3.1528 $$\int \frac{(d+e x)^5}{(a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=138 $-\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6}$

[Out]

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

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Rubi [A]  time = 0.116599, antiderivative size = 138, 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} $-\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.0886145, size = 171, normalized size = 1.24 $\frac{e^5 \log (a+b x)}{b^6}-\frac{(b d-a e) \left (a^2 b^2 e^2 \left (47 d^2+325 d e x+1100 e^2 x^2\right )+a^3 b e^3 (77 d+625 e x)+137 a^4 e^4+a b^3 e \left (175 d^2 e x+27 d^3+500 d e^2 x^2+900 e^3 x^3\right )+b^4 \left (200 d^2 e^2 x^2+75 d^3 e x+12 d^4+300 d e^3 x^3+300 e^4 x^4\right )\right )}{60 b^6 (a+b x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-((b*d - a*e)*(137*a^4*e^4 + a^3*b*e^3*(77*d + 625*e*x) + a^2*b^2*e^2*(47*d^2 + 325*d*e*x + 1100*e^2*x^2) + a*
b^3*e*(27*d^3 + 175*d^2*e*x + 500*d*e^2*x^2 + 900*e^3*x^3) + b^4*(12*d^4 + 75*d^3*e*x + 200*d^2*e^2*x^2 + 300*
d*e^3*x^3 + 300*e^4*x^4)))/(60*b^6*(a + b*x)^5) + (e^5*Log[a + b*x])/b^6

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Maple [B]  time = 0.052, size = 377, normalized size = 2.7 \begin{align*} -5\,{\frac{{a}^{2}{e}^{5}}{{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{d{e}^{4}a}{{b}^{5} \left ( bx+a \right ) ^{2}}}-5\,{\frac{{d}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}{e}^{5}}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{d{e}^{4}{a}^{4}}{{b}^{5} \left ( bx+a \right ) ^{5}}}+2\,{\frac{{d}^{2}{e}^{3}{a}^{3}}{{b}^{4} \left ( bx+a \right ) ^{5}}}-2\,{\frac{{d}^{3}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{{d}^{4}ea}{{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{5}}{5\,b \left ( bx+a \right ) ^{5}}}+{\frac{10\,{a}^{3}{e}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-10\,{\frac{{a}^{2}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+10\,{\frac{{d}^{2}{e}^{3}a}{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{10\,{d}^{3}{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{5}\ln \left ( bx+a \right ) }{{b}^{6}}}-{\frac{5\,{e}^{5}{a}^{4}}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+5\,{\frac{d{e}^{4}{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{15\,{d}^{2}{e}^{3}{a}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{4}}}+5\,{\frac{{d}^{3}{e}^{2}a}{{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{5\,{d}^{4}e}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}+5\,{\frac{{e}^{5}a}{{b}^{6} \left ( bx+a \right ) }}-5\,{\frac{d{e}^{4}}{{b}^{5} \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)^3,x)

[Out]

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

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Maxima [B]  time = 1.18595, size = 419, normalized size = 3.04 \begin{align*} -\frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac{e^{5} \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)^3,x, algorithm="maxima")

[Out]

-1/60*(12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5 +
300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 + 100*(2*b^5*d^3*e^2 +
3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 + 25*(3*b^5*d^4*e + 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e
^3 + 12*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x)/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^
7*x + a^5*b^6) + e^5*log(b*x + a)/b^6

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Fricas [B]  time = 2.09062, size = 775, normalized size = 5.62 \begin{align*} -\frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} 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)^3,x, algorithm="fricas")

[Out]

-1/60*(12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5 +
300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 + 100*(2*b^5*d^3*e^2 +
3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 + 25*(3*b^5*d^4*e + 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e
^3 + 12*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*
e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log(b*x + a))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 +
5*a^4*b^7*x + a^5*b^6)

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Sympy [B]  time = 14.3819, size = 326, normalized size = 2.36 \begin{align*} \frac{137 a^{5} e^{5} - 60 a^{4} b d e^{4} - 30 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 15 a b^{4} d^{4} e - 12 b^{5} d^{5} + x^{4} \left (300 a b^{4} e^{5} - 300 b^{5} d e^{4}\right ) + x^{3} \left (900 a^{2} b^{3} e^{5} - 600 a b^{4} d e^{4} - 300 b^{5} d^{2} e^{3}\right ) + x^{2} \left (1100 a^{3} b^{2} e^{5} - 600 a^{2} b^{3} d e^{4} - 300 a b^{4} d^{2} e^{3} - 200 b^{5} d^{3} e^{2}\right ) + x \left (625 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} - 150 a^{2} b^{3} d^{2} e^{3} - 100 a b^{4} d^{3} e^{2} - 75 b^{5} d^{4} e\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} + \frac{e^{5} \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)**3,x)

[Out]

(137*a**5*e**5 - 60*a**4*b*d*e**4 - 30*a**3*b**2*d**2*e**3 - 20*a**2*b**3*d**3*e**2 - 15*a*b**4*d**4*e - 12*b*
*5*d**5 + x**4*(300*a*b**4*e**5 - 300*b**5*d*e**4) + x**3*(900*a**2*b**3*e**5 - 600*a*b**4*d*e**4 - 300*b**5*d
**2*e**3) + x**2*(1100*a**3*b**2*e**5 - 600*a**2*b**3*d*e**4 - 300*a*b**4*d**2*e**3 - 200*b**5*d**3*e**2) + x*
(625*a**4*b*e**5 - 300*a**3*b**2*d*e**4 - 150*a**2*b**3*d**2*e**3 - 100*a*b**4*d**3*e**2 - 75*b**5*d**4*e))/(6
0*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) +
e**5*log(a + b*x)/b**6

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Giac [A]  time = 1.1774, size = 335, normalized size = 2.43 \begin{align*} \frac{e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{300 \,{\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} x^{4} + 300 \,{\left (b^{4} d^{2} e^{3} + 2 \, a b^{3} d e^{4} - 3 \, a^{2} b^{2} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} + 6 \, a^{2} b^{2} d e^{4} - 11 \, a^{3} b e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{4} d^{4} e + 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} + 12 \, a^{3} b d e^{4} - 25 \, a^{4} e^{5}\right )} x + \frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5}}{b}}{60 \,{\left (b x + a\right )}^{5} b^{5}} \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)^3,x, algorithm="giac")

[Out]

e^5*log(abs(b*x + a))/b^6 - 1/60*(300*(b^4*d*e^4 - a*b^3*e^5)*x^4 + 300*(b^4*d^2*e^3 + 2*a*b^3*d*e^4 - 3*a^2*b
^2*e^5)*x^3 + 100*(2*b^4*d^3*e^2 + 3*a*b^3*d^2*e^3 + 6*a^2*b^2*d*e^4 - 11*a^3*b*e^5)*x^2 + 25*(3*b^4*d^4*e + 4
*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 + 12*a^3*b*d*e^4 - 25*a^4*e^5)*x + (12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^
3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5)/b)/((b*x + a)^5*b^5)