### 3.1517 $$\int \frac{(d+e x)^4}{(a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=103 $-\frac{6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac{4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac{2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac{(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac{e^4 x}{b^4}$

[Out]

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

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Rubi [A]  time = 0.0884639, antiderivative size = 103, 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} $-\frac{6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac{4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac{2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac{(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac{e^4 x}{b^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0606032, size = 166, normalized size = 1.61 $\frac{-3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a^3 b e^3 (22 d-27 e x)-13 a^4 e^4+a b^3 e \left (-18 d^2 e x-2 d^3+36 d e^2 x^2+9 e^3 x^3\right )-12 e^3 (a+b x)^3 (a e-b d) \log (a+b x)+b^4 \left (-\left (18 d^2 e^2 x^2+6 d^3 e x+d^4-3 e^4 x^4\right )\right )}{3 b^5 (a+b x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(-13*a^4*e^4 + a^3*b*e^3*(22*d - 27*e*x) - 3*a^2*b^2*e^2*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*b^3*e*(-2*d^3 - 18
*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - b^4*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4*x^4) - 12*e^3*(-(b*d) + a
*e)*(a + b*x)^3*Log[a + b*x])/(3*b^5*(a + b*x)^3)

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Maple [B]  time = 0.049, size = 255, normalized size = 2.5 \begin{align*}{\frac{{e}^{4}x}{{b}^{4}}}+2\,{\frac{{a}^{3}{e}^{4}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{a}^{2}{e}^{3}d}{{b}^{4} \left ( bx+a \right ) ^{2}}}+6\,{\frac{a{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-2\,{\frac{e{d}^{3}}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{4}{e}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{4\,{a}^{3}d{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-2\,{\frac{{d}^{2}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{4\,a{d}^{3}e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{4}}{3\,b \left ( bx+a \right ) ^{3}}}-4\,{\frac{{e}^{4}\ln \left ( bx+a \right ) a}{{b}^{5}}}+4\,{\frac{{e}^{3}\ln \left ( bx+a \right ) d}{{b}^{4}}}-6\,{\frac{{a}^{2}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+12\,{\frac{ad{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-6\,{\frac{{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

e^4*x/b^4+2/b^5*e^4/(b*x+a)^2*a^3-6/b^4*e^3/(b*x+a)^2*a^2*d+6/b^3*e^2/(b*x+a)^2*a*d^2-2/b^2*e/(b*x+a)^2*d^3-1/
3/b^5/(b*x+a)^3*a^4*e^4+4/3/b^4/(b*x+a)^3*a^3*d*e^3-2/b^3/(b*x+a)^3*d^2*e^2*a^2+4/3/b^2/(b*x+a)^3*a*d^3*e-1/3/
b/(b*x+a)^3*d^4-4/b^5*e^4*ln(b*x+a)*a+4/b^4*e^3*ln(b*x+a)*d-6/b^5*e^4/(b*x+a)*a^2+12/b^4*e^3/(b*x+a)*a*d-6/b^3
*e^2/(b*x+a)*d^2

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Maxima [A]  time = 1.12441, size = 271, normalized size = 2.63 \begin{align*} \frac{e^{4} x}{b^{4}} - \frac{b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac{4 \,{\left (b d e^{3} - a e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

e^4*x/b^4 - 1/3*(b^4*d^4 + 2*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 22*a^3*b*d*e^3 + 13*a^4*e^4 + 18*(b^4*d^2*e^2 -
2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x)/(b^8*x^
3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5) + 4*(b*d*e^3 - a*e^4)*log(b*x + a)/b^5

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Fricas [B]  time = 1.69989, size = 581, normalized size = 5.64 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} + 9 \, a b^{3} e^{4} x^{3} - b^{4} d^{4} - 2 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} + 22 \, a^{3} b d e^{3} - 13 \, a^{4} e^{4} - 9 \,{\left (2 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} - 3 \,{\left (2 \, b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 18 \, a^{2} b^{2} d e^{3} + 9 \, a^{3} b e^{4}\right )} x + 12 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/3*(3*b^4*e^4*x^4 + 9*a*b^3*e^4*x^3 - b^4*d^4 - 2*a*b^3*d^3*e - 6*a^2*b^2*d^2*e^2 + 22*a^3*b*d*e^3 - 13*a^4*e
^4 - 9*(2*b^4*d^2*e^2 - 4*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 - 3*(2*b^4*d^3*e + 6*a*b^3*d^2*e^2 - 18*a^2*b^2*d*e^3
+ 9*a^3*b*e^4)*x + 12*(a^3*b*d*e^3 - a^4*e^4 + (b^4*d*e^3 - a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 - a^2*b^2*e^4)*x^
2 + 3*(a^2*b^2*d*e^3 - a^3*b*e^4)*x)*log(b*x + a))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)

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Sympy [B]  time = 2.46601, size = 209, normalized size = 2.03 \begin{align*} - \frac{13 a^{4} e^{4} - 22 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 2 a b^{3} d^{3} e + b^{4} d^{4} + x^{2} \left (18 a^{2} b^{2} e^{4} - 36 a b^{3} d e^{3} + 18 b^{4} d^{2} e^{2}\right ) + x \left (30 a^{3} b e^{4} - 54 a^{2} b^{2} d e^{3} + 18 a b^{3} d^{2} e^{2} + 6 b^{4} d^{3} e\right )}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac{e^{4} x}{b^{4}} - \frac{4 e^{3} \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-(13*a**4*e**4 - 22*a**3*b*d*e**3 + 6*a**2*b**2*d**2*e**2 + 2*a*b**3*d**3*e + b**4*d**4 + x**2*(18*a**2*b**2*e
**4 - 36*a*b**3*d*e**3 + 18*b**4*d**2*e**2) + x*(30*a**3*b*e**4 - 54*a**2*b**2*d*e**3 + 18*a*b**3*d**2*e**2 +
6*b**4*d**3*e))/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + e**4*x/b**4 - 4*e**3*(a*e - b*d)
*log(a + b*x)/b**5

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Giac [A]  time = 1.14, size = 225, normalized size = 2.18 \begin{align*} \frac{x e^{4}}{b^{4}} + \frac{4 \,{\left (b d e^{3} - a e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \,{\left (b x + a\right )}^{3} b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

x*e^4/b^4 + 4*(b*d*e^3 - a*e^4)*log(abs(b*x + a))/b^5 - 1/3*(b^4*d^4 + 2*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 22*
a^3*b*d*e^3 + 13*a^4*e^4 + 18*(b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 3*a*b^3*d^2*e^2
- 9*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x)/((b*x + a)^3*b^5)