3.1913 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^2} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.146515, 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} \[ -\frac{5 b^4 (d+e x)^3 (b d-a e)}{3 e^6}+\frac{5 b^3 (d+e x)^2 (b d-a e)^2}{e^6}-\frac{10 b^2 x (b d-a e)^3}{e^5}+\frac{(b d-a e)^5}{e^6 (d+e x)}+\frac{5 b (b d-a e)^4 \log (d+e x)}{e^6}+\frac{b^5 (d+e x)^4}{4 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.072889, size = 228, normalized size = 1.75 \[ \frac{60 a^2 b^3 e^2 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+120 a^3 b^2 e^3 \left (-d^2+d e x+e^2 x^2\right )+60 a^4 b d e^4-12 a^5 e^5+20 a b^4 e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+60 b (d+e x) (b d-a e)^4 \log (d+e x)+b^5 \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )}{12 e^6 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a^4*b*d*e^4 - 12*a^5*e^5 + 120*a^3*b^2*e^3*(-d^2 + d*e*x + e^2*x^2) + 60*a^2*b^3*e^2*(2*d^3 - 4*d^2*e*x -
3*d*e^2*x^2 + e^3*x^3) + 20*a*b^4*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^5*(12*d^5
- 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + 60*b*(b*d - a*e)^4*(d + e*x)*Log[
d + e*x])/(12*e^6*(d + e*x))

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 326, normalized size = 2.5 \begin{align*}{\frac{{b}^{5}{x}^{4}}{4\,{e}^{2}}}+{\frac{5\,{b}^{4}{x}^{3}a}{3\,{e}^{2}}}-{\frac{2\,{b}^{5}{x}^{3}d}{3\,{e}^{3}}}+5\,{\frac{{b}^{3}{x}^{2}{a}^{2}}{{e}^{2}}}-5\,{\frac{{b}^{4}{x}^{2}ad}{{e}^{3}}}+{\frac{3\,{b}^{5}{x}^{2}{d}^{2}}{2\,{e}^{4}}}+10\,{\frac{{a}^{3}{b}^{2}x}{{e}^{2}}}-20\,{\frac{{a}^{2}d{b}^{3}x}{{e}^{3}}}+15\,{\frac{a{d}^{2}{b}^{4}x}{{e}^{4}}}-4\,{\frac{{b}^{5}{d}^{3}x}{{e}^{5}}}+5\,{\frac{b\ln \left ( ex+d \right ){a}^{4}}{{e}^{2}}}-20\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{3}d}{{e}^{3}}}+30\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{2}{d}^{2}}{{e}^{4}}}-20\,{\frac{{b}^{4}\ln \left ( ex+d \right ) a{d}^{3}}{{e}^{5}}}+5\,{\frac{{b}^{5}\ln \left ( ex+d \right ){d}^{4}}{{e}^{6}}}-{\frac{{a}^{5}}{e \left ( ex+d \right ) }}+5\,{\frac{{a}^{4}db}{{e}^{2} \left ( ex+d \right ) }}-10\,{\frac{{a}^{3}{d}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+10\,{\frac{{a}^{2}{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-5\,{\frac{a{d}^{4}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{b}^{5}{d}^{5}}{{e}^{6} \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)^2/(e*x+d)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.11113, size = 356, normalized size = 2.74 \begin{align*} \frac{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{e^{7} x + d e^{6}} + \frac{3 \, b^{5} e^{3} x^{4} - 4 \,{\left (2 \, b^{5} d e^{2} - 5 \, a b^{4} e^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{5} d^{2} e - 10 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{2} - 12 \,{\left (4 \, b^{5} d^{3} - 15 \, a b^{4} d^{2} e + 20 \, a^{2} b^{3} d e^{2} - 10 \, a^{3} b^{2} e^{3}\right )} x}{12 \, e^{5}} + \frac{5 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\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)^2,x, algorithm="maxima")

[Out]

(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/(e^7*x + d*e^6)
+ 1/12*(3*b^5*e^3*x^4 - 4*(2*b^5*d*e^2 - 5*a*b^4*e^3)*x^3 + 6*(3*b^5*d^2*e - 10*a*b^4*d*e^2 + 10*a^2*b^3*e^3)*
x^2 - 12*(4*b^5*d^3 - 15*a*b^4*d^2*e + 20*a^2*b^3*d*e^2 - 10*a^3*b^2*e^3)*x)/e^5 + 5*(b^5*d^4 - 4*a*b^4*d^3*e
+ 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*log(e*x + d)/e^6

________________________________________________________________________________________

Fricas [B]  time = 1.49006, size = 767, normalized size = 5.9 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} + 12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e + 120 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} - 5 \,{\left (b^{5} d e^{4} - 4 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \,{\left (b^{5} d^{3} e^{2} - 4 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 4 \, a^{3} b^{2} e^{5}\right )} x^{2} - 12 \,{\left (4 \, b^{5} d^{4} e - 15 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4}\right )} x + 60 \,{\left (b^{5} d^{5} - 4 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} - 4 \, a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} +{\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d 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)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*e^5*x^5 + 12*b^5*d^5 - 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4
- 12*a^5*e^5 - 5*(b^5*d*e^4 - 4*a*b^4*e^5)*x^4 + 10*(b^5*d^2*e^3 - 4*a*b^4*d*e^4 + 6*a^2*b^3*e^5)*x^3 - 30*(b
^5*d^3*e^2 - 4*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 4*a^3*b^2*e^5)*x^2 - 12*(4*b^5*d^4*e - 15*a*b^4*d^3*e^2 + 20*
a^2*b^3*d^2*e^3 - 10*a^3*b^2*d*e^4)*x + 60*(b^5*d^5 - 4*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 - 4*a^3*b^2*d^2*e^3 +
a^4*b*d*e^4 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(e*x + d))
/(e^7*x + d*e^6)

________________________________________________________________________________________

Sympy [A]  time = 1.16755, size = 224, normalized size = 1.72 \begin{align*} \frac{b^{5} x^{4}}{4 e^{2}} + \frac{5 b \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{6}} - \frac{a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{d e^{6} + e^{7} x} + \frac{x^{3} \left (5 a b^{4} e - 2 b^{5} d\right )}{3 e^{3}} + \frac{x^{2} \left (10 a^{2} b^{3} e^{2} - 10 a b^{4} d e + 3 b^{5} d^{2}\right )}{2 e^{4}} + \frac{x \left (10 a^{3} b^{2} e^{3} - 20 a^{2} b^{3} d e^{2} + 15 a b^{4} d^{2} e - 4 b^{5} d^{3}\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)**2,x)

[Out]

b**5*x**4/(4*e**2) + 5*b*(a*e - b*d)**4*log(d + e*x)/e**6 - (a**5*e**5 - 5*a**4*b*d*e**4 + 10*a**3*b**2*d**2*e
**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/(d*e**6 + e**7*x) + x**3*(5*a*b**4*e - 2*b**5*d)/(
3*e**3) + x**2*(10*a**2*b**3*e**2 - 10*a*b**4*d*e + 3*b**5*d**2)/(2*e**4) + x*(10*a**3*b**2*e**3 - 20*a**2*b**
3*d*e**2 + 15*a*b**4*d**2*e - 4*b**5*d**3)/e**5

________________________________________________________________________________________

Giac [B]  time = 1.10945, size = 443, normalized size = 3.41 \begin{align*} \frac{1}{12} \,{\left (3 \, b^{5} - \frac{20 \,{\left (b^{5} d e - a b^{4} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{60 \,{\left (b^{5} d^{2} e^{2} - 2 \, a b^{4} d e^{3} + a^{2} b^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{120 \,{\left (b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{4} e^{\left (-6\right )} - 5 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} e^{\left (-6\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{b^{5} d^{5} e^{4}}{x e + d} - \frac{5 \, a b^{4} d^{4} e^{5}}{x e + d} + \frac{10 \, a^{2} b^{3} d^{3} e^{6}}{x e + d} - \frac{10 \, a^{3} b^{2} d^{2} e^{7}}{x e + d} + \frac{5 \, a^{4} b d e^{8}}{x e + d} - \frac{a^{5} e^{9}}{x e + d}\right )} e^{\left (-10\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)^2,x, algorithm="giac")

[Out]

1/12*(3*b^5 - 20*(b^5*d*e - a*b^4*e^2)*e^(-1)/(x*e + d) + 60*(b^5*d^2*e^2 - 2*a*b^4*d*e^3 + a^2*b^3*e^4)*e^(-2
)/(x*e + d)^2 - 120*(b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 - a^3*b^2*e^6)*e^(-3)/(x*e + d)^3)*(x*e +
d)^4*e^(-6) - 5*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*e^(-6)*log(abs(x*
e + d)*e^(-1)/(x*e + d)^2) + (b^5*d^5*e^4/(x*e + d) - 5*a*b^4*d^4*e^5/(x*e + d) + 10*a^2*b^3*d^3*e^6/(x*e + d)
- 10*a^3*b^2*d^2*e^7/(x*e + d) + 5*a^4*b*d*e^8/(x*e + d) - a^5*e^9/(x*e + d))*e^(-10)