### 3.1470 $$\int \frac{(a^2+2 a b x+b^2 x^2)^2}{d+e x} \, dx$$

Optimal. Leaf size=98 $-\frac{b x (b d-a e)^3}{e^4}+\frac{(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac{(a+b x)^3 (b d-a e)}{3 e^2}+\frac{(b d-a e)^4 \log (d+e x)}{e^5}+\frac{(a+b x)^4}{4 e}$

[Out]

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

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Rubi [A]  time = 0.0423296, antiderivative size = 98, 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} $-\frac{b x (b d-a e)^3}{e^4}+\frac{(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac{(a+b x)^3 (b d-a e)}{3 e^2}+\frac{(b d-a e)^4 \log (d+e x)}{e^5}+\frac{(a+b x)^4}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0452689, size = 115, normalized size = 1.17 $\frac{b e x \left (36 a^2 b e^2 (e x-2 d)+48 a^3 e^3+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)}{12 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*e*x*(48*a^3*e^3 + 36*a^2*b*e^2*(-2*d + e*x) + 8*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^3*(-12*d^3 + 6*d^
2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 12*(b*d - a*e)^4*Log[d + e*x])/(12*e^5)

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Maple [B]  time = 0.042, size = 209, normalized size = 2.1 \begin{align*}{\frac{{b}^{4}{x}^{4}}{4\,e}}+{\frac{4\,a{b}^{3}{x}^{3}}{3\,e}}-{\frac{{b}^{4}{x}^{3}d}{3\,{e}^{2}}}+3\,{\frac{{b}^{2}{x}^{2}{a}^{2}}{e}}-2\,{\frac{{b}^{3}{x}^{2}ad}{{e}^{2}}}+{\frac{{b}^{4}{x}^{2}{d}^{2}}{2\,{e}^{3}}}+4\,{\frac{x{a}^{3}b}{e}}-6\,{\frac{{b}^{2}{a}^{2}dx}{{e}^{2}}}+4\,{\frac{{d}^{2}a{b}^{3}x}{{e}^{3}}}-{\frac{{b}^{4}{d}^{3}x}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){a}^{4}}{e}}-4\,{\frac{\ln \left ( ex+d \right ){a}^{3}bd}{{e}^{2}}}+6\,{\frac{\ln \left ( ex+d \right ){d}^{2}{b}^{2}{a}^{2}}{{e}^{3}}}-4\,{\frac{\ln \left ( ex+d \right ){d}^{3}a{b}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){b}^{4}{d}^{4}}{{e}^{5}}} \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),x)

[Out]

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

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Maxima [A]  time = 1.15193, size = 239, normalized size = 2.44 \begin{align*} \frac{3 \, b^{4} e^{3} x^{4} - 4 \,{\left (b^{4} d e^{2} - 4 \, a b^{3} e^{3}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e - 4 \, a b^{3} d e^{2} + 6 \, a^{2} b^{2} e^{3}\right )} x^{2} - 12 \,{\left (b^{4} d^{3} - 4 \, a b^{3} d^{2} e + 6 \, a^{2} b^{2} d e^{2} - 4 \, a^{3} b e^{3}\right )} x}{12 \, e^{4}} + \frac{{\left (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}\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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.78922, size = 369, normalized size = 3.77 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} - 4 \,{\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \,{\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (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}\right )} \log \left (e x + d\right )}{12 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.706945, size = 134, normalized size = 1.37 \begin{align*} \frac{b^{4} x^{4}}{4 e} + \frac{x^{3} \left (4 a b^{3} e - b^{4} d\right )}{3 e^{2}} + \frac{x^{2} \left (6 a^{2} b^{2} e^{2} - 4 a b^{3} d e + b^{4} d^{2}\right )}{2 e^{3}} + \frac{x \left (4 a^{3} b e^{3} - 6 a^{2} b^{2} d e^{2} + 4 a b^{3} d^{2} e - b^{4} d^{3}\right )}{e^{4}} + \frac{\left (a e - b d\right )^{4} \log{\left (d + e x \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),x)

[Out]

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

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Giac [A]  time = 1.15053, size = 238, normalized size = 2.43 \begin{align*}{\left (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}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{4} x^{4} e^{3} - 4 \, b^{4} d x^{3} e^{2} + 6 \, b^{4} d^{2} x^{2} e - 12 \, b^{4} d^{3} x + 16 \, a b^{3} x^{3} e^{3} - 24 \, a b^{3} d x^{2} e^{2} + 48 \, a b^{3} d^{2} x e + 36 \, a^{2} b^{2} x^{2} e^{3} - 72 \, a^{2} b^{2} d x e^{2} + 48 \, a^{3} b x e^{3}\right )} e^{\left (-4\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),x, algorithm="giac")

[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)*e^(-5)*log(abs(x*e + d)) + 1/12*(3*b^4
*x^4*e^3 - 4*b^4*d*x^3*e^2 + 6*b^4*d^2*x^2*e - 12*b^4*d^3*x + 16*a*b^3*x^3*e^3 - 24*a*b^3*d*x^2*e^2 + 48*a*b^3
*d^2*x*e + 36*a^2*b^2*x^2*e^3 - 72*a^2*b^2*d*x*e^2 + 48*a^3*b*x*e^3)*e^(-4)