3.1915 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=130 \[ -\frac{b^4 x (4 b d-5 a e)}{e^5}+\frac{10 b^2 (b d-a e)^3}{e^6 (d+e x)}+\frac{10 b^3 (b d-a e)^2 \log (d+e x)}{e^6}-\frac{5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^5}{3 e^6 (d+e x)^3}+\frac{b^5 x^2}{2 e^4} \]

[Out]

-((b^4*(4*b*d - 5*a*e)*x)/e^5) + (b^5*x^2)/(2*e^4) + (b*d - a*e)^5/(3*e^6*(d + e*x)^3) - (5*b*(b*d - a*e)^4)/(
2*e^6*(d + e*x)^2) + (10*b^2*(b*d - a*e)^3)/(e^6*(d + e*x)) + (10*b^3*(b*d - a*e)^2*Log[d + e*x])/e^6

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Rubi [A]  time = 0.115316, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{b^4 x (4 b d-5 a e)}{e^5}+\frac{10 b^2 (b d-a e)^3}{e^6 (d+e x)}+\frac{10 b^3 (b d-a e)^2 \log (d+e x)}{e^6}-\frac{5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^5}{3 e^6 (d+e x)^3}+\frac{b^5 x^2}{2 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^4*(4*b*d - 5*a*e)*x)/e^5) + (b^5*x^2)/(2*e^4) + (b*d - a*e)^5/(3*e^6*(d + e*x)^3) - (5*b*(b*d - a*e)^4)/(
2*e^6*(d + e*x)^2) + (10*b^2*(b*d - a*e)^3)/(e^6*(d + e*x)) + (10*b^3*(b*d - a*e)^2*Log[d + e*x])/e^6

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 )^2}{(d+e x)^4} \, dx &=\int \frac{(a+b x)^5}{(d+e x)^4} \, dx\\ &=\int \left (-\frac{b^4 (4 b d-5 a e)}{e^5}+\frac{b^5 x}{e^4}+\frac{(-b d+a e)^5}{e^5 (d+e x)^4}+\frac{5 b (b d-a e)^4}{e^5 (d+e x)^3}-\frac{10 b^2 (b d-a e)^3}{e^5 (d+e x)^2}+\frac{10 b^3 (b d-a e)^2}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{b^4 (4 b d-5 a e) x}{e^5}+\frac{b^5 x^2}{2 e^4}+\frac{(b d-a e)^5}{3 e^6 (d+e x)^3}-\frac{5 b (b d-a e)^4}{2 e^6 (d+e x)^2}+\frac{10 b^2 (b d-a e)^3}{e^6 (d+e x)}+\frac{10 b^3 (b d-a e)^2 \log (d+e x)}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.0821586, size = 229, normalized size = 1.76 \[ \frac{10 a^2 b^3 d e^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )-20 a^3 b^2 e^3 \left (d^2+3 d e x+3 e^2 x^2\right )-5 a^4 b e^4 (d+3 e x)-2 a^5 e^5+10 a b^4 e \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )+60 b^3 (d+e x)^3 (b d-a e)^2 \log (d+e x)+b^5 \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )}{6 e^6 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^5*e^5 - 5*a^4*b*e^4*(d + 3*e*x) - 20*a^3*b^2*e^3*(d^2 + 3*d*e*x + 3*e^2*x^2) + 10*a^2*b^3*d*e^2*(11*d^2
+ 27*d*e*x + 18*e^2*x^2) + 10*a*b^4*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + b^5*(
47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + 60*b^3*(b*d - a*e)^2*(d + e
*x)^3*Log[d + e*x])/(6*e^6*(d + e*x)^3)

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Maple [B]  time = 0.009, size = 361, normalized size = 2.8 \begin{align*}{\frac{{b}^{5}{x}^{2}}{2\,{e}^{4}}}+5\,{\frac{a{b}^{4}x}{{e}^{4}}}-4\,{\frac{{b}^{5}dx}{{e}^{5}}}-{\frac{5\,{a}^{4}b}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{a}^{3}d{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}-15\,{\frac{{a}^{2}{d}^{2}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+10\,{\frac{a{d}^{3}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{5\,{b}^{5}{d}^{4}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{2}}{{e}^{4}}}-20\,{\frac{{b}^{4}\ln \left ( ex+d \right ) ad}{{e}^{5}}}+10\,{\frac{{b}^{5}\ln \left ( ex+d \right ){d}^{2}}{{e}^{6}}}-10\,{\frac{{a}^{3}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+30\,{\frac{{a}^{2}d{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-30\,{\frac{a{d}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}+10\,{\frac{{b}^{5}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{a}^{5}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{5\,{a}^{4}db}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{10\,{a}^{3}{d}^{2}{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{10\,{a}^{2}{d}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{5\,a{d}^{4}{b}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{5}{d}^{5}}{3\,{e}^{6} \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)^2/(e*x+d)^4,x)

[Out]

1/2*b^5*x^2/e^4+5*b^4/e^4*a*x-4*b^5/e^5*d*x-5/2*b/e^2/(e*x+d)^2*a^4+10*b^2/e^3/(e*x+d)^2*a^3*d-15*b^3/e^4/(e*x
+d)^2*a^2*d^2+10*b^4/e^5/(e*x+d)^2*a*d^3-5/2*b^5/e^6/(e*x+d)^2*d^4+10*b^3/e^4*ln(e*x+d)*a^2-20*b^4/e^5*ln(e*x+
d)*a*d+10*b^5/e^6*ln(e*x+d)*d^2-10*b^2/e^3/(e*x+d)*a^3+30*b^3/e^4/(e*x+d)*a^2*d-30*b^4/e^5/(e*x+d)*a*d^2+10*b^
5/e^6/(e*x+d)*d^3-1/3/e/(e*x+d)^3*a^5+5/3/e^2/(e*x+d)^3*d*a^4*b-10/3/e^3/(e*x+d)^3*d^2*a^3*b^2+10/3/e^4/(e*x+d
)^3*d^3*a^2*b^3-5/3/e^5/(e*x+d)^3*a*b^4*d^4+1/3/e^6/(e*x+d)^3*b^5*d^5

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Maxima [B]  time = 1.13565, size = 381, normalized size = 2.93 \begin{align*} \frac{47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 60 \,{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \,{\left (7 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 18 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{b^{5} e x^{2} - 2 \,{\left (4 \, b^{5} d - 5 \, a b^{4} e\right )} x}{2 \, e^{5}} + \frac{10 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \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)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(47*b^5*d^5 - 130*a*b^4*d^4*e + 110*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 - 2*a^5*e^5 + 60*
(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(7*b^5*d^4*e - 20*a*b^4*d^3*e^2 + 18*
a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 - a^4*b*e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) + 1/2*(b^5*e
*x^2 - 2*(4*b^5*d - 5*a*b^4*e)*x)/e^5 + 10*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*log(e*x + d)/e^6

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Fricas [B]  time = 1.49269, size = 867, normalized size = 6.67 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} + 47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} - 15 \,{\left (b^{5} d e^{4} - 2 \, a b^{4} e^{5}\right )} x^{4} - 9 \,{\left (7 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, b^{5} d^{3} e^{2} + 30 \, a b^{4} d^{2} e^{3} - 60 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 3 \,{\left (27 \, b^{5} d^{4} e - 90 \, a b^{4} d^{3} e^{2} + 90 \, a^{2} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x + 60 \,{\left (b^{5} d^{5} - 2 \, a b^{4} d^{4} e + a^{2} b^{3} d^{3} e^{2} +{\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \,{\left (b^{5} d^{3} e^{2} - 2 \, a b^{4} d^{2} e^{3} + a^{2} b^{3} d e^{4}\right )} x^{2} + 3 \,{\left (b^{5} d^{4} e - 2 \, a b^{4} d^{3} e^{2} + a^{2} b^{3} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{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)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*e^5*x^5 + 47*b^5*d^5 - 130*a*b^4*d^4*e + 110*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 -
2*a^5*e^5 - 15*(b^5*d*e^4 - 2*a*b^4*e^5)*x^4 - 9*(7*b^5*d^2*e^3 - 10*a*b^4*d*e^4)*x^3 - 3*(3*b^5*d^3*e^2 + 30
*a*b^4*d^2*e^3 - 60*a^2*b^3*d*e^4 + 20*a^3*b^2*e^5)*x^2 + 3*(27*b^5*d^4*e - 90*a*b^4*d^3*e^2 + 90*a^2*b^3*d^2*
e^3 - 20*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x + 60*(b^5*d^5 - 2*a*b^4*d^4*e + a^2*b^3*d^3*e^2 + (b^5*d^2*e^3 - 2*a*b
^4*d*e^4 + a^2*b^3*e^5)*x^3 + 3*(b^5*d^3*e^2 - 2*a*b^4*d^2*e^3 + a^2*b^3*d*e^4)*x^2 + 3*(b^5*d^4*e - 2*a*b^4*d
^3*e^2 + a^2*b^3*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [B]  time = 3.60213, size = 282, normalized size = 2.17 \begin{align*} \frac{b^{5} x^{2}}{2 e^{4}} + \frac{10 b^{3} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{6}} - \frac{2 a^{5} e^{5} + 5 a^{4} b d e^{4} + 20 a^{3} b^{2} d^{2} e^{3} - 110 a^{2} b^{3} d^{3} e^{2} + 130 a b^{4} d^{4} e - 47 b^{5} d^{5} + x^{2} \left (60 a^{3} b^{2} e^{5} - 180 a^{2} b^{3} d e^{4} + 180 a b^{4} d^{2} e^{3} - 60 b^{5} d^{3} e^{2}\right ) + x \left (15 a^{4} b e^{5} + 60 a^{3} b^{2} d e^{4} - 270 a^{2} b^{3} d^{2} e^{3} + 300 a b^{4} d^{3} e^{2} - 105 b^{5} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} + \frac{x \left (5 a b^{4} e - 4 b^{5} d\right )}{e^{5}} \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)**2/(e*x+d)**4,x)

[Out]

b**5*x**2/(2*e**4) + 10*b**3*(a*e - b*d)**2*log(d + e*x)/e**6 - (2*a**5*e**5 + 5*a**4*b*d*e**4 + 20*a**3*b**2*
d**2*e**3 - 110*a**2*b**3*d**3*e**2 + 130*a*b**4*d**4*e - 47*b**5*d**5 + x**2*(60*a**3*b**2*e**5 - 180*a**2*b*
*3*d*e**4 + 180*a*b**4*d**2*e**3 - 60*b**5*d**3*e**2) + x*(15*a**4*b*e**5 + 60*a**3*b**2*d*e**4 - 270*a**2*b**
3*d**2*e**3 + 300*a*b**4*d**3*e**2 - 105*b**5*d**4*e))/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9
*x**3) + x*(5*a*b**4*e - 4*b**5*d)/e**5

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Giac [B]  time = 1.09528, size = 336, normalized size = 2.58 \begin{align*} 10 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{5} x^{2} e^{4} - 8 \, b^{5} d x e^{3} + 10 \, a b^{4} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, b^{5} d^{5} - 130 \, a b^{4} d^{4} e + 110 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 2 \, a^{5} e^{5} + 60 \,{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \,{\left (7 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 18 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x\right )} e^{\left (-6\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)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

10*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*e^(-6)*log(abs(x*e + d)) + 1/2*(b^5*x^2*e^4 - 8*b^5*d*x*e^3 + 10*a*b^
4*x*e^4)*e^(-8) + 1/6*(47*b^5*d^5 - 130*a*b^4*d^4*e + 110*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4
- 2*a^5*e^5 + 60*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(7*b^5*d^4*e - 20*a
*b^4*d^3*e^2 + 18*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 - a^4*b*e^5)*x)*e^(-6)/(x*e + d)^3