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

Optimal. Leaf size=109 \[ \frac {e^2}{(d+e x) (b d-a e)^3}+\frac {3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac {3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac {2 b e}{(a+b x) (b d-a e)^3}-\frac {b}{2 (a+b x)^2 (b d-a e)^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 109, 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^2}{(d+e x) (b d-a e)^3}+\frac {3 b e^2 \log (a+b x)}{(b d-a e)^4}-\frac {3 b e^2 \log (d+e x)}{(b d-a e)^4}+\frac {2 b e}{(a+b x) (b d-a e)^3}-\frac {b}{2 (a+b x)^2 (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)^2),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 98, normalized size = 0.90 \begin {gather*} \frac {\frac {2 e^2 (b d-a e)}{d+e x}+\frac {4 b e (b d-a e)}{a+b x}-\frac {b (b d-a e)^2}{(a+b x)^2}+6 b e^2 \log (a+b x)-6 b e^2 \log (d+e x)}{2 (b d-a e)^4} \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)^2),x]

[Out]

(-((b*(b*d - a*e)^2)/(a + b*x)^2) + (4*b*e*(b*d - a*e))/(a + b*x) + (2*e^2*(b*d - a*e))/(d + e*x) + 6*b*e^2*Lo
g[a + b*x] - 6*b*e^2*Log[d + e*x])/(2*(b*d - a*e)^4)

________________________________________________________________________________________

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 )^2} \, 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)^2),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 494, normalized size = 4.53 \begin {gather*} -\frac {b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} + {\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + {\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} + {\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} + {\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} + {\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\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)^2,x, algorithm="fricas")

[Out]

-1/2*(b^3*d^3 - 6*a*b^2*d^2*e + 3*a^2*b*d*e^2 + 2*a^3*e^3 - 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e + 2*a
*b^2*d*e^2 - 3*a^2*b*e^3)*x - 6*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 +
a^2*b*e^3)*x)*log(b*x + a) + 6*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 + a
^2*b*e^3)*x)*log(e*x + d))/(a^2*b^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 +
(b^6*d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x^3 + (b^6*d^5 - 2*a*b^5*d^4
*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4
*e + 8*a^3*b^3*d^3*e^2 - 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*e^5)*x)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 212, normalized size = 1.94 \begin {gather*} \frac {3 \, b e^{3} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac {e^{5}}{{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} {\left (x e + d\right )}} + \frac {5 \, b^{3} e^{2} - \frac {6 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} e^{\left (-1\right )}}{x e + d}}{2 \, {\left (b d - a e\right )}^{4} {\left (b - \frac {b d}{x e + d} + \frac {a e}{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)^2/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 109, normalized size = 1.00 \begin {gather*} \frac {3 b \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {3 b \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {2 b e}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {e^{2}}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {b}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}} \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)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.66, size = 386, normalized size = 3.54 \begin {gather*} \frac {3 \, b e^{2} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {3 \, b e^{2} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {6 \, b^{2} e^{2} x^{2} - b^{2} d^{2} + 5 \, a b d e + 2 \, a^{2} e^{2} + 3 \, {\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{2 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.21, size = 330, normalized size = 3.03 \begin {gather*} \frac {6\,b\,e^2\,\mathrm {atanh}\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,e\,x\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {2\,a^2\,e^2+5\,a\,b\,d\,e-b^2\,d^2}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {3\,e\,x\,\left (d\,b^2+3\,a\,e\,b\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {3\,b^2\,e^2\,x^2}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{x\,\left (e\,a^2+2\,b\,d\,a\right )+a^2\,d+x^2\,\left (d\,b^2+2\,a\,e\,b\right )+b^2\,e\,x^3} \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)^2),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.73, size = 634, normalized size = 5.82 \begin {gather*} - \frac {3 b e^{2} \log {\left (x + \frac {- \frac {3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} + \frac {15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} - \frac {30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} + \frac {30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} - \frac {15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} + \frac {3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac {3 b e^{2} \log {\left (x + \frac {\frac {3 a^{5} b e^{7}}{\left (a e - b d\right )^{4}} - \frac {15 a^{4} b^{2} d e^{6}}{\left (a e - b d\right )^{4}} + \frac {30 a^{3} b^{3} d^{2} e^{5}}{\left (a e - b d\right )^{4}} - \frac {30 a^{2} b^{4} d^{3} e^{4}}{\left (a e - b d\right )^{4}} + \frac {15 a b^{5} d^{4} e^{3}}{\left (a e - b d\right )^{4}} + 3 a b e^{3} - \frac {3 b^{6} d^{5} e^{2}}{\left (a e - b d\right )^{4}} + 3 b^{2} d e^{2}}{6 b^{2} e^{3}} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 2 a^{2} e^{2} - 5 a b d e + b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (- 9 a b e^{2} - 3 b^{2} d e\right )}{2 a^{5} d e^{3} - 6 a^{4} b d^{2} e^{2} + 6 a^{3} b^{2} d^{3} e - 2 a^{2} b^{3} d^{4} + x^{3} \left (2 a^{3} b^{2} e^{4} - 6 a^{2} b^{3} d e^{3} + 6 a b^{4} d^{2} e^{2} - 2 b^{5} d^{3} e\right ) + x^{2} \left (4 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 6 a^{2} b^{3} d^{2} e^{2} + 2 a b^{4} d^{3} e - 2 b^{5} d^{4}\right ) + x \left (2 a^{5} e^{4} - 2 a^{4} b d e^{3} - 6 a^{3} b^{2} d^{2} e^{2} + 10 a^{2} b^{3} d^{3} e - 4 a b^{4} d^{4}\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)**2,x)

[Out]

-3*b*e**2*log(x + (-3*a**5*b*e**7/(a*e - b*d)**4 + 15*a**4*b**2*d*e**6/(a*e - b*d)**4 - 30*a**3*b**3*d**2*e**5
/(a*e - b*d)**4 + 30*a**2*b**4*d**3*e**4/(a*e - b*d)**4 - 15*a*b**5*d**4*e**3/(a*e - b*d)**4 + 3*a*b*e**3 + 3*
b**6*d**5*e**2/(a*e - b*d)**4 + 3*b**2*d*e**2)/(6*b**2*e**3))/(a*e - b*d)**4 + 3*b*e**2*log(x + (3*a**5*b*e**7
/(a*e - b*d)**4 - 15*a**4*b**2*d*e**6/(a*e - b*d)**4 + 30*a**3*b**3*d**2*e**5/(a*e - b*d)**4 - 30*a**2*b**4*d*
*3*e**4/(a*e - b*d)**4 + 15*a*b**5*d**4*e**3/(a*e - b*d)**4 + 3*a*b*e**3 - 3*b**6*d**5*e**2/(a*e - b*d)**4 + 3
*b**2*d*e**2)/(6*b**2*e**3))/(a*e - b*d)**4 + (-2*a**2*e**2 - 5*a*b*d*e + b**2*d**2 - 6*b**2*e**2*x**2 + x*(-9
*a*b*e**2 - 3*b**2*d*e))/(2*a**5*d*e**3 - 6*a**4*b*d**2*e**2 + 6*a**3*b**2*d**3*e - 2*a**2*b**3*d**4 + x**3*(2
*a**3*b**2*e**4 - 6*a**2*b**3*d*e**3 + 6*a*b**4*d**2*e**2 - 2*b**5*d**3*e) + x**2*(4*a**4*b*e**4 - 10*a**3*b**
2*d*e**3 + 6*a**2*b**3*d**2*e**2 + 2*a*b**4*d**3*e - 2*b**5*d**4) + x*(2*a**5*e**4 - 2*a**4*b*d*e**3 - 6*a**3*
b**2*d**2*e**2 + 10*a**2*b**3*d**3*e - 4*a*b**4*d**4))

________________________________________________________________________________________