3.1903 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=78 \[ -\frac{3 b^2 (b d-a e) \log (d+e x)}{e^4}-\frac{3 b (b d-a e)^2}{e^4 (d+e x)}+\frac{(b d-a e)^3}{2 e^4 (d+e x)^2}+\frac{b^3 x}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.0546046, antiderivative size = 78, 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} \[ -\frac{3 b^2 (b d-a e) \log (d+e x)}{e^4}-\frac{3 b (b d-a e)^2}{e^4 (d+e x)}+\frac{(b d-a e)^3}{2 e^4 (d+e x)^2}+\frac{b^3 x}{e^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0438245, size = 114, normalized size = 1.46 \[ \frac{-3 a^2 b e^2 (d+2 e x)-a^3 e^3+3 a b^2 d e (3 d+4 e x)-6 b^2 (d+e x)^2 (b d-a e) \log (d+e x)+b^3 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^3,x]

[Out]

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

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Maple [B]  time = 0.009, size = 160, normalized size = 2.1 \begin{align*}{\frac{{b}^{3}x}{{e}^{3}}}-{\frac{{a}^{3}}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{3\,{a}^{2}db}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{2}{d}^{2}a}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{3}{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}\ln \left ( ex+d \right ) a}{{e}^{3}}}-3\,{\frac{{b}^{3}\ln \left ( ex+d \right ) d}{{e}^{4}}}-3\,{\frac{{a}^{2}b}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{{b}^{2}da}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{{b}^{3}{d}^{2}}{{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)^3,x)

[Out]

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

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Maxima [A]  time = 0.977461, size = 169, normalized size = 2.17 \begin{align*} \frac{b^{3} x}{e^{3}} - \frac{5 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac{3 \,{\left (b^{3} d - a b^{2} e\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.4407, size = 375, normalized size = 4.81 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d e^{2} x^{2} - 5 \, b^{3} d^{3} + 9 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 2 \,{\left (2 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} d^{3} - a b^{2} d^{2} e +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (b^{3} d^{2} e - a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.04912, size = 128, normalized size = 1.64 \begin{align*} \frac{b^{3} x}{e^{3}} + \frac{3 b^{2} \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{3} + 3 a^{2} b d e^{2} - 9 a b^{2} d^{2} e + 5 b^{3} d^{3} + x \left (6 a^{2} b e^{3} - 12 a b^{2} d e^{2} + 6 b^{3} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \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)**3,x)

[Out]

b**3*x/e**3 + 3*b**2*(a*e - b*d)*log(d + e*x)/e**4 - (a**3*e**3 + 3*a**2*b*d*e**2 - 9*a*b**2*d**2*e + 5*b**3*d
**3 + x*(6*a**2*b*e**3 - 12*a*b**2*d*e**2 + 6*b**3*d**2*e))/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)

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Giac [A]  time = 1.14758, size = 149, normalized size = 1.91 \begin{align*} b^{3} x e^{\left (-3\right )} - 3 \,{\left (b^{3} d - a b^{2} e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

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