B a^{5} - 26 \, A a^{4} b\right )} e^{4} + 3 \, {\left (8 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} e^{4}\right )} x^{4} + 9 \, {\left (8 \, B a b^{4} d e^{3} - {\left (7 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} e^{4}\right )} x^{3} - 3 \, {\left (8 \, B b^{5} d^{3} e - 12 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 24 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + 3 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 3 \, {\left (B b^{5} d^{4} + 4 \, {\left (2 \, B a b^{4} + A b^{5}\right )} d^{3} e - 6 \, {\left (9 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} + 36 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - 9 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (3 \, B a^{3} b^{2} d^{2} e^{2} - 2 \, {\left (4 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (5 \, B a^{5} - 2 \, A a^{4} b\right )} e^{4} + {\left (3 \, B b^{5} d^{2} e^{2} - 2 \, {\left (4 \, B a b^{4} - A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (3 \, B a b^{4} d^{2} e^{2} - 2 \, {\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + {\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 3 \, {\left (3 \, B a^{2} b^{3} d^{2} e^{2} - 2 \, {\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} + {\left (5 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/6*(3*B*b^5*e^4*x^5 - (B*a*b^4 + 2*A*b^5)*d^4 - 4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e + 6*(11*B*a^3*b^2 - 2*A*a^2*b
^3)*d^2*e^2 - 4*(26*B*a^4*b - 11*A*a^3*b^2)*d*e^3 + (47*B*a^5 - 26*A*a^4*b)*e^4 + 3*(8*B*b^5*d*e^3 - (5*B*a*b^
4 - 2*A*b^5)*e^4)*x^4 + 9*(8*B*a*b^4*d*e^3 - (7*B*a^2*b^3 - 2*A*a*b^4)*e^4)*x^3 - 3*(8*B*b^5*d^3*e - 12*(3*B*a
*b^4 - A*b^5)*d^2*e^2 + 24*(B*a^2*b^3 - A*a*b^4)*d*e^3 + 3*(B*a^3*b^2 + 2*A*a^2*b^3)*e^4)*x^2 - 3*(B*b^5*d^4 +
4*(2*B*a*b^4 + A*b^5)*d^3*e - 6*(9*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 + 36*(2*B*a^3*b^2 - A*a^2*b^3)*d*e^3 - 9*(3
*B*a^4*b - 2*A*a^3*b^2)*e^4)*x + 12*(3*B*a^3*b^2*d^2*e^2 - 2*(4*B*a^4*b - A*a^3*b^2)*d*e^3 + (5*B*a^5 - 2*A*a^
4*b)*e^4 + (3*B*b^5*d^2*e^2 - 2*(4*B*a*b^4 - A*b^5)*d*e^3 + (5*B*a^2*b^3 - 2*A*a*b^4)*e^4)*x^3 + 3*(3*B*a*b^4*
d^2*e^2 - 2*(4*B*a^2*b^3 - A*a*b^4)*d*e^3 + (5*B*a^3*b^2 - 2*A*a^2*b^3)*e^4)*x^2 + 3*(3*B*a^2*b^3*d^2*e^2 - 2*
(4*B*a^3*b^2 - A*a^2*b^3)*d*e^3 + (5*B*a^4*b - 2*A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a
^2*b^7*x + a^3*b^6)

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giac [B]  time = 0.18, size = 415, normalized size = 2.23 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} d^{2} e^{2} - 8 \, B a b d e^{3} + 2 \, A b^{2} d e^{3} + 5 \, B a^{2} e^{4} - 2 \, A a b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {B b^{4} x^{2} e^{4} + 8 \, B b^{4} d x e^{3} - 8 \, B a b^{3} x e^{4} + 2 \, A b^{4} x e^{4}}{2 \, b^{8}} - \frac {B a b^{4} d^{4} + 2 \, A b^{5} d^{4} + 8 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e - 66 \, B a^{3} b^{2} d^{2} e^{2} + 12 \, A a^{2} b^{3} d^{2} e^{2} + 104 \, B a^{4} b d e^{3} - 44 \, A a^{3} b^{2} d e^{3} - 47 \, B a^{5} e^{4} + 26 \, A a^{4} b e^{4} + 12 \, {\left (2 \, B b^{5} d^{3} e - 9 \, B a b^{4} d^{2} e^{2} + 3 \, A b^{5} d^{2} e^{2} + 12 \, B a^{2} b^{3} d e^{3} - 6 \, A a b^{4} d e^{3} - 5 \, B a^{3} b^{2} e^{4} + 3 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 3 \, {\left (B b^{5} d^{4} + 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e - 54 \, B a^{2} b^{3} d^{2} e^{2} + 12 \, A a b^{4} d^{2} e^{2} + 80 \, B a^{3} b^{2} d e^{3} - 36 \, A a^{2} b^{3} d e^{3} - 35 \, B a^{4} b e^{4} + 20 \, A a^{3} b^{2} e^{4}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2*(3*B*b^2*d^2*e^2 - 8*B*a*b*d*e^3 + 2*A*b^2*d*e^3 + 5*B*a^2*e^4 - 2*A*a*b*e^4)*log(abs(b*x + a))/b^6 + 1/2*(B
*b^4*x^2*e^4 + 8*B*b^4*d*x*e^3 - 8*B*a*b^3*x*e^4 + 2*A*b^4*x*e^4)/b^8 - 1/6*(B*a*b^4*d^4 + 2*A*b^5*d^4 + 8*B*a
^2*b^3*d^3*e + 4*A*a*b^4*d^3*e - 66*B*a^3*b^2*d^2*e^2 + 12*A*a^2*b^3*d^2*e^2 + 104*B*a^4*b*d*e^3 - 44*A*a^3*b^
2*d*e^3 - 47*B*a^5*e^4 + 26*A*a^4*b*e^4 + 12*(2*B*b^5*d^3*e - 9*B*a*b^4*d^2*e^2 + 3*A*b^5*d^2*e^2 + 12*B*a^2*b
^3*d*e^3 - 6*A*a*b^4*d*e^3 - 5*B*a^3*b^2*e^4 + 3*A*a^2*b^3*e^4)*x^2 + 3*(B*b^5*d^4 + 8*B*a*b^4*d^3*e + 4*A*b^5
*d^3*e - 54*B*a^2*b^3*d^2*e^2 + 12*A*a*b^4*d^2*e^2 + 80*B*a^3*b^2*d*e^3 - 36*A*a^2*b^3*d*e^3 - 35*B*a^4*b*e^4
+ 20*A*a^3*b^2*e^4)*x)/((b*x + a)^3*b^6)

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maple [B]  time = 0.06, size = 626, normalized size = 3.37 \begin {gather*} -\frac {A \,a^{4} e^{4}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {4 A \,a^{3} d \,e^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {2 A \,a^{2} d^{2} e^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {4 A a \,d^{3} e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {A \,d^{4}}{3 \left (b x +a \right )^{3} b}+\frac {B \,a^{5} e^{4}}{3 \left (b x +a \right )^{3} b^{6}}-\frac {4 B \,a^{4} d \,e^{3}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {2 B \,a^{3} d^{2} e^{2}}{\left (b x +a \right )^{3} b^{4}}-\frac {4 B \,a^{2} d^{3} e}{3 \left (b x +a \right )^{3} b^{3}}+\frac {B a \,d^{4}}{3 \left (b x +a \right )^{3} b^{2}}+\frac {2 A \,a^{3} e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {6 A \,a^{2} d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}+\frac {6 A a \,d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {2 A \,d^{3} e}{\left (b x +a \right )^{2} b^{2}}-\frac {5 B \,a^{4} e^{4}}{2 \left (b x +a \right )^{2} b^{6}}+\frac {8 B \,a^{3} d \,e^{3}}{\left (b x +a \right )^{2} b^{5}}-\frac {9 B \,a^{2} d^{2} e^{2}}{\left (b x +a \right )^{2} b^{4}}+\frac {4 B a \,d^{3} e}{\left (b x +a \right )^{2} b^{3}}-\frac {B \,d^{4}}{2 \left (b x +a \right )^{2} b^{2}}+\frac {B \,e^{4} x^{2}}{2 b^{4}}-\frac {6 A \,a^{2} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {12 A a d \,e^{3}}{\left (b x +a \right ) b^{4}}-\frac {4 A a \,e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {6 A \,d^{2} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {4 A d \,e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {A \,e^{4} x}{b^{4}}+\frac {10 B \,a^{3} e^{4}}{\left (b x +a \right ) b^{6}}-\frac {24 B \,a^{2} d \,e^{3}}{\left (b x +a \right ) b^{5}}+\frac {10 B \,a^{2} e^{4} \ln \left (b x +a \right )}{b^{6}}+\frac {18 B a \,d^{2} e^{2}}{\left (b x +a \right ) b^{4}}-\frac {16 B a d \,e^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {4 B a \,e^{4} x}{b^{5}}-\frac {4 B \,d^{3} e}{\left (b x +a \right ) b^{3}}+\frac {6 B \,d^{2} e^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {4 B d \,e^{3} x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

6/b^3/(b*x+a)^2*A*d^2*a*e^2+8/b^5/(b*x+a)^2*B*d*a^3*e^3-4/3/b^3/(b*x+a)^3*B*a^2*d^3*e-6/b^4/(b*x+a)^2*A*a^2*d*
e^3+2/b^4/(b*x+a)^3*B*a^3*d^2*e^2-9/b^4/(b*x+a)^2*B*a^2*d^2*e^2+4/b^3/(b*x+a)^2*B*d^3*a*e+12*e^3/b^4/(b*x+a)*A
*a*d-16/b^5*e^3*ln(b*x+a)*B*d*a-24*e^3/b^5/(b*x+a)*B*a^2*d+18*e^2/b^4/(b*x+a)*B*a*d^2-4/3/b^5/(b*x+a)^3*B*a^4*
d*e^3+4/3/b^4/(b*x+a)^3*A*a^3*d*e^3-1/2/b^2/(b*x+a)^2*B*d^4-1/3/b/(b*x+a)^3*A*d^4+e^4/b^4*A*x+4/3/b^2/(b*x+a)^
3*A*a*d^3*e-2/b^3/(b*x+a)^3*A*a^2*d^2*e^2+10*e^4/b^6/(b*x+a)*B*a^3-4*e/b^3/(b*x+a)*B*d^3+1/3/b^2/(b*x+a)^3*B*a
*d^4+2/b^5/(b*x+a)^2*A*a^3*e^4-2/b^2/(b*x+a)^2*A*d^3*e-5/2/b^6/(b*x+a)^2*B*a^4*e^4-4*e^4/b^5*a*B*x+4*e^3/b^4*B
*d*x-1/3/b^5/(b*x+a)^3*A*a^4*e^4+1/3/b^6/(b*x+a)^3*B*a^5*e^4+10/b^6*e^4*ln(b*x+a)*B*a^2+6/b^4*e^2*ln(b*x+a)*B*
d^2-6*e^4/b^5/(b*x+a)*A*a^2-6*e^2/b^3/(b*x+a)*A*d^2-4/b^5*e^4*ln(b*x+a)*A*a+4/b^4*e^3*ln(b*x+a)*A*d+1/2*B*e^4*
x^2/b^4

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maxima [B]  time = 0.61, size = 434, normalized size = 2.33 \begin {gather*} -\frac {{\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e - 6 \, {\left (11 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 4 \, {\left (26 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} d e^{3} - {\left (47 \, B a^{5} - 26 \, A a^{4} b\right )} e^{4} + 12 \, {\left (2 \, B b^{5} d^{3} e - 3 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 3 \, {\left (B b^{5} d^{4} + 4 \, {\left (2 \, B a b^{4} + A b^{5}\right )} d^{3} e - 6 \, {\left (9 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} + 4 \, {\left (20 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} d e^{3} - 5 \, {\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac {B b e^{4} x^{2} + 2 \, {\left (4 \, B b d e^{3} - {\left (4 \, B a - A b\right )} e^{4}\right )} x}{2 \, b^{5}} + \frac {2 \, {\left (3 \, B b^{2} d^{2} e^{2} - 2 \, {\left (4 \, B a b - A b^{2}\right )} d e^{3} + {\left (5 \, B a^{2} - 2 \, A a b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-1/6*((B*a*b^4 + 2*A*b^5)*d^4 + 4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e - 6*(11*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 + 4*(
26*B*a^4*b - 11*A*a^3*b^2)*d*e^3 - (47*B*a^5 - 26*A*a^4*b)*e^4 + 12*(2*B*b^5*d^3*e - 3*(3*B*a*b^4 - A*b^5)*d^2
*e^2 + 6*(2*B*a^2*b^3 - A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 3*(B*b^5*d^4 + 4*(2*B*a*b^4 +
A*b^5)*d^3*e - 6*(9*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 + 4*(20*B*a^3*b^2 - 9*A*a^2*b^3)*d*e^3 - 5*(7*B*a^4*b - 4*A
*a^3*b^2)*e^4)*x)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(B*b*e^4*x^2 + 2*(4*B*b*d*e^3 - (4*B*a
- A*b)*e^4)*x)/b^5 + 2*(3*B*b^2*d^2*e^2 - 2*(4*B*a*b - A*b^2)*d*e^3 + (5*B*a^2 - 2*A*a*b)*e^4)*log(b*x + a)/b
^6

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mupad [B]  time = 0.19, size = 451, normalized size = 2.42 \begin {gather*} x\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^4}-\frac {4\,B\,a\,e^4}{b^5}\right )-\frac {\frac {-47\,B\,a^5\,e^4+104\,B\,a^4\,b\,d\,e^3+26\,A\,a^4\,b\,e^4-66\,B\,a^3\,b^2\,d^2\,e^2-44\,A\,a^3\,b^2\,d\,e^3+8\,B\,a^2\,b^3\,d^3\,e+12\,A\,a^2\,b^3\,d^2\,e^2+B\,a\,b^4\,d^4+4\,A\,a\,b^4\,d^3\,e+2\,A\,b^5\,d^4}{6\,b}+x\,\left (-\frac {35\,B\,a^4\,e^4}{2}+40\,B\,a^3\,b\,d\,e^3+10\,A\,a^3\,b\,e^4-27\,B\,a^2\,b^2\,d^2\,e^2-18\,A\,a^2\,b^2\,d\,e^3+4\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+\frac {B\,b^4\,d^4}{2}+2\,A\,b^4\,d^3\,e\right )+x^2\,\left (-10\,B\,a^3\,b\,e^4+24\,B\,a^2\,b^2\,d\,e^3+6\,A\,a^2\,b^2\,e^4-18\,B\,a\,b^3\,d^2\,e^2-12\,A\,a\,b^3\,d\,e^3+4\,B\,b^4\,d^3\,e+6\,A\,b^4\,d^2\,e^2\right )}{a^3\,b^5+3\,a^2\,b^6\,x+3\,a\,b^7\,x^2+b^8\,x^3}+\frac {\ln \left (a+b\,x\right )\,\left (10\,B\,a^2\,e^4-16\,B\,a\,b\,d\,e^3-4\,A\,a\,b\,e^4+6\,B\,b^2\,d^2\,e^2+4\,A\,b^2\,d\,e^3\right )}{b^6}+\frac {B\,e^4\,x^2}{2\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^4)/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x*((A*e^4 + 4*B*d*e^3)/b^4 - (4*B*a*e^4)/b^5) - ((2*A*b^5*d^4 - 47*B*a^5*e^4 + 26*A*a^4*b*e^4 + B*a*b^4*d^4 -
44*A*a^3*b^2*d*e^3 + 8*B*a^2*b^3*d^3*e + 12*A*a^2*b^3*d^2*e^2 - 66*B*a^3*b^2*d^2*e^2 + 4*A*a*b^4*d^3*e + 104*B
*a^4*b*d*e^3)/(6*b) + x*((B*b^4*d^4)/2 - (35*B*a^4*e^4)/2 + 10*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2
- 18*A*a^2*b^2*d*e^3 - 27*B*a^2*b^2*d^2*e^2 + 4*B*a*b^3*d^3*e + 40*B*a^3*b*d*e^3) + x^2*(4*B*b^4*d^3*e - 10*B
*a^3*b*e^4 + 6*A*a^2*b^2*e^4 + 6*A*b^4*d^2*e^2 - 18*B*a*b^3*d^2*e^2 + 24*B*a^2*b^2*d*e^3 - 12*A*a*b^3*d*e^3))/
(a^3*b^5 + b^8*x^3 + 3*a^2*b^6*x + 3*a*b^7*x^2) + (log(a + b*x)*(10*B*a^2*e^4 - 4*A*a*b*e^4 + 4*A*b^2*d*e^3 +
6*B*b^2*d^2*e^2 - 16*B*a*b*d*e^3))/b^6 + (B*e^4*x^2)/(2*b^4)

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sympy [B]  time = 21.38, size = 486, normalized size = 2.61 \begin {gather*} \frac {B e^{4} x^{2}}{2 b^{4}} + x \left (\frac {A e^{4}}{b^{4}} - \frac {4 B a e^{4}}{b^{5}} + \frac {4 B d e^{3}}{b^{4}}\right ) + \frac {- 26 A a^{4} b e^{4} + 44 A a^{3} b^{2} d e^{3} - 12 A a^{2} b^{3} d^{2} e^{2} - 4 A a b^{4} d^{3} e - 2 A b^{5} d^{4} + 47 B a^{5} e^{4} - 104 B a^{4} b d e^{3} + 66 B a^{3} b^{2} d^{2} e^{2} - 8 B a^{2} b^{3} d^{3} e - B a b^{4} d^{4} + x^{2} \left (- 36 A a^{2} b^{3} e^{4} + 72 A a b^{4} d e^{3} - 36 A b^{5} d^{2} e^{2} + 60 B a^{3} b^{2} e^{4} - 144 B a^{2} b^{3} d e^{3} + 108 B a b^{4} d^{2} e^{2} - 24 B b^{5} d^{3} e\right ) + x \left (- 60 A a^{3} b^{2} e^{4} + 108 A a^{2} b^{3} d e^{3} - 36 A a b^{4} d^{2} e^{2} - 12 A b^{5} d^{3} e + 105 B a^{4} b e^{4} - 240 B a^{3} b^{2} d e^{3} + 162 B a^{2} b^{3} d^{2} e^{2} - 24 B a b^{4} d^{3} e - 3 B b^{5} d^{4}\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac {2 e^{2} \left (a e - b d\right ) \left (- 2 A b e + 5 B a e - 3 B b d\right ) \log {\left (a + b x \right )}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

B*e**4*x**2/(2*b**4) + x*(A*e**4/b**4 - 4*B*a*e**4/b**5 + 4*B*d*e**3/b**4) + (-26*A*a**4*b*e**4 + 44*A*a**3*b*
*2*d*e**3 - 12*A*a**2*b**3*d**2*e**2 - 4*A*a*b**4*d**3*e - 2*A*b**5*d**4 + 47*B*a**5*e**4 - 104*B*a**4*b*d*e**
3 + 66*B*a**3*b**2*d**2*e**2 - 8*B*a**2*b**3*d**3*e - B*a*b**4*d**4 + x**2*(-36*A*a**2*b**3*e**4 + 72*A*a*b**4
*d*e**3 - 36*A*b**5*d**2*e**2 + 60*B*a**3*b**2*e**4 - 144*B*a**2*b**3*d*e**3 + 108*B*a*b**4*d**2*e**2 - 24*B*b
**5*d**3*e) + x*(-60*A*a**3*b**2*e**4 + 108*A*a**2*b**3*d*e**3 - 36*A*a*b**4*d**2*e**2 - 12*A*b**5*d**3*e + 10
5*B*a**4*b*e**4 - 240*B*a**3*b**2*d*e**3 + 162*B*a**2*b**3*d**2*e**2 - 24*B*a*b**4*d**3*e - 3*B*b**5*d**4))/(6
*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 2*e**2*(a*e - b*d)*(-2*A*b*e + 5*B*a*e - 3*B*b*d
)*log(a + b*x)/b**6

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