### 3.1506 $$\int \frac{(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx$$

Optimal. Leaf size=104 $\frac{2 e^3 (a+b x)^2 (b d-a e)}{b^5}+\frac{6 e^2 x (b d-a e)^2}{b^4}-\frac{(b d-a e)^4}{b^5 (a+b x)}+\frac{4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac{e^4 (a+b x)^3}{3 b^5}$

[Out]

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

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Rubi [A]  time = 0.102178, antiderivative size = 104, 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} $\frac{2 e^3 (a+b x)^2 (b d-a e)}{b^5}+\frac{6 e^2 x (b d-a e)^2}{b^4}-\frac{(b d-a e)^4}{b^5 (a+b x)}+\frac{4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac{e^4 (a+b x)^3}{3 b^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0637961, size = 166, normalized size = 1.6 $\frac{6 a^2 b^2 e^2 \left (-3 d^2-4 d e x+e^2 x^2\right )+3 a^3 b e^3 (4 d+3 e x)-3 a^4 e^4-2 a b^3 e \left (-9 d^2 e x-6 d^3+9 d e^2 x^2+e^3 x^3\right )-12 e (a+b x) (a e-b d)^3 \log (a+b x)+b^4 \left (18 d^2 e^2 x^2-3 d^4+6 d e^3 x^3+e^4 x^4\right )}{3 b^5 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.048, size = 230, normalized size = 2.2 \begin{align*}{\frac{{e}^{4}{x}^{3}}{3\,{b}^{2}}}-{\frac{{e}^{4}{x}^{2}a}{{b}^{3}}}+2\,{\frac{{e}^{3}{x}^{2}d}{{b}^{2}}}+3\,{\frac{{a}^{2}{e}^{4}x}{{b}^{4}}}-8\,{\frac{ad{e}^{3}x}{{b}^{3}}}+6\,{\frac{{d}^{2}{e}^{2}x}{{b}^{2}}}-4\,{\frac{{e}^{4}\ln \left ( bx+a \right ){a}^{3}}{{b}^{5}}}+12\,{\frac{{e}^{3}\ln \left ( bx+a \right ){a}^{2}d}{{b}^{4}}}-12\,{\frac{{e}^{2}\ln \left ( bx+a \right ) a{d}^{2}}{{b}^{3}}}+4\,{\frac{e\ln \left ( bx+a \right ){d}^{3}}{{b}^{2}}}-{\frac{{a}^{4}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+4\,{\frac{{a}^{3}d{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-6\,{\frac{{d}^{2}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+4\,{\frac{a{d}^{3}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{4}}{b \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),x)

[Out]

1/3*e^4/b^2*x^3-e^4/b^3*x^2*a+2*e^3/b^2*x^2*d+3*e^4/b^4*a^2*x-8*e^3/b^3*a*d*x+6*e^2/b^2*d^2*x-4/b^5*e^4*ln(b*x
+a)*a^3+12/b^4*e^3*ln(b*x+a)*a^2*d-12/b^3*e^2*ln(b*x+a)*a*d^2+4/b^2*e*ln(b*x+a)*d^3-1/b^5/(b*x+a)*a^4*e^4+4/b^
4/(b*x+a)*a^3*d*e^3-6/b^3/(b*x+a)*d^2*e^2*a^2+4/b^2/(b*x+a)*a*d^3*e-1/b/(b*x+a)*d^4

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Maxima [A]  time = 1.1323, size = 248, normalized size = 2.38 \begin{align*} -\frac{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}}{b^{6} x + a b^{5}} + \frac{b^{2} e^{4} x^{3} + 3 \,{\left (2 \, b^{2} d e^{3} - a b e^{4}\right )} x^{2} + 3 \,{\left (6 \, b^{2} d^{2} e^{2} - 8 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} x}{3 \, b^{4}} + \frac{4 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} 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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.75765, size = 540, normalized size = 5.19 \begin{align*} \frac{b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 2 \,{\left (3 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (3 \, b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (6 \, a b^{3} d^{2} e^{2} - 8 \, a^{2} b^{2} d e^{3} + 3 \, a^{3} b e^{4}\right )} x + 12 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.978266, size = 151, normalized size = 1.45 \begin{align*} - \frac{a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}}{a b^{5} + b^{6} x} + \frac{e^{4} x^{3}}{3 b^{2}} - \frac{x^{2} \left (a e^{4} - 2 b d e^{3}\right )}{b^{3}} + \frac{x \left (3 a^{2} e^{4} - 8 a b d e^{3} + 6 b^{2} d^{2} e^{2}\right )}{b^{4}} - \frac{4 e \left (a e - b d\right )^{3} \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),x)

[Out]

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

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Giac [A]  time = 1.19414, size = 240, normalized size = 2.31 \begin{align*} \frac{4 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{b^{4} x^{3} e^{4} + 6 \, b^{4} d x^{2} e^{3} + 18 \, b^{4} d^{2} x e^{2} - 3 \, a b^{3} x^{2} e^{4} - 24 \, a b^{3} d x e^{3} + 9 \, a^{2} b^{2} x e^{4}}{3 \, b^{6}} - \frac{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}}{{\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),x, algorithm="giac")

[Out]

4*(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4)*log(abs(b*x + a))/b^5 + 1/3*(b^4*x^3*e^4 + 6*b^4*d*x
^2*e^3 + 18*b^4*d^2*x*e^2 - 3*a*b^3*x^2*e^4 - 24*a*b^3*d*x*e^3 + 9*a^2*b^2*x*e^4)/b^6 - (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)/((b*x + a)*b^5)