3.1474 \(\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 b^3 (b d-a e)}{e^5 (d+e x)}-\frac{3 b^2 (b d-a e)^2}{e^5 (d+e x)^2}+\frac{4 b (b d-a e)^3}{3 e^5 (d+e x)^3}-\frac{(b d-a e)^4}{4 e^5 (d+e x)^4}+\frac{b^4 \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.0765808, antiderivative size = 111, 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} \[ \frac{4 b^3 (b d-a e)}{e^5 (d+e x)}-\frac{3 b^2 (b d-a e)^2}{e^5 (d+e x)^2}+\frac{4 b (b d-a e)^3}{3 e^5 (d+e x)^3}-\frac{(b d-a e)^4}{4 e^5 (d+e x)^4}+\frac{b^4 \log (d+e x)}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.0649222, size = 119, normalized size = 1.07 \[ \frac{\frac{(b d-a e) \left (a^2 b e^2 (7 d+16 e x)+3 a^3 e^3+a b^2 e \left (13 d^2+40 d e x+36 e^2 x^2\right )+b^3 \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )}{(d+e x)^4}+12 b^4 \log (d+e x)}{12 e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^5,x]

[Out]

(((b*d - a*e)*(3*a^3*e^3 + a^2*b*e^2*(7*d + 16*e*x) + a*b^2*e*(13*d^2 + 40*d*e*x + 36*e^2*x^2) + b^3*(25*d^3 +
88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)))/(d + e*x)^4 + 12*b^4*Log[d + e*x])/(12*e^5)

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Maple [B]  time = 0.047, size = 260, normalized size = 2.3 \begin{align*} -{\frac{4\,{a}^{3}b}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{b}^{2}{a}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{b}^{4}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{4}}{4\,e \left ( ex+d \right ) ^{4}}}+{\frac{{a}^{3}bd}{{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{b}^{2}{a}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{a{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{4}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-3\,{\frac{{b}^{2}{a}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{ad{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{b}^{4}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{4}\ln \left ( ex+d \right ) }{{e}^{5}}}-4\,{\frac{a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{b}^{4}d}{{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)^5,x)

[Out]

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

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Maxima [B]  time = 1.22636, size = 297, normalized size = 2.68 \begin{align*} \frac{25 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} - 2 \, a^{3} b e^{4}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{b^{4} \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)^5,x, algorithm="maxima")

[Out]

1/12*(25*b^4*d^4 - 12*a*b^3*d^3*e - 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 - 3*a^4*e^4 + 48*(b^4*d*e^3 - a*b^3*e^4)
*x^3 + 36*(3*b^4*d^2*e^2 - 2*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 8*(11*b^4*d^3*e - 6*a*b^3*d^2*e^2 - 3*a^2*b^2*d*
e^3 - 2*a^3*b*e^4)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5) + b^4*log(e*x + d)/e^5

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Fricas [B]  time = 1.99869, size = 544, normalized size = 4.9 \begin{align*} \frac{25 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} - 2 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, b^{4} d e^{3} x^{3} + 6 \, b^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} d^{3} e x + b^{4} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/12*(25*b^4*d^4 - 12*a*b^3*d^3*e - 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 - 3*a^4*e^4 + 48*(b^4*d*e^3 - a*b^3*e^4)
*x^3 + 36*(3*b^4*d^2*e^2 - 2*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 8*(11*b^4*d^3*e - 6*a*b^3*d^2*e^2 - 3*a^2*b^2*d*
e^3 - 2*a^3*b*e^4)*x + 12*(b^4*e^4*x^4 + 4*b^4*d*e^3*x^3 + 6*b^4*d^2*e^2*x^2 + 4*b^4*d^3*e*x + b^4*d^4)*log(e*
x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

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Sympy [B]  time = 3.70738, size = 230, normalized size = 2.07 \begin{align*} \frac{b^{4} \log{\left (d + e x \right )}}{e^{5}} - \frac{3 a^{4} e^{4} + 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 12 a b^{3} d^{3} e - 25 b^{4} d^{4} + x^{3} \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \left (36 a^{2} b^{2} e^{4} + 72 a b^{3} d e^{3} - 108 b^{4} d^{2} e^{2}\right ) + x \left (16 a^{3} b e^{4} + 24 a^{2} b^{2} d e^{3} + 48 a b^{3} d^{2} e^{2} - 88 b^{4} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \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)**5,x)

[Out]

b**4*log(d + e*x)/e**5 - (3*a**4*e**4 + 4*a**3*b*d*e**3 + 6*a**2*b**2*d**2*e**2 + 12*a*b**3*d**3*e - 25*b**4*d
**4 + x**3*(48*a*b**3*e**4 - 48*b**4*d*e**3) + x**2*(36*a**2*b**2*e**4 + 72*a*b**3*d*e**3 - 108*b**4*d**2*e**2
) + x*(16*a**3*b*e**4 + 24*a**2*b**2*d*e**3 + 48*a*b**3*d**2*e**2 - 88*b**4*d**3*e))/(12*d**4*e**5 + 48*d**3*e
**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)

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Giac [B]  time = 1.17618, size = 377, normalized size = 3.4 \begin{align*} -b^{4} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, b^{4} d e^{15}}{x e + d} - \frac{36 \, b^{4} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, b^{4} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{4} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{48 \, a b^{3} e^{16}}{x e + d} + \frac{72 \, a b^{3} d e^{16}}{{\left (x e + d\right )}^{2}} - \frac{48 \, a b^{3} d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac{12 \, a b^{3} d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac{36 \, a^{2} b^{2} e^{17}}{{\left (x e + d\right )}^{2}} + \frac{48 \, a^{2} b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{18 \, a^{2} b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{16 \, a^{3} b e^{18}}{{\left (x e + d\right )}^{3}} + \frac{12 \, a^{3} b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{4} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\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)^5,x, algorithm="giac")

[Out]

-b^4*e^(-5)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*b^4*d*e^15/(x*e + d) - 36*b^4*d^2*e^15/(x*e + d)^2
+ 16*b^4*d^3*e^15/(x*e + d)^3 - 3*b^4*d^4*e^15/(x*e + d)^4 - 48*a*b^3*e^16/(x*e + d) + 72*a*b^3*d*e^16/(x*e +
d)^2 - 48*a*b^3*d^2*e^16/(x*e + d)^3 + 12*a*b^3*d^3*e^16/(x*e + d)^4 - 36*a^2*b^2*e^17/(x*e + d)^2 + 48*a^2*b
^2*d*e^17/(x*e + d)^3 - 18*a^2*b^2*d^2*e^17/(x*e + d)^4 - 16*a^3*b*e^18/(x*e + d)^3 + 12*a^3*b*d*e^18/(x*e + d
)^4 - 3*a^4*e^19/(x*e + d)^4)*e^(-20)