3.1473 $$\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^4} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0798661, 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} $-\frac{6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac{4 b^3 (b d-a e) \log (d+e x)}{e^5}+\frac{2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac{(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac{b^4 x}{e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0746552, size = 163, normalized size = 1.58 $-\frac{6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+2 a^3 b e^3 (d+3 e x)+a^4 e^4-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+12 b^3 (d+e x)^3 (b d-a e) \log (d+e x)+b^4 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )}{3 e^5 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^4*e^4 + 2*a^3*b*e^3*(d + 3*e*x) + 6*a^2*b^2*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2) - 2*a*b^3*d*e*(11*d^2 + 27*d*e
*x + 18*e^2*x^2) + b^4*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4) + 12*b^3*(b*d - a*e)*(d
+ e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)

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Maple [B]  time = 0.05, size = 255, normalized size = 2.5 \begin{align*}{\frac{{b}^{4}x}{{e}^{4}}}-{\frac{{a}^{4}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{4\,d{a}^{3}b}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-2\,{\frac{{b}^{2}{d}^{2}{a}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{d}^{3}a{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{4}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-2\,{\frac{{a}^{3}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{2}{a}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-6\,{\frac{{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+4\,{\frac{{b}^{3}\ln \left ( ex+d \right ) a}{{e}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( ex+d \right ) d}{{e}^{5}}}-6\,{\frac{{b}^{2}{a}^{2}}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{ad{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{b}^{4}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04734, size = 271, normalized size = 2.63 \begin{align*} \frac{b^{4} x}{e^{4}} - \frac{13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + 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 (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac{4 \,{\left (b^{4} d - a b^{3} e\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.03765, size = 581, normalized size = 5.64 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} + 9 \, b^{4} d e^{3} x^{3} - 13 \, b^{4} d^{4} + 22 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} - a^{4} e^{4} - 9 \,{\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 2 \, a^{2} b^{2} e^{4}\right )} x^{2} - 3 \,{\left (9 \, b^{4} d^{3} e - 18 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 2 \, a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} d^{4} - a b^{3} d^{3} e +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} - a b^{3} d e^{3}\right )} x^{2} + 3 \,{\left (b^{4} d^{3} e - a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.67267, size = 209, normalized size = 2.03 \begin{align*} \frac{b^{4} x}{e^{4}} + \frac{4 b^{3} \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{4} e^{4} + 2 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 22 a b^{3} d^{3} e + 13 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 (6 a^{3} b e^{4} + 18 a^{2} b^{2} d e^{3} - 54 a b^{3} d^{2} e^{2} + 30 b^{4} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17378, size = 230, normalized size = 2.23 \begin{align*} b^{4} x e^{\left (-4\right )} - 4 \,{\left (b^{4} d - a b^{3} e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + 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 (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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