3.18.26 \(\int \frac {a+b x}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=159 \[ \frac {e^4}{(d+e x) (b d-a e)^5}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac {4 b e^3}{(a+b x) (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac {2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac {b}{4 (a+b x)^4 (b d-a e)^2} \]

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Rubi [A]  time = 0.14, antiderivative size = 159, 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, 44} \begin {gather*} \frac {e^4}{(d+e x) (b d-a e)^5}+\frac {4 b e^3}{(a+b x) (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac {2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac {b}{4 (a+b x)^4 (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-b/(4*(b*d - a*e)^2*(a + b*x)^4) + (2*b*e)/(3*(b*d - a*e)^3*(a + b*x)^3) - (3*b*e^2)/(2*(b*d - a*e)^4*(a + b*x
)^2) + (4*b*e^3)/((b*d - a*e)^5*(a + b*x)) + e^4/((b*d - a*e)^5*(d + e*x)) + (5*b*e^4*Log[a + b*x])/(b*d - a*e
)^6 - (5*b*e^4*Log[d + e*x])/(b*d - a*e)^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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^5}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^4}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)^3}-\frac {4 b^2 e^3}{(b d-a e)^5 (a+b x)^2}+\frac {5 b^2 e^4}{(b d-a e)^6 (a+b x)}-\frac {e^5}{(b d-a e)^5 (d+e x)^2}-\frac {5 b e^5}{(b d-a e)^6 (d+e x)}\right ) \, dx\\ &=-\frac {b}{4 (b d-a e)^2 (a+b x)^4}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^3}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x)^2}+\frac {4 b e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^5 (d+e x)}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 144, normalized size = 0.91 \begin {gather*} \frac {\frac {12 e^4 (b d-a e)}{d+e x}+\frac {48 b e^3 (b d-a e)}{a+b x}-\frac {18 b e^2 (b d-a e)^2}{(a+b x)^2}+\frac {8 b e (b d-a e)^3}{(a+b x)^3}-\frac {3 b (b d-a e)^4}{(a+b x)^4}+60 b e^4 \log (a+b x)-60 b e^4 \log (d+e x)}{12 (b d-a e)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b*(b*d - a*e)^4)/(a + b*x)^4 + (8*b*e*(b*d - a*e)^3)/(a + b*x)^3 - (18*b*e^2*(b*d - a*e)^2)/(a + b*x)^2 +
(48*b*e^3*(b*d - a*e))/(a + b*x) + (12*e^4*(b*d - a*e))/(d + e*x) + 60*b*e^4*Log[a + b*x] - 60*b*e^4*Log[d +
e*x])/(12*(b*d - a*e)^6)

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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)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 1083, normalized size = 6.81 \begin {gather*} -\frac {3 \, b^{5} d^{5} - 20 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 65 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} - 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (b^{5} d^{3} e^{2} - 12 \, a b^{4} d^{2} e^{3} - 15 \, a^{2} b^{3} d e^{4} + 26 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 8 \, a b^{4} d^{3} e^{2} + 36 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{6} d^{7} - 6 \, a^{5} b^{5} d^{6} e + 15 \, a^{6} b^{4} d^{5} e^{2} - 20 \, a^{7} b^{3} d^{4} e^{3} + 15 \, a^{8} b^{2} d^{3} e^{4} - 6 \, a^{9} b d^{2} e^{5} + a^{10} d e^{6} + {\left (b^{10} d^{6} e - 6 \, a b^{9} d^{5} e^{2} + 15 \, a^{2} b^{8} d^{4} e^{3} - 20 \, a^{3} b^{7} d^{3} e^{4} + 15 \, a^{4} b^{6} d^{2} e^{5} - 6 \, a^{5} b^{5} d e^{6} + a^{6} b^{4} e^{7}\right )} x^{5} + {\left (b^{10} d^{7} - 2 \, a b^{9} d^{6} e - 9 \, a^{2} b^{8} d^{5} e^{2} + 40 \, a^{3} b^{7} d^{4} e^{3} - 65 \, a^{4} b^{6} d^{3} e^{4} + 54 \, a^{5} b^{5} d^{2} e^{5} - 23 \, a^{6} b^{4} d e^{6} + 4 \, a^{7} b^{3} e^{7}\right )} x^{4} + 2 \, {\left (2 \, a b^{9} d^{7} - 9 \, a^{2} b^{8} d^{6} e + 12 \, a^{3} b^{7} d^{5} e^{2} + 5 \, a^{4} b^{6} d^{4} e^{3} - 30 \, a^{5} b^{5} d^{3} e^{4} + 33 \, a^{6} b^{4} d^{2} e^{5} - 16 \, a^{7} b^{3} d e^{6} + 3 \, a^{8} b^{2} e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{8} d^{7} - 16 \, a^{3} b^{7} d^{6} e + 33 \, a^{4} b^{6} d^{5} e^{2} - 30 \, a^{5} b^{5} d^{4} e^{3} + 5 \, a^{6} b^{4} d^{3} e^{4} + 12 \, a^{7} b^{3} d^{2} e^{5} - 9 \, a^{8} b^{2} d e^{6} + 2 \, a^{9} b e^{7}\right )} x^{2} + {\left (4 \, a^{3} b^{7} d^{7} - 23 \, a^{4} b^{6} d^{6} e + 54 \, a^{5} b^{5} d^{5} e^{2} - 65 \, a^{6} b^{4} d^{4} e^{3} + 40 \, a^{7} b^{3} d^{3} e^{4} - 9 \, a^{8} b^{2} d^{2} e^{5} - 2 \, a^{9} b d e^{6} + a^{10} e^{7}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/12*(3*b^5*d^5 - 20*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 65*a^4*b*d*e^4 + 12*a^5*e^5 - 6
0*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^3 + 6*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 + 10*(b^5*d^3*e^2 - 12*a*
b^4*d^2*e^3 - 15*a^2*b^3*d*e^4 + 26*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 8*a*b^4*d^3*e^2 + 36*a^2*b^3*d^2*e^3 - 4
*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*
d*e^4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(b*
x + a) + 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3
+ 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(e*x + d))/(a^4*b^6*d^7 - 6*a
^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b^3*d^4*e^3 + 15*a^8*b^2*d^3*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (
b^10*d^6*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^6*d^2*e^5 - 6*a^5*b^5*d*e^6
+ a^6*b^4*e^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 65*a^4*b^6*d^3*e^4 +
54*a^5*b^5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 + 2*(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*a^3*b^7*d^
5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*a^5*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 +
2*(3*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5*a^6*b^4*d^3*e^4 + 12*a^7*b^3
*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 + (4*a^3*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5*b^5*d^5*e^2 - 65*a^
6*b^4*d^4*e^3 + 40*a^7*b^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a^9*b*d*e^6 + a^10*e^7)*x)

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giac [B]  time = 0.21, size = 358, normalized size = 2.25 \begin {gather*} \frac {5 \, b e^{5} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} + \frac {e^{9}}{{\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )} {\left (x e + d\right )}} + \frac {77 \, b^{5} e^{4} - \frac {260 \, {\left (b^{5} d e^{5} - a b^{4} e^{6}\right )} e^{\left (-1\right )}}{x e + d} + \frac {300 \, {\left (b^{5} d^{2} e^{6} - 2 \, a b^{4} d e^{7} + a^{2} b^{3} e^{8}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

5*b*e^5*log(abs(b - b*d/(x*e + d) + a*e/(x*e + d)))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^4*d^4*e^3 - 20*a^3
*b^3*d^3*e^4 + 15*a^4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) + e^9/((b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^
3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^9 - a^5*e^10)*(x*e + d)) + 1/12*(77*b^5*e^4 - 260*(b^5*d*e^5 - a*
b^4*e^6)*e^(-1)/(x*e + d) + 300*(b^5*d^2*e^6 - 2*a*b^4*d*e^7 + a^2*b^3*e^8)*e^(-2)/(x*e + d)^2 - 120*(b^5*d^3*
e^7 - 3*a*b^4*d^2*e^8 + 3*a^2*b^3*d*e^9 - a^3*b^2*e^10)*e^(-3)/(x*e + d)^3)/((b*d - a*e)^6*(b - b*d/(x*e + d)
+ a*e/(x*e + d))^4)

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maple [A]  time = 0.06, size = 155, normalized size = 0.97 \begin {gather*} \frac {5 b \,e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}-\frac {4 b \,e^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {e^{4}}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {3 b \,e^{2}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}-\frac {2 b e}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}-\frac {b}{4 \left (a e -b d \right )^{2} \left (b x +a \right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

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maxima [B]  time = 0.86, size = 858, normalized size = 5.40 \begin {gather*} \frac {5 \, b e^{4} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} - \frac {5 \, b e^{4} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {60 \, b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 17 \, a b^{3} d^{3} e - 43 \, a^{2} b^{2} d^{2} e^{2} + 77 \, a^{3} b d e^{3} + 12 \, a^{4} e^{4} + 30 \, {\left (b^{4} d e^{3} + 7 \, a b^{3} e^{4}\right )} x^{3} - 10 \, {\left (b^{4} d^{2} e^{2} - 11 \, a b^{3} d e^{3} - 26 \, a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e - 7 \, a b^{3} d^{2} e^{2} + 29 \, a^{2} b^{2} d e^{3} + 25 \, a^{3} b e^{4}\right )} x}{12 \, {\left (a^{4} b^{5} d^{6} - 5 \, a^{5} b^{4} d^{5} e + 10 \, a^{6} b^{3} d^{4} e^{2} - 10 \, a^{7} b^{2} d^{3} e^{3} + 5 \, a^{8} b d^{2} e^{4} - a^{9} d e^{5} + {\left (b^{9} d^{5} e - 5 \, a b^{8} d^{4} e^{2} + 10 \, a^{2} b^{7} d^{3} e^{3} - 10 \, a^{3} b^{6} d^{2} e^{4} + 5 \, a^{4} b^{5} d e^{5} - a^{5} b^{4} e^{6}\right )} x^{5} + {\left (b^{9} d^{6} - a b^{8} d^{5} e - 10 \, a^{2} b^{7} d^{4} e^{2} + 30 \, a^{3} b^{6} d^{3} e^{3} - 35 \, a^{4} b^{5} d^{2} e^{4} + 19 \, a^{5} b^{4} d e^{5} - 4 \, a^{6} b^{3} e^{6}\right )} x^{4} + 2 \, {\left (2 \, a b^{8} d^{6} - 7 \, a^{2} b^{7} d^{5} e + 5 \, a^{3} b^{6} d^{4} e^{2} + 10 \, a^{4} b^{5} d^{3} e^{3} - 20 \, a^{5} b^{4} d^{2} e^{4} + 13 \, a^{6} b^{3} d e^{5} - 3 \, a^{7} b^{2} e^{6}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{7} d^{6} - 13 \, a^{3} b^{6} d^{5} e + 20 \, a^{4} b^{5} d^{4} e^{2} - 10 \, a^{5} b^{4} d^{3} e^{3} - 5 \, a^{6} b^{3} d^{2} e^{4} + 7 \, a^{7} b^{2} d e^{5} - 2 \, a^{8} b e^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{6} d^{6} - 19 \, a^{4} b^{5} d^{5} e + 35 \, a^{5} b^{4} d^{4} e^{2} - 30 \, a^{6} b^{3} d^{3} e^{3} + 10 \, a^{7} b^{2} d^{2} e^{4} + a^{8} b d e^{5} - a^{9} e^{6}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

5*b*e^4*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 -
6*a^5*b*d*e^5 + a^6*e^6) - 5*b*e^4*log(e*x + d)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^
3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 1/12*(60*b^4*e^4*x^4 - 3*b^4*d^4 + 17*a*b^3*d^3*e - 43
*a^2*b^2*d^2*e^2 + 77*a^3*b*d*e^3 + 12*a^4*e^4 + 30*(b^4*d*e^3 + 7*a*b^3*e^4)*x^3 - 10*(b^4*d^2*e^2 - 11*a*b^3
*d*e^3 - 26*a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e - 7*a*b^3*d^2*e^2 + 29*a^2*b^2*d*e^3 + 25*a^3*b*e^4)*x)/(a^4*b^5*d
^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8*b*d^2*e^4 - a^9*d*e^5 + (b^9*d^5*e - 5*
a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6)*x^5 + (b^9*d^6 - a*b^
8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4 + 19*a^5*b^4*d*e^5 - 4*a^6*b^3*e^6)*x^4
+ 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d^3*e^3 - 20*a^5*b^4*d^2*e^4 + 13*a^6*b^3
*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20*a^4*b^5*d^4*e^2 - 10*a^5*b^4*d^3*e^3 -
5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^6 - 19*a^4*b^5*d^5*e + 35*a^5*b^4*d^4*e^
2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*x)

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mupad [B]  time = 0.52, size = 763, normalized size = 4.80 \begin {gather*} \frac {10\,b\,e^4\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {12\,a^4\,e^4+77\,a^3\,b\,d\,e^3-43\,a^2\,b^2\,d^2\,e^2+17\,a\,b^3\,d^3\,e-3\,b^4\,d^4}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e\,x\,\left (25\,a^3\,b\,e^3+29\,a^2\,b^2\,d\,e^2-7\,a\,b^3\,d^2\,e+b^4\,d^3\right )}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {5\,e^3\,x^3\,\left (d\,b^4+7\,a\,e\,b^3\right )}{2\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e^2\,x^2\,\left (26\,a^2\,b^2\,e^2+11\,a\,b^3\,d\,e-b^4\,d^2\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{x^4\,\left (d\,b^4+4\,a\,e\,b^3\right )+a^4\,d+x\,\left (e\,a^4+4\,b\,d\,a^3\right )+x^2\,\left (4\,e\,a^3\,b+6\,d\,a^2\,b^2\right )+x^3\,\left (6\,e\,a^2\,b^2+4\,d\,a\,b^3\right )+b^4\,e\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/((d + e*x)^2*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

(10*b*e^4*atanh((a^6*e^6 - b^6*d^6 - 5*a^2*b^4*d^4*e^2 + 5*a^4*b^2*d^2*e^4 + 4*a*b^5*d^5*e - 4*a^5*b*d*e^5)/(a
*e - b*d)^6 + (2*b*e*x*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*
d*e^4))/(a*e - b*d)^6))/(a*e - b*d)^6 - ((12*a^4*e^4 - 3*b^4*d^4 - 43*a^2*b^2*d^2*e^2 + 17*a*b^3*d^3*e + 77*a^
3*b*d*e^3)/(12*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))
+ (5*e*x*(b^4*d^3 + 25*a^3*b*e^3 + 29*a^2*b^2*d*e^2 - 7*a*b^3*d^2*e))/(12*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*
e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (5*b^4*e^4*x^4)/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d
^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (5*e^3*x^3*(b^4*d + 7*a*b^3*e))/(2*(a^5*e^5 - b
^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (5*e^2*x^2*(26*a^2*b^2*e^
2 - b^4*d^2 + 11*a*b^3*d*e))/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e -
5*a^4*b*d*e^4)))/(x^4*(b^4*d + 4*a*b^3*e) + a^4*d + x*(a^4*e + 4*a^3*b*d) + x^2*(6*a^2*b^2*d + 4*a^3*b*e) + x
^3*(6*a^2*b^2*e + 4*a*b^3*d) + b^4*e*x^5)

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sympy [B]  time = 3.55, size = 1178, normalized size = 7.41 \begin {gather*} - \frac {5 b e^{4} \log {\left (x + \frac {- \frac {5 a^{7} b e^{11}}{\left (a e - b d\right )^{6}} + \frac {35 a^{6} b^{2} d e^{10}}{\left (a e - b d\right )^{6}} - \frac {105 a^{5} b^{3} d^{2} e^{9}}{\left (a e - b d\right )^{6}} + \frac {175 a^{4} b^{4} d^{3} e^{8}}{\left (a e - b d\right )^{6}} - \frac {175 a^{3} b^{5} d^{4} e^{7}}{\left (a e - b d\right )^{6}} + \frac {105 a^{2} b^{6} d^{5} e^{6}}{\left (a e - b d\right )^{6}} - \frac {35 a b^{7} d^{6} e^{5}}{\left (a e - b d\right )^{6}} + 5 a b e^{5} + \frac {5 b^{8} d^{7} e^{4}}{\left (a e - b d\right )^{6}} + 5 b^{2} d e^{4}}{10 b^{2} e^{5}} \right )}}{\left (a e - b d\right )^{6}} + \frac {5 b e^{4} \log {\left (x + \frac {\frac {5 a^{7} b e^{11}}{\left (a e - b d\right )^{6}} - \frac {35 a^{6} b^{2} d e^{10}}{\left (a e - b d\right )^{6}} + \frac {105 a^{5} b^{3} d^{2} e^{9}}{\left (a e - b d\right )^{6}} - \frac {175 a^{4} b^{4} d^{3} e^{8}}{\left (a e - b d\right )^{6}} + \frac {175 a^{3} b^{5} d^{4} e^{7}}{\left (a e - b d\right )^{6}} - \frac {105 a^{2} b^{6} d^{5} e^{6}}{\left (a e - b d\right )^{6}} + \frac {35 a b^{7} d^{6} e^{5}}{\left (a e - b d\right )^{6}} + 5 a b e^{5} - \frac {5 b^{8} d^{7} e^{4}}{\left (a e - b d\right )^{6}} + 5 b^{2} d e^{4}}{10 b^{2} e^{5}} \right )}}{\left (a e - b d\right )^{6}} + \frac {- 12 a^{4} e^{4} - 77 a^{3} b d e^{3} + 43 a^{2} b^{2} d^{2} e^{2} - 17 a b^{3} d^{3} e + 3 b^{4} d^{4} - 60 b^{4} e^{4} x^{4} + x^{3} \left (- 210 a b^{3} e^{4} - 30 b^{4} d e^{3}\right ) + x^{2} \left (- 260 a^{2} b^{2} e^{4} - 110 a b^{3} d e^{3} + 10 b^{4} d^{2} e^{2}\right ) + x \left (- 125 a^{3} b e^{4} - 145 a^{2} b^{2} d e^{3} + 35 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{12 a^{9} d e^{5} - 60 a^{8} b d^{2} e^{4} + 120 a^{7} b^{2} d^{3} e^{3} - 120 a^{6} b^{3} d^{4} e^{2} + 60 a^{5} b^{4} d^{5} e - 12 a^{4} b^{5} d^{6} + x^{5} \left (12 a^{5} b^{4} e^{6} - 60 a^{4} b^{5} d e^{5} + 120 a^{3} b^{6} d^{2} e^{4} - 120 a^{2} b^{7} d^{3} e^{3} + 60 a b^{8} d^{4} e^{2} - 12 b^{9} d^{5} e\right ) + x^{4} \left (48 a^{6} b^{3} e^{6} - 228 a^{5} b^{4} d e^{5} + 420 a^{4} b^{5} d^{2} e^{4} - 360 a^{3} b^{6} d^{3} e^{3} + 120 a^{2} b^{7} d^{4} e^{2} + 12 a b^{8} d^{5} e - 12 b^{9} d^{6}\right ) + x^{3} \left (72 a^{7} b^{2} e^{6} - 312 a^{6} b^{3} d e^{5} + 480 a^{5} b^{4} d^{2} e^{4} - 240 a^{4} b^{5} d^{3} e^{3} - 120 a^{3} b^{6} d^{4} e^{2} + 168 a^{2} b^{7} d^{5} e - 48 a b^{8} d^{6}\right ) + x^{2} \left (48 a^{8} b e^{6} - 168 a^{7} b^{2} d e^{5} + 120 a^{6} b^{3} d^{2} e^{4} + 240 a^{5} b^{4} d^{3} e^{3} - 480 a^{4} b^{5} d^{4} e^{2} + 312 a^{3} b^{6} d^{5} e - 72 a^{2} b^{7} d^{6}\right ) + x \left (12 a^{9} e^{6} - 12 a^{8} b d e^{5} - 120 a^{7} b^{2} d^{2} e^{4} + 360 a^{6} b^{3} d^{3} e^{3} - 420 a^{5} b^{4} d^{4} e^{2} + 228 a^{4} b^{5} d^{5} e - 48 a^{3} b^{6} d^{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-5*b*e**4*log(x + (-5*a**7*b*e**11/(a*e - b*d)**6 + 35*a**6*b**2*d*e**10/(a*e - b*d)**6 - 105*a**5*b**3*d**2*e
**9/(a*e - b*d)**6 + 175*a**4*b**4*d**3*e**8/(a*e - b*d)**6 - 175*a**3*b**5*d**4*e**7/(a*e - b*d)**6 + 105*a**
2*b**6*d**5*e**6/(a*e - b*d)**6 - 35*a*b**7*d**6*e**5/(a*e - b*d)**6 + 5*a*b*e**5 + 5*b**8*d**7*e**4/(a*e - b*
d)**6 + 5*b**2*d*e**4)/(10*b**2*e**5))/(a*e - b*d)**6 + 5*b*e**4*log(x + (5*a**7*b*e**11/(a*e - b*d)**6 - 35*a
**6*b**2*d*e**10/(a*e - b*d)**6 + 105*a**5*b**3*d**2*e**9/(a*e - b*d)**6 - 175*a**4*b**4*d**3*e**8/(a*e - b*d)
**6 + 175*a**3*b**5*d**4*e**7/(a*e - b*d)**6 - 105*a**2*b**6*d**5*e**6/(a*e - b*d)**6 + 35*a*b**7*d**6*e**5/(a
*e - b*d)**6 + 5*a*b*e**5 - 5*b**8*d**7*e**4/(a*e - b*d)**6 + 5*b**2*d*e**4)/(10*b**2*e**5))/(a*e - b*d)**6 +
(-12*a**4*e**4 - 77*a**3*b*d*e**3 + 43*a**2*b**2*d**2*e**2 - 17*a*b**3*d**3*e + 3*b**4*d**4 - 60*b**4*e**4*x**
4 + x**3*(-210*a*b**3*e**4 - 30*b**4*d*e**3) + x**2*(-260*a**2*b**2*e**4 - 110*a*b**3*d*e**3 + 10*b**4*d**2*e*
*2) + x*(-125*a**3*b*e**4 - 145*a**2*b**2*d*e**3 + 35*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(12*a**9*d*e**5 - 60*
a**8*b*d**2*e**4 + 120*a**7*b**2*d**3*e**3 - 120*a**6*b**3*d**4*e**2 + 60*a**5*b**4*d**5*e - 12*a**4*b**5*d**6
+ x**5*(12*a**5*b**4*e**6 - 60*a**4*b**5*d*e**5 + 120*a**3*b**6*d**2*e**4 - 120*a**2*b**7*d**3*e**3 + 60*a*b*
*8*d**4*e**2 - 12*b**9*d**5*e) + x**4*(48*a**6*b**3*e**6 - 228*a**5*b**4*d*e**5 + 420*a**4*b**5*d**2*e**4 - 36
0*a**3*b**6*d**3*e**3 + 120*a**2*b**7*d**4*e**2 + 12*a*b**8*d**5*e - 12*b**9*d**6) + x**3*(72*a**7*b**2*e**6 -
312*a**6*b**3*d*e**5 + 480*a**5*b**4*d**2*e**4 - 240*a**4*b**5*d**3*e**3 - 120*a**3*b**6*d**4*e**2 + 168*a**2
*b**7*d**5*e - 48*a*b**8*d**6) + x**2*(48*a**8*b*e**6 - 168*a**7*b**2*d*e**5 + 120*a**6*b**3*d**2*e**4 + 240*a
**5*b**4*d**3*e**3 - 480*a**4*b**5*d**4*e**2 + 312*a**3*b**6*d**5*e - 72*a**2*b**7*d**6) + x*(12*a**9*e**6 - 1
2*a**8*b*d*e**5 - 120*a**7*b**2*d**2*e**4 + 360*a**6*b**3*d**3*e**3 - 420*a**5*b**4*d**4*e**2 + 228*a**4*b**5*
d**5*e - 48*a**3*b**6*d**6))

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