3.1902 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac{b^3 x^2}{2 e^2} \]

[Out]

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

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Rubi [A]  time = 0.0617914, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ -\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac{b^3 x^2}{2 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0390258, size = 114, normalized size = 1.52 \[ \frac{3 a^2 b d e^2-a^3 e^3-3 a b^2 d^2 e+b^3 d^3}{e^4 (d+e x)}+\frac{3 \left (a^2 b e^2-2 a b^2 d e+b^3 d^2\right ) \log (d+e x)}{e^4}-\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{b^3 x^2}{2 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.008, size = 149, normalized size = 2. \begin{align*}{\frac{{b}^{3}{x}^{2}}{2\,{e}^{2}}}+3\,{\frac{{b}^{2}ax}{{e}^{2}}}-2\,{\frac{{b}^{3}dx}{{e}^{3}}}+3\,{\frac{b\ln \left ( ex+d \right ){a}^{2}}{{e}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( ex+d \right ) ad}{{e}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{4}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }}+3\,{\frac{{a}^{2}db}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}{d}^{2}a}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959432, size = 158, normalized size = 2.11 \begin{align*} \frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{e^{5} x + d e^{4}} + \frac{b^{3} e x^{2} - 2 \,{\left (2 \, b^{3} d - 3 \, a b^{2} e\right )} x}{2 \, e^{3}} + \frac{3 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.52022, size = 354, normalized size = 4.72 \begin{align*} \frac{b^{3} e^{3} x^{3} + 2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} - 3 \,{\left (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \,{\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.712109, size = 100, normalized size = 1.33 \begin{align*} \frac{b^{3} x^{2}}{2 e^{2}} + \frac{3 b \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (3 a b^{2} e - 2 b^{3} d\right )}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17438, size = 221, normalized size = 2.95 \begin{align*} \frac{1}{2} \,{\left (b^{3} - \frac{6 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} e^{\left (-4\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{b^{3} d^{3} e^{2}}{x e + d} - \frac{3 \, a b^{2} d^{2} e^{3}}{x e + d} + \frac{3 \, a^{2} b d e^{4}}{x e + d} - \frac{a^{3} e^{5}}{x e + d}\right )} e^{\left (-6\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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