3.1914 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=133 \[ -\frac{5 b^4 (d+e x)^2 (b d-a e)}{2 e^6}+\frac{10 b^3 x (b d-a e)^2}{e^5}-\frac{10 b^2 (b d-a e)^3 \log (d+e x)}{e^6}-\frac{5 b (b d-a e)^4}{e^6 (d+e x)}+\frac{(b d-a e)^5}{2 e^6 (d+e x)^2}+\frac{b^5 (d+e x)^3}{3 e^6} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.127147, antiderivative size = 133, 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} \[ -\frac{5 b^4 (d+e x)^2 (b d-a e)}{2 e^6}+\frac{10 b^3 x (b d-a e)^2}{e^5}-\frac{10 b^2 (b d-a e)^3 \log (d+e x)}{e^6}-\frac{5 b (b d-a e)^4}{e^6 (d+e x)}+\frac{(b d-a e)^5}{2 e^6 (d+e x)^2}+\frac{b^5 (d+e x)^3}{3 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0816614, size = 230, normalized size = 1.73 \[ \frac{30 a^2 b^3 e^2 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+30 a^3 b^2 d e^3 (3 d+4 e x)-15 a^4 b e^4 (d+2 e x)-3 a^5 e^5+15 a b^4 e \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )-60 b^2 (d+e x)^2 (b d-a e)^3 \log (d+e x)+b^5 \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )}{6 e^6 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^5*e^5 - 15*a^4*b*e^4*(d + 2*e*x) + 30*a^3*b^2*d*e^3*(3*d + 4*e*x) + 30*a^2*b^3*e^2*(-5*d^3 - 4*d^2*e*x +
4*d*e^2*x^2 + 2*e^3*x^3) + 15*a*b^4*e*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + b^5*(-27
*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) - 60*b^2*(b*d - a*e)^3*(d + e*x)
^2*Log[d + e*x])/(6*e^6*(d + e*x)^2)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 346, normalized size = 2.6 \begin{align*}{\frac{{b}^{5}{x}^{3}}{3\,{e}^{3}}}+{\frac{5\,{b}^{4}{x}^{2}a}{2\,{e}^{3}}}-{\frac{3\,{b}^{5}{x}^{2}d}{2\,{e}^{4}}}+10\,{\frac{{a}^{2}{b}^{3}x}{{e}^{3}}}-15\,{\frac{ad{b}^{4}x}{{e}^{4}}}+6\,{\frac{{b}^{5}{d}^{2}x}{{e}^{5}}}-{\frac{{a}^{5}}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{5\,{a}^{4}db}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-5\,{\frac{{a}^{3}{d}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+5\,{\frac{{a}^{2}{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{5\,a{d}^{4}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{5}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{3}}{{e}^{3}}}-30\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{2}d}{{e}^{4}}}+30\,{\frac{{b}^{4}\ln \left ( ex+d \right ) a{d}^{2}}{{e}^{5}}}-10\,{\frac{{b}^{5}\ln \left ( ex+d \right ){d}^{3}}{{e}^{6}}}-5\,{\frac{{a}^{4}b}{{e}^{2} \left ( ex+d \right ) }}+20\,{\frac{{a}^{3}d{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-30\,{\frac{{a}^{2}{d}^{2}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+20\,{\frac{a{d}^{3}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-5\,{\frac{{b}^{5}{d}^{4}}{{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)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.05559, size = 366, normalized size = 2.75 \begin{align*} -\frac{9 \, b^{5} d^{5} - 35 \, a b^{4} d^{4} e + 50 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} + a^{5} e^{5} + 10 \,{\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}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, b^{5} e^{2} x^{3} - 3 \,{\left (3 \, b^{5} d e - 5 \, a b^{4} e^{2}\right )} x^{2} + 6 \,{\left (6 \, b^{5} d^{2} - 15 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x}{6 \, e^{5}} - \frac{10 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\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)^3,x, algorithm="maxima")

[Out]

-1/2*(9*b^5*d^5 - 35*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 + a^5*e^5 + 10*(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)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)
+ 1/6*(2*b^5*e^2*x^3 - 3*(3*b^5*d*e - 5*a*b^4*e^2)*x^2 + 6*(6*b^5*d^2 - 15*a*b^4*d*e + 10*a^2*b^3*e^2)*x)/e^5
- 10*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*log(e*x + d)/e^6

________________________________________________________________________________________

Fricas [B]  time = 1.54776, size = 840, normalized size = 6.32 \begin{align*} \frac{2 \, b^{5} e^{5} x^{5} - 27 \, b^{5} d^{5} + 105 \, a b^{4} d^{4} e - 150 \, a^{2} b^{3} d^{3} e^{2} + 90 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} - 3 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 3 \,{\left (21 \, b^{5} d^{3} e^{2} - 55 \, a b^{4} d^{2} e^{3} + 40 \, a^{2} b^{3} d e^{4}\right )} x^{2} + 6 \,{\left (b^{5} d^{4} e + 5 \, a b^{4} d^{3} e^{2} - 20 \, 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} - 3 \, a b^{4} d^{4} e + 3 \, a^{2} b^{3} d^{3} e^{2} - a^{3} b^{2} d^{2} e^{3} +{\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} + 2 \,{\left (b^{5} d^{4} e - 3 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3} - a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*e^5*x^5 - 27*b^5*d^5 + 105*a*b^4*d^4*e - 150*a^2*b^3*d^3*e^2 + 90*a^3*b^2*d^2*e^3 - 15*a^4*b*d*e^4
- 3*a^5*e^5 - 5*(b^5*d*e^4 - 3*a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 - 3*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 3*(21*b
^5*d^3*e^2 - 55*a*b^4*d^2*e^3 + 40*a^2*b^3*d*e^4)*x^2 + 6*(b^5*d^4*e + 5*a*b^4*d^3*e^2 - 20*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 - 3*a*b^4*d^4*e + 3*a^2*b^3*d^3*e^2 - a^3*b^2*d^2*e^3 + (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 + 2*(b^5*d^4*e - 3*a*b^4*d^3*e^2 + 3*a^2*b^3*d^2
*e^3 - a^3*b^2*d*e^4)*x)*log(e*x + d))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)

________________________________________________________________________________________

Sympy [B]  time = 2.10048, size = 253, normalized size = 1.9 \begin{align*} \frac{b^{5} x^{3}}{3 e^{3}} + \frac{10 b^{2} \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{6}} - \frac{a^{5} e^{5} + 5 a^{4} b d e^{4} - 30 a^{3} b^{2} d^{2} e^{3} + 50 a^{2} b^{3} d^{3} e^{2} - 35 a b^{4} d^{4} e + 9 b^{5} d^{5} + x \left (10 a^{4} b e^{5} - 40 a^{3} b^{2} d e^{4} + 60 a^{2} b^{3} d^{2} e^{3} - 40 a b^{4} d^{3} e^{2} + 10 b^{5} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (5 a b^{4} e - 3 b^{5} d\right )}{2 e^{4}} + \frac{x \left (10 a^{2} b^{3} e^{2} - 15 a b^{4} d e + 6 b^{5} d^{2}\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)**3,x)

[Out]

b**5*x**3/(3*e**3) + 10*b**2*(a*e - b*d)**3*log(d + e*x)/e**6 - (a**5*e**5 + 5*a**4*b*d*e**4 - 30*a**3*b**2*d*
*2*e**3 + 50*a**2*b**3*d**3*e**2 - 35*a*b**4*d**4*e + 9*b**5*d**5 + x*(10*a**4*b*e**5 - 40*a**3*b**2*d*e**4 +
60*a**2*b**3*d**2*e**3 - 40*a*b**4*d**3*e**2 + 10*b**5*d**4*e))/(2*d**2*e**6 + 4*d*e**7*x + 2*e**8*x**2) + x**
2*(5*a*b**4*e - 3*b**5*d)/(2*e**4) + x*(10*a**2*b**3*e**2 - 15*a*b**4*d*e + 6*b**5*d**2)/e**5

________________________________________________________________________________________

Giac [A]  time = 1.1173, size = 338, normalized size = 2.54 \begin{align*} -10 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, b^{5} x^{3} e^{6} - 9 \, b^{5} d x^{2} e^{5} + 36 \, b^{5} d^{2} x e^{4} + 15 \, a b^{4} x^{2} e^{6} - 90 \, a b^{4} d x e^{5} + 60 \, a^{2} b^{3} x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, b^{5} d^{5} - 35 \, a b^{4} d^{4} e + 50 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} + a^{5} e^{5} + 10 \,{\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 )} e^{\left (-6\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)^2/(e*x+d)^3,x, algorithm="giac")

[Out]

-10*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*e^(-6)*log(abs(x*e + d)) + 1/6*(2*b^5*x^3*e^6 -
9*b^5*d*x^2*e^5 + 36*b^5*d^2*x*e^4 + 15*a*b^4*x^2*e^6 - 90*a*b^4*d*x*e^5 + 60*a^2*b^3*x*e^6)*e^(-9) - 1/2*(9*b
^5*d^5 - 35*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 + a^5*e^5 + 10*(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)*e^(-6)/(x*e + d)^2