### 3.1516 $$\int \frac{(d+e x)^5}{(a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=129 $\frac{e^4 x (5 b d-4 a e)}{b^5}-\frac{10 e^2 (b d-a e)^3}{b^6 (a+b x)}+\frac{10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}-\frac{5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{(b d-a e)^5}{3 b^6 (a+b x)^3}+\frac{e^5 x^2}{2 b^4}$

[Out]

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

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Rubi [A]  time = 0.124658, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 43} $\frac{e^4 x (5 b d-4 a e)}{b^5}-\frac{10 e^2 (b d-a e)^3}{b^6 (a+b x)}+\frac{10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}-\frac{5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{(b d-a e)^5}{3 b^6 (a+b x)^3}+\frac{e^5 x^2}{2 b^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(e^4*(5*b*d - 4*a*e)*x)/b^5 + (e^5*x^2)/(2*b^4) - (b*d - a*e)^5/(3*b^6*(a + b*x)^3) - (5*e*(b*d - a*e)^4)/(2*b
^6*(a + b*x)^2) - (10*e^2*(b*d - a*e)^3)/(b^6*(a + b*x)) + (10*e^3*(b*d - a*e)^2*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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0818163, size = 228, normalized size = 1.77 $\frac{-a^2 b^3 e^2 \left (-270 d^2 e x+20 d^3+90 d e^2 x^2+63 e^3 x^3\right )+a^3 b^2 e^3 \left (110 d^2-270 d e x-9 e^2 x^2\right )+a^4 b e^4 (81 e x-130 d)+47 a^5 e^5-5 a b^4 e \left (-36 d^2 e^2 x^2+12 d^3 e x+d^4-18 d e^3 x^3+3 e^4 x^4\right )+60 e^3 (a+b x)^3 (b d-a e)^2 \log (a+b x)+b^5 \left (-60 d^3 e^2 x^2-15 d^4 e x-2 d^5+30 d e^4 x^4+3 e^5 x^5\right )}{6 b^6 (a+b x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(47*a^5*e^5 + a^4*b*e^4*(-130*d + 81*e*x) + a^3*b^2*e^3*(110*d^2 - 270*d*e*x - 9*e^2*x^2) - a^2*b^3*e^2*(20*d^
3 - 270*d^2*e*x + 90*d*e^2*x^2 + 63*e^3*x^3) - 5*a*b^4*e*(d^4 + 12*d^3*e*x - 36*d^2*e^2*x^2 - 18*d*e^3*x^3 + 3
*e^4*x^4) + b^5*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) + 60*e^3*(b*d - a*e)^2*(a +
b*x)^3*Log[a + b*x])/(6*b^6*(a + b*x)^3)

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Maple [B]  time = 0.049, size = 361, normalized size = 2.8 \begin{align*}{\frac{{e}^{5}{x}^{2}}{2\,{b}^{4}}}-4\,{\frac{a{e}^{5}x}{{b}^{5}}}+5\,{\frac{{e}^{4}xd}{{b}^{4}}}-{\frac{5\,{a}^{4}{e}^{5}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{e}^{4}{a}^{3}d}{{b}^{5} \left ( bx+a \right ) ^{2}}}-15\,{\frac{{e}^{3}{d}^{2}{a}^{2}}{{b}^{4} \left ( bx+a \right ) ^{2}}}+10\,{\frac{a{e}^{2}{d}^{3}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{5\,e{d}^{4}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}{e}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-{\frac{5\,{a}^{4}d{e}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{10\,{a}^{3}{d}^{2}{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{10\,{a}^{2}{d}^{3}{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{5\,a{d}^{4}e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{5}}{3\,b \left ( bx+a \right ) ^{3}}}+10\,{\frac{{e}^{5}\ln \left ( bx+a \right ){a}^{2}}{{b}^{6}}}-20\,{\frac{{e}^{4}\ln \left ( bx+a \right ) ad}{{b}^{5}}}+10\,{\frac{{e}^{3}\ln \left ( bx+a \right ){d}^{2}}{{b}^{4}}}+10\,{\frac{{a}^{3}{e}^{5}}{{b}^{6} \left ( bx+a \right ) }}-30\,{\frac{{a}^{2}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+30\,{\frac{a{e}^{3}{d}^{2}}{{b}^{4} \left ( bx+a \right ) }}-10\,{\frac{{e}^{2}{d}^{3}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/2*e^5*x^2/b^4-4*e^5/b^5*a*x+5*e^4/b^4*x*d-5/2/b^6*e^5/(b*x+a)^2*a^4+10/b^5*e^4/(b*x+a)^2*a^3*d-15/b^4*e^3/(b
*x+a)^2*d^2*a^2+10/b^3*e^2/(b*x+a)^2*a*d^3-5/2/b^2*e/(b*x+a)^2*d^4+1/3/b^6/(b*x+a)^3*a^5*e^5-5/3/b^5/(b*x+a)^3
*a^4*d*e^4+10/3/b^4/(b*x+a)^3*a^3*d^2*e^3-10/3/b^3/(b*x+a)^3*a^2*d^3*e^2+5/3/b^2/(b*x+a)^3*a*d^4*e-1/3/b/(b*x+
a)^3*d^5+10/b^6*e^5*ln(b*x+a)*a^2-20/b^5*e^4*ln(b*x+a)*a*d+10/b^4*e^3*ln(b*x+a)*d^2+10/b^6*e^5/(b*x+a)*a^3-30/
b^5*e^4/(b*x+a)*a^2*d+30/b^4*e^3/(b*x+a)*a*d^2-10/b^3*e^2/(b*x+a)*d^3

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Maxima [B]  time = 1.14849, size = 379, normalized size = 2.94 \begin{align*} -\frac{2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \,{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \,{\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\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 e^{5} x^{2} + 2 \,{\left (5 \, b d e^{4} - 4 \, a e^{5}\right )} x}{2 \, b^{5}} + \frac{10 \,{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-1/6*(2*b^5*d^5 + 5*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 - 110*a^3*b^2*d^2*e^3 + 130*a^4*b*d*e^4 - 47*a^5*e^5 + 60
*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(b^5*d^4*e + 4*a*b^4*d^3*e^2 - 18*a^
2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 - 7*a^4*b*e^5)*x)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(b*e^
5*x^2 + 2*(5*b*d*e^4 - 4*a*e^5)*x)/b^5 + 10*(b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*log(b*x + a)/b^6

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Fricas [B]  time = 1.66791, size = 867, normalized size = 6.72 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} - 2 \, b^{5} d^{5} - 5 \, a b^{4} d^{4} e - 20 \, a^{2} b^{3} d^{3} e^{2} + 110 \, a^{3} b^{2} d^{2} e^{3} - 130 \, a^{4} b d e^{4} + 47 \, a^{5} e^{5} + 15 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 9 \,{\left (10 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \,{\left (20 \, b^{5} d^{3} e^{2} - 60 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 3 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (5 \, b^{5} d^{4} e + 20 \, a b^{4} d^{3} e^{2} - 90 \, a^{2} b^{3} d^{2} e^{3} + 90 \, a^{3} b^{2} d e^{4} - 27 \, a^{4} b e^{5}\right )} x + 60 \,{\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} +{\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \,{\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \,{\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\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{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/6*(3*b^5*e^5*x^5 - 2*b^5*d^5 - 5*a*b^4*d^4*e - 20*a^2*b^3*d^3*e^2 + 110*a^3*b^2*d^2*e^3 - 130*a^4*b*d*e^4 +
47*a^5*e^5 + 15*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 9*(10*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 - 3*(20*b^5*d^3*e^2 - 6
0*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 3*a^3*b^2*e^5)*x^2 - 3*(5*b^5*d^4*e + 20*a*b^4*d^3*e^2 - 90*a^2*b^3*d^2*e
^3 + 90*a^3*b^2*d*e^4 - 27*a^4*b*e^5)*x + 60*(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5 + (b^5*d^2*e^3 - 2*a*b
^4*d*e^4 + a^2*b^3*e^5)*x^3 + 3*(a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + 3*(a^2*b^3*d^2*e^3 - 2*a
^3*b^2*d*e^4 + a^4*b*e^5)*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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Sympy [B]  time = 3.74392, size = 282, normalized size = 2.19 \begin{align*} \frac{47 a^{5} e^{5} - 130 a^{4} b d e^{4} + 110 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 5 a b^{4} d^{4} e - 2 b^{5} d^{5} + x^{2} \left (60 a^{3} b^{2} e^{5} - 180 a^{2} b^{3} d e^{4} + 180 a b^{4} d^{2} e^{3} - 60 b^{5} d^{3} e^{2}\right ) + x \left (105 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} + 270 a^{2} b^{3} d^{2} e^{3} - 60 a b^{4} d^{3} e^{2} - 15 b^{5} d^{4} e\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac{e^{5} x^{2}}{2 b^{4}} - \frac{x \left (4 a e^{5} - 5 b d e^{4}\right )}{b^{5}} + \frac{10 e^{3} \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

(47*a**5*e**5 - 130*a**4*b*d*e**4 + 110*a**3*b**2*d**2*e**3 - 20*a**2*b**3*d**3*e**2 - 5*a*b**4*d**4*e - 2*b**
5*d**5 + x**2*(60*a**3*b**2*e**5 - 180*a**2*b**3*d*e**4 + 180*a*b**4*d**2*e**3 - 60*b**5*d**3*e**2) + x*(105*a
**4*b*e**5 - 300*a**3*b**2*d*e**4 + 270*a**2*b**3*d**2*e**3 - 60*a*b**4*d**3*e**2 - 15*b**5*d**4*e))/(6*a**3*b
**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + e**5*x**2/(2*b**4) - x*(4*a*e**5 - 5*b*d*e**4)/b**5 + 1
0*e**3*(a*e - b*d)**2*log(a + b*x)/b**6

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Giac [A]  time = 1.12406, size = 332, normalized size = 2.57 \begin{align*} \frac{10 \,{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{b^{4} x^{2} e^{5} + 10 \, b^{4} d x e^{4} - 8 \, a b^{3} x e^{5}}{2 \, b^{8}} - \frac{2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \,{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \,{\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

10*(b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*log(abs(b*x + a))/b^6 + 1/2*(b^4*x^2*e^5 + 10*b^4*d*x*e^4 - 8*a*b^3*x
*e^5)/b^8 - 1/6*(2*b^5*d^5 + 5*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 - 110*a^3*b^2*d^2*e^3 + 130*a^4*b*d*e^4 - 47*a
^5*e^5 + 60*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(b^5*d^4*e + 4*a*b^4*d^3*
e^2 - 18*a^2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 - 7*a^4*b*e^5)*x)/((b*x + a)^3*b^6)