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

Optimal. Leaf size=192 \[ \frac {15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac {15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac {10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac {b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac {b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac {5 b e^4}{(d+e x) (b d-a e)^6}+\frac {e^4}{2 (d+e x)^2 (b d-a e)^5} \]

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Rubi [A]  time = 0.19, antiderivative size = 192, 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 {10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac {3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac {15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac {15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac {b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac {b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac {5 b e^4}{(d+e x) (b d-a e)^6}+\frac {e^4}{2 (d+e x)^2 (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(4*(b*d - a*e)^3*(a + b*x)^4) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^3) - (3*b^2*e^2)/((b*d - a*e)^5*(a + b*x
)^2) + (10*b^2*e^3)/((b*d - a*e)^6*(a + b*x)) + e^4/(2*(b*d - a*e)^5*(d + e*x)^2) + (5*b*e^4)/((b*d - a*e)^6*(
d + e*x)) + (15*b^2*e^4*Log[a + b*x])/(b*d - a*e)^7 - (15*b^2*e^4*Log[d + e*x])/(b*d - a*e)^7

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)^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^5}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^4}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^3}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)^2}+\frac {15 b^3 e^4}{(b d-a e)^7 (a+b x)}-\frac {e^5}{(b d-a e)^5 (d+e x)^3}-\frac {5 b e^5}{(b d-a e)^6 (d+e x)^2}-\frac {15 b^2 e^5}{(b d-a e)^7 (d+e x)}\right ) \, dx\\ &=-\frac {b^2}{4 (b d-a e)^3 (a+b x)^4}+\frac {b^2 e}{(b d-a e)^4 (a+b x)^3}-\frac {3 b^2 e^2}{(b d-a e)^5 (a+b x)^2}+\frac {10 b^2 e^3}{(b d-a e)^6 (a+b x)}+\frac {e^4}{2 (b d-a e)^5 (d+e x)^2}+\frac {5 b e^4}{(b d-a e)^6 (d+e x)}+\frac {15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac {15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 179, normalized size = 0.93 \begin {gather*} \frac {\frac {40 b^2 e^3 (b d-a e)}{a+b x}-\frac {12 b^2 e^2 (b d-a e)^2}{(a+b x)^2}+\frac {4 b^2 e (b d-a e)^3}{(a+b x)^3}-\frac {b^2 (b d-a e)^4}{(a+b x)^4}+60 b^2 e^4 \log (a+b x)+\frac {20 b e^4 (b d-a e)}{d+e x}+\frac {2 e^4 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^4 \log (d+e x)}{4 (b d-a e)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b^2*(b*d - a*e)^4)/(a + b*x)^4) + (4*b^2*e*(b*d - a*e)^3)/(a + b*x)^3 - (12*b^2*e^2*(b*d - a*e)^2)/(a + b*
x)^2 + (40*b^2*e^3*(b*d - a*e))/(a + b*x) + (2*e^4*(b*d - a*e)^2)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e))/(d + e*
x) + 60*b^2*e^4*Log[a + b*x] - 60*b^2*e^4*Log[d + e*x])/(4*(b*d - a*e)^7)

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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)^3 \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)^3*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

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

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fricas [B]  time = 0.45, size = 1565, normalized size = 8.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a^5*b*d*e^5
- 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^
6*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3*e^3 - 66*
a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^
3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x - 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5
+ 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 +
a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*
e^5)*x)*log(b*x + a) + 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*
a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4
+ 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(a^4*b^7*d^9 - 7
*a^5*b^6*d^8*e + 21*a^6*b^5*d^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b*
d^3*e^6 - a^11*d^2*e^7 + (b^11*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4 - 35*a^3*b^8*d^4*e^5 + 35*a^4*b
^7*d^3*e^6 - 21*a^5*b^6*d^2*e^7 + 7*a^6*b^5*d*e^8 - a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^
2*b^9*d^6*e^3 + 7*a^3*b^8*d^5*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 + 13*a^7*b^4*
d*e^8 - 2*a^8*b^3*e^9)*x^5 + (b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*
d^5*e^4 + 49*a^5*b^6*d^4*e^5 + 49*a^6*b^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4
+ 4*(a*b^10*d^9 - 4*a^2*b^9*d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^5*e^4 + 49*a^6*b^5*d^
4*e^5 - 21*a^7*b^4*d^3*e^6 - a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8
*d^8*e + 71*a^4*b^7*d^7*e^2 - 49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*e
^6 + 29*a^9*b^2*d^2*e^7 - a^10*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*a^4*b^7*d^8*e + 35*a^5*b^6*d^7*
e^2 - 49*a^6*b^5*d^6*e^3 + 35*a^7*b^4*d^5*e^4 - 7*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^1
1*d*e^8)*x)

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giac [B]  time = 0.17, size = 547, normalized size = 2.85 \begin {gather*} \frac {15 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} - 7 \, a b^{7} d^{6} e + 21 \, a^{2} b^{6} d^{5} e^{2} - 35 \, a^{3} b^{5} d^{4} e^{3} + 35 \, a^{4} b^{4} d^{3} e^{4} - 21 \, a^{5} b^{3} d^{2} e^{5} + 7 \, a^{6} b^{2} d e^{6} - a^{7} b e^{7}} - \frac {15 \, b^{2} e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{7} d^{7} e - 7 \, a b^{6} d^{6} e^{2} + 21 \, a^{2} b^{5} d^{5} e^{3} - 35 \, a^{3} b^{4} d^{4} e^{4} + 35 \, a^{4} b^{3} d^{3} e^{5} - 21 \, a^{5} b^{2} d^{2} e^{6} + 7 \, a^{6} b d e^{7} - a^{7} e^{8}} - \frac {b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \, {\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \, {\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \, {\left (b d - a e\right )}^{7} {\left (b x + a\right )}^{4} {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*b^3*e^4*log(abs(b*x + a))/(b^8*d^7 - 7*a*b^7*d^6*e + 21*a^2*b^6*d^5*e^2 - 35*a^3*b^5*d^4*e^3 + 35*a^4*b^4*d
^3*e^4 - 21*a^5*b^3*d^2*e^5 + 7*a^6*b^2*d*e^6 - a^7*b*e^7) - 15*b^2*e^5*log(abs(x*e + d))/(b^7*d^7*e - 7*a*b^6
*d^6*e^2 + 21*a^2*b^5*d^5*e^3 - 35*a^3*b^4*d^4*e^4 + 35*a^4*b^3*d^3*e^5 - 21*a^5*b^2*d^2*e^6 + 7*a^6*b*d*e^7 -
a^7*e^8) - 1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a
^5*b*d*e^5 - 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x
^4 - 20*(b^6*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^
3*e^3 - 66*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2
*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x)/((b*d - a*e)^7*(b*x + a)^4*(x*e + d)^2)

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maple [A]  time = 0.06, size = 189, normalized size = 0.98 \begin {gather*} -\frac {15 b^{2} e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{7}}+\frac {15 b^{2} e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{7}}+\frac {10 b^{2} e^{3}}{\left (a e -b d \right )^{6} \left (b x +a \right )}+\frac {5 b \,e^{4}}{\left (a e -b d \right )^{6} \left (e x +d \right )}+\frac {3 b^{2} e^{2}}{\left (a e -b d \right )^{5} \left (b x +a \right )^{2}}-\frac {e^{4}}{2 \left (a e -b d \right )^{5} \left (e x +d \right )^{2}}+\frac {b^{2} e}{\left (a e -b d \right )^{4} \left (b x +a \right )^{3}}+\frac {b^{2}}{4 \left (a e -b d \right )^{3} \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)^3/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/4/(a*e-b*d)^3*b^2/(b*x+a)^4-15*b^2/(a*e-b*d)^7*e^4*ln(b*x+a)+10*b^2/(a*e-b*d)^6*e^3/(b*x+a)+3*b^2/(a*e-b*d)^
5*e^2/(b*x+a)^2+b^2/(a*e-b*d)^4*e/(b*x+a)^3-1/2*e^4/(a*e-b*d)^5/(e*x+d)^2+15*b^2/(a*e-b*d)^7*e^4*ln(e*x+d)+5*e
^4/(a*e-b*d)^6*b/(e*x+d)

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maxima [B]  time = 1.27, size = 1200, normalized size = 6.25 \begin {gather*} \frac {15 \, b^{2} e^{4} \log \left (b x + a\right )}{b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}} - \frac {15 \, b^{2} e^{4} \log \left (e x + d\right )}{b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}} + \frac {60 \, b^{5} e^{5} x^{5} - b^{5} d^{5} + 7 \, a b^{4} d^{4} e - 23 \, a^{2} b^{3} d^{3} e^{2} + 57 \, a^{3} b^{2} d^{2} e^{3} + 22 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 30 \, {\left (3 \, b^{5} d e^{4} + 7 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} + 16 \, a b^{4} d e^{4} + 13 \, a^{2} b^{3} e^{5}\right )} x^{3} - 5 \, {\left (b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} - 81 \, a^{2} b^{3} d e^{4} - 25 \, a^{3} b^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{5} d^{4} e - 9 \, a b^{4} d^{3} e^{2} + 51 \, a^{2} b^{3} d^{2} e^{3} + 101 \, a^{3} b^{2} d e^{4} + 6 \, a^{4} b e^{5}\right )} x}{4 \, {\left (a^{4} b^{6} d^{8} - 6 \, a^{5} b^{5} d^{7} e + 15 \, a^{6} b^{4} d^{6} e^{2} - 20 \, a^{7} b^{3} d^{5} e^{3} + 15 \, a^{8} b^{2} d^{4} e^{4} - 6 \, a^{9} b d^{3} e^{5} + a^{10} d^{2} e^{6} + {\left (b^{10} d^{6} e^{2} - 6 \, a b^{9} d^{5} e^{3} + 15 \, a^{2} b^{8} d^{4} e^{4} - 20 \, a^{3} b^{7} d^{3} e^{5} + 15 \, a^{4} b^{6} d^{2} e^{6} - 6 \, a^{5} b^{5} d e^{7} + a^{6} b^{4} e^{8}\right )} x^{6} + 2 \, {\left (b^{10} d^{7} e - 4 \, a b^{9} d^{6} e^{2} + 3 \, a^{2} b^{8} d^{5} e^{3} + 10 \, a^{3} b^{7} d^{4} e^{4} - 25 \, a^{4} b^{6} d^{3} e^{5} + 24 \, a^{5} b^{5} d^{2} e^{6} - 11 \, a^{6} b^{4} d e^{7} + 2 \, a^{7} b^{3} e^{8}\right )} x^{5} + {\left (b^{10} d^{8} + 2 \, a b^{9} d^{7} e - 27 \, a^{2} b^{8} d^{6} e^{2} + 64 \, a^{3} b^{7} d^{5} e^{3} - 55 \, a^{4} b^{6} d^{4} e^{4} - 6 \, a^{5} b^{5} d^{3} e^{5} + 43 \, a^{6} b^{4} d^{2} e^{6} - 28 \, a^{7} b^{3} d e^{7} + 6 \, a^{8} b^{2} e^{8}\right )} x^{4} + 4 \, {\left (a b^{9} d^{8} - 3 \, a^{2} b^{8} d^{7} e - 2 \, a^{3} b^{7} d^{6} e^{2} + 19 \, a^{4} b^{6} d^{5} e^{3} - 30 \, a^{5} b^{5} d^{4} e^{4} + 19 \, a^{6} b^{4} d^{3} e^{5} - 2 \, a^{7} b^{3} d^{2} e^{6} - 3 \, a^{8} b^{2} d e^{7} + a^{9} b e^{8}\right )} x^{3} + {\left (6 \, a^{2} b^{8} d^{8} - 28 \, a^{3} b^{7} d^{7} e + 43 \, a^{4} b^{6} d^{6} e^{2} - 6 \, a^{5} b^{5} d^{5} e^{3} - 55 \, a^{6} b^{4} d^{4} e^{4} + 64 \, a^{7} b^{3} d^{3} e^{5} - 27 \, a^{8} b^{2} d^{2} e^{6} + 2 \, a^{9} b d e^{7} + a^{10} e^{8}\right )} x^{2} + 2 \, {\left (2 \, a^{3} b^{7} d^{8} - 11 \, a^{4} b^{6} d^{7} e + 24 \, a^{5} b^{5} d^{6} e^{2} - 25 \, a^{6} b^{4} d^{5} e^{3} + 10 \, a^{7} b^{3} d^{4} e^{4} + 3 \, a^{8} b^{2} d^{3} e^{5} - 4 \, a^{9} b d^{2} e^{6} + a^{10} d 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)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

15*b^2*e^4*log(b*x + a)/(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^
4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7) - 15*b^2*e^4*log(e*x + d)/(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*
b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7) + 1/4*(6
0*b^5*e^5*x^5 - b^5*d^5 + 7*a*b^4*d^4*e - 23*a^2*b^3*d^3*e^2 + 57*a^3*b^2*d^2*e^3 + 22*a^4*b*d*e^4 - 2*a^5*e^5
+ 30*(3*b^5*d*e^4 + 7*a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 + 16*a*b^4*d*e^4 + 13*a^2*b^3*e^5)*x^3 - 5*(b^5*d^3*e^
2 - 15*a*b^4*d^2*e^3 - 81*a^2*b^3*d*e^4 - 25*a^3*b^2*e^5)*x^2 + 2*(b^5*d^4*e - 9*a*b^4*d^3*e^2 + 51*a^2*b^3*d^
2*e^3 + 101*a^3*b^2*d*e^4 + 6*a^4*b*e^5)*x)/(a^4*b^6*d^8 - 6*a^5*b^5*d^7*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3*d
^5*e^3 + 15*a^8*b^2*d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6*a*b^9*d^5*e^3 + 15*a^2*b^8*d^
4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)*x^6 + 2*(b^10*d^7*e - 4*a*b^9
*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 + 24*a^5*b^5*d^2*e^6 - 11*a^6*b^4*d*e^7
+ 2*a^7*b^3*e^8)*x^5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^3 - 55*a^4*b^6*d^4*e
^4 - 6*a^5*b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b^8
*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 19*a^6*b^4*d^3*e^5 - 2*a^7*b^3*d^2*e^6
- 3*a^8*b^2*d*e^7 + a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e + 43*a^4*b^6*d^6*e^2 - 6*a^5*b^5*d^5*e^
3 - 55*a^6*b^4*d^4*e^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2*e^6 + 2*a^9*b*d*e^7 + a^10*e^8)*x^2 + 2*(2*a^3*b^
7*d^8 - 11*a^4*b^6*d^7*e + 24*a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*a^7*b^3*d^4*e^4 + 3*a^8*b^2*d^3*e^5 -
4*a^9*b*d^2*e^6 + a^10*d*e^7)*x)

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mupad [B]  time = 2.69, size = 1098, normalized size = 5.72 \begin {gather*} \frac {\frac {5\,e^3\,x^3\,\left (13\,a^2\,b^3\,e^2+16\,a\,b^4\,d\,e+b^5\,d^2\right )}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}-\frac {2\,a^5\,e^5-22\,a^4\,b\,d\,e^4-57\,a^3\,b^2\,d^2\,e^3+23\,a^2\,b^3\,d^3\,e^2-7\,a\,b^4\,d^4\,e+b^5\,d^5}{4\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {5\,e^2\,x^2\,\left (25\,a^3\,b^2\,e^3+81\,a^2\,b^3\,d\,e^2+15\,a\,b^4\,d^2\,e-b^5\,d^3\right )}{4\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {15\,b^5\,e^5\,x^5}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {e\,x\,\left (6\,a^4\,b\,e^4+101\,a^3\,b^2\,d\,e^3+51\,a^2\,b^3\,d^2\,e^2-9\,a\,b^4\,d^3\,e+b^5\,d^4\right )}{2\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {15\,b\,e^3\,x^4\,\left (3\,d\,b^4\,e+7\,a\,b^3\,e^2\right )}{2\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}}{x\,\left (2\,e\,a^4\,d+4\,b\,a^3\,d^2\right )+x^2\,\left (a^4\,e^2+8\,a^3\,b\,d\,e+6\,a^2\,b^2\,d^2\right )+x^4\,\left (6\,a^2\,b^2\,e^2+8\,a\,b^3\,d\,e+b^4\,d^2\right )+x^5\,\left (2\,d\,b^4\,e+4\,a\,b^3\,e^2\right )+x^3\,\left (4\,a^3\,b\,e^2+12\,a^2\,b^2\,d\,e+4\,a\,b^3\,d^2\right )+a^4\,d^2+b^4\,e^2\,x^6}-\frac {30\,b^2\,e^4\,\mathrm {atanh}\left (\frac {a^7\,e^7-5\,a^6\,b\,d\,e^6+9\,a^5\,b^2\,d^2\,e^5-5\,a^4\,b^3\,d^3\,e^4-5\,a^3\,b^4\,d^4\,e^3+9\,a^2\,b^5\,d^5\,e^2-5\,a\,b^6\,d^6\,e+b^7\,d^7}{{\left (a\,e-b\,d\right )}^7}+\frac {2\,b\,e\,x\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^7}\right )}{{\left (a\,e-b\,d\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*e^3*x^3*(b^5*d^2 + 13*a^2*b^3*e^2 + 16*a*b^4*d*e))/(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5) - (2*a^5*e^5 + b^5*d^5 + 23*a^2*b^3*d^3*e^2 - 57*a
^3*b^2*d^2*e^3 - 7*a*b^4*d^4*e - 22*a^4*b*d*e^4)/(4*(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5)) + (5*e^2*x^2*(25*a^3*b^2*e^3 - b^5*d^3 + 81*a^2*b^3*
d*e^2 + 15*a*b^4*d^2*e))/(4*(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5)) + (15*b^5*e^5*x^5)/(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5) + (e*x*(b^5*d^4 + 6*a^4*b*e^4 + 101*a^3*b^2*d*e^3 +
51*a^2*b^3*d^2*e^2 - 9*a*b^4*d^3*e))/(2*(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5)) + (15*b*e^3*x^4*(7*a*b^3*e^2 + 3*b^4*d*e))/(2*(a^6*e^6 + b^6*d^6
+ 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*b^5*d^5*e - 6*a^5*b*d*e^5)))/(x*(4*a^3*b
*d^2 + 2*a^4*d*e) + x^2*(a^4*e^2 + 6*a^2*b^2*d^2 + 8*a^3*b*d*e) + x^4*(b^4*d^2 + 6*a^2*b^2*e^2 + 8*a*b^3*d*e)
+ x^5*(4*a*b^3*e^2 + 2*b^4*d*e) + x^3*(4*a*b^3*d^2 + 4*a^3*b*e^2 + 12*a^2*b^2*d*e) + a^4*d^2 + b^4*e^2*x^6) -
(30*b^2*e^4*atanh((a^7*e^7 + b^7*d^7 + 9*a^2*b^5*d^5*e^2 - 5*a^3*b^4*d^4*e^3 - 5*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d
^2*e^5 - 5*a*b^6*d^6*e - 5*a^6*b*d*e^6)/(a*e - b*d)^7 + (2*b*e*x*(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5))/(a*e - b*d)^7))/(a*e - b*d)^7

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sympy [B]  time = 5.06, size = 1571, normalized size = 8.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15*b**2*e**4*log(x + (-15*a**8*b**2*e**12/(a*e - b*d)**7 + 120*a**7*b**3*d*e**11/(a*e - b*d)**7 - 420*a**6*b**
4*d**2*e**10/(a*e - b*d)**7 + 840*a**5*b**5*d**3*e**9/(a*e - b*d)**7 - 1050*a**4*b**6*d**4*e**8/(a*e - b*d)**7
+ 840*a**3*b**7*d**5*e**7/(a*e - b*d)**7 - 420*a**2*b**8*d**6*e**6/(a*e - b*d)**7 + 120*a*b**9*d**7*e**5/(a*e
- b*d)**7 + 15*a*b**2*e**5 - 15*b**10*d**8*e**4/(a*e - b*d)**7 + 15*b**3*d*e**4)/(30*b**3*e**5))/(a*e - b*d)*
*7 - 15*b**2*e**4*log(x + (15*a**8*b**2*e**12/(a*e - b*d)**7 - 120*a**7*b**3*d*e**11/(a*e - b*d)**7 + 420*a**6
*b**4*d**2*e**10/(a*e - b*d)**7 - 840*a**5*b**5*d**3*e**9/(a*e - b*d)**7 + 1050*a**4*b**6*d**4*e**8/(a*e - b*d
)**7 - 840*a**3*b**7*d**5*e**7/(a*e - b*d)**7 + 420*a**2*b**8*d**6*e**6/(a*e - b*d)**7 - 120*a*b**9*d**7*e**5/
(a*e - b*d)**7 + 15*a*b**2*e**5 + 15*b**10*d**8*e**4/(a*e - b*d)**7 + 15*b**3*d*e**4)/(30*b**3*e**5))/(a*e - b
*d)**7 + (-2*a**5*e**5 + 22*a**4*b*d*e**4 + 57*a**3*b**2*d**2*e**3 - 23*a**2*b**3*d**3*e**2 + 7*a*b**4*d**4*e
- b**5*d**5 + 60*b**5*e**5*x**5 + x**4*(210*a*b**4*e**5 + 90*b**5*d*e**4) + x**3*(260*a**2*b**3*e**5 + 320*a*b
**4*d*e**4 + 20*b**5*d**2*e**3) + x**2*(125*a**3*b**2*e**5 + 405*a**2*b**3*d*e**4 + 75*a*b**4*d**2*e**3 - 5*b*
*5*d**3*e**2) + x*(12*a**4*b*e**5 + 202*a**3*b**2*d*e**4 + 102*a**2*b**3*d**2*e**3 - 18*a*b**4*d**3*e**2 + 2*b
**5*d**4*e))/(4*a**10*d**2*e**6 - 24*a**9*b*d**3*e**5 + 60*a**8*b**2*d**4*e**4 - 80*a**7*b**3*d**5*e**3 + 60*a
**6*b**4*d**6*e**2 - 24*a**5*b**5*d**7*e + 4*a**4*b**6*d**8 + x**6*(4*a**6*b**4*e**8 - 24*a**5*b**5*d*e**7 + 6
0*a**4*b**6*d**2*e**6 - 80*a**3*b**7*d**3*e**5 + 60*a**2*b**8*d**4*e**4 - 24*a*b**9*d**5*e**3 + 4*b**10*d**6*e
**2) + x**5*(16*a**7*b**3*e**8 - 88*a**6*b**4*d*e**7 + 192*a**5*b**5*d**2*e**6 - 200*a**4*b**6*d**3*e**5 + 80*
a**3*b**7*d**4*e**4 + 24*a**2*b**8*d**5*e**3 - 32*a*b**9*d**6*e**2 + 8*b**10*d**7*e) + x**4*(24*a**8*b**2*e**8
- 112*a**7*b**3*d*e**7 + 172*a**6*b**4*d**2*e**6 - 24*a**5*b**5*d**3*e**5 - 220*a**4*b**6*d**4*e**4 + 256*a**
3*b**7*d**5*e**3 - 108*a**2*b**8*d**6*e**2 + 8*a*b**9*d**7*e + 4*b**10*d**8) + x**3*(16*a**9*b*e**8 - 48*a**8*
b**2*d*e**7 - 32*a**7*b**3*d**2*e**6 + 304*a**6*b**4*d**3*e**5 - 480*a**5*b**5*d**4*e**4 + 304*a**4*b**6*d**5*
e**3 - 32*a**3*b**7*d**6*e**2 - 48*a**2*b**8*d**7*e + 16*a*b**9*d**8) + x**2*(4*a**10*e**8 + 8*a**9*b*d*e**7 -
108*a**8*b**2*d**2*e**6 + 256*a**7*b**3*d**3*e**5 - 220*a**6*b**4*d**4*e**4 - 24*a**5*b**5*d**5*e**3 + 172*a*
*4*b**6*d**6*e**2 - 112*a**3*b**7*d**7*e + 24*a**2*b**8*d**8) + x*(8*a**10*d*e**7 - 32*a**9*b*d**2*e**6 + 24*a
**8*b**2*d**3*e**5 + 80*a**7*b**3*d**4*e**4 - 200*a**6*b**4*d**5*e**3 + 192*a**5*b**5*d**6*e**2 - 88*a**4*b**6
*d**7*e + 16*a**3*b**7*d**8))

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