3.1904 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=86 \[ \frac{3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac{b^3 \log (d+e x)}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.0572354, antiderivative size = 86, 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} \[ \frac{3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac{b^3 \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0436969, size = 79, normalized size = 0.92 \[ \frac{\frac{(b d-a e) \left (2 a^2 e^2+a b e (5 d+9 e x)+b^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 b^3 \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.006, size = 166, normalized size = 1.9 \begin{align*} -{\frac{3\,{a}^{2}b}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}da}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{3}{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( ex+d \right ) }{{e}^{4}}}-3\,{\frac{{b}^{2}a}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{{b}^{3}d}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{a}^{3}}{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 ) ^{3}}}+{\frac{{b}^{3}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \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)^4,x)

[Out]

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

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Maxima [A]  time = 0.987621, size = 193, normalized size = 2.24 \begin{align*} \frac{11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{b^{3} \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)^4,x, algorithm="maxima")

[Out]

1/6*(11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^3*e^3 + 18*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 9*(3*b^3*d^2*e
- 2*a*b^2*d*e^2 - a^2*b*e^3)*x)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + b^3*log(e*x + d)/e^4

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Fricas [B]  time = 1.45371, size = 359, normalized size = 4.17 \begin{align*} \frac{11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/6*(11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^3*e^3 + 18*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 9*(3*b^3*d^2*e
- 2*a*b^2*d*e^2 - a^2*b*e^3)*x + 6*(b^3*e^3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(e*x + d))/(e^
7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)

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Sympy [A]  time = 1.33141, size = 148, normalized size = 1.72 \begin{align*} \frac{b^{3} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a^{3} e^{3} + 3 a^{2} b d e^{2} + 6 a b^{2} d^{2} e - 11 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (9 a^{2} b e^{3} + 18 a b^{2} d e^{2} - 27 b^{3} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{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)**4,x)

[Out]

b**3*log(d + e*x)/e**4 - (2*a**3*e**3 + 3*a**2*b*d*e**2 + 6*a*b**2*d**2*e - 11*b**3*d**3 + x**2*(18*a*b**2*e**
3 - 18*b**3*d*e**2) + x*(9*a**2*b*e**3 + 18*a*b**2*d*e**2 - 27*b**3*d**2*e))/(6*d**3*e**4 + 18*d**2*e**5*x + 1
8*d*e**6*x**2 + 6*e**7*x**3)

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Giac [A]  time = 1.18953, size = 158, normalized size = 1.84 \begin{align*} b^{3} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (18 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} - 2 \, a b^{2} d e - a^{2} b e^{2}\right )} x +{\left (11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{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)^4,x, algorithm="giac")

[Out]

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