3.1490 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{d+e x} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b x (b d-a e)^5}{e^6}+\frac{(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac{(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac{(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac{(a+b x)^5 (b d-a e)}{5 e^2}+\frac{(b d-a e)^6 \log (d+e x)}{e^7}+\frac{(a+b x)^6}{6 e} \]

[Out]

-((b*(b*d - a*e)^5*x)/e^6) + ((b*d - a*e)^4*(a + b*x)^2)/(2*e^5) - ((b*d - a*e)^3*(a + b*x)^3)/(3*e^4) + ((b*d
- a*e)^2*(a + b*x)^4)/(4*e^3) - ((b*d - a*e)*(a + b*x)^5)/(5*e^2) + (a + b*x)^6/(6*e) + ((b*d - a*e)^6*Log[d
+ e*x])/e^7

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Rubi [A]  time = 0.0642648, antiderivative size = 146, 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} \[ -\frac{b x (b d-a e)^5}{e^6}+\frac{(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac{(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac{(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac{(a+b x)^5 (b d-a e)}{5 e^2}+\frac{(b d-a e)^6 \log (d+e x)}{e^7}+\frac{(a+b x)^6}{6 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(b*d - a*e)^5*x)/e^6) + ((b*d - a*e)^4*(a + b*x)^2)/(2*e^5) - ((b*d - a*e)^3*(a + b*x)^3)/(3*e^4) + ((b*d
- a*e)^2*(a + b*x)^4)/(4*e^3) - ((b*d - a*e)*(a + b*x)^5)/(5*e^2) + (a + b*x)^6/(6*e) + ((b*d - a*e)^6*Log[d
+ e*x])/e^7

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 )^3}{d+e x} \, dx &=\int \frac{(a+b x)^6}{d+e x} \, dx\\ &=\int \left (-\frac{b (b d-a e)^5}{e^6}+\frac{b (b d-a e)^4 (a+b x)}{e^5}-\frac{b (b d-a e)^3 (a+b x)^2}{e^4}+\frac{b (b d-a e)^2 (a+b x)^3}{e^3}-\frac{b (b d-a e) (a+b x)^4}{e^2}+\frac{b (a+b x)^5}{e}+\frac{(-b d+a e)^6}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{b (b d-a e)^5 x}{e^6}+\frac{(b d-a e)^4 (a+b x)^2}{2 e^5}-\frac{(b d-a e)^3 (a+b x)^3}{3 e^4}+\frac{(b d-a e)^2 (a+b x)^4}{4 e^3}-\frac{(b d-a e) (a+b x)^5}{5 e^2}+\frac{(a+b x)^6}{6 e}+\frac{(b d-a e)^6 \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.0867949, size = 230, normalized size = 1.58 \[ \frac{b e x \left (75 a^2 b^3 e^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+450 a^4 b e^4 (e x-2 d)+360 a^5 e^5+6 a b^4 e \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)}{60 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x*(360*a^5*e^5 + 450*a^4*b*e^4*(-2*d + e*x) + 200*a^3*b^2*e^3*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 75*a^2*b^3*
e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 6*a*b^4*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e
^3*x^3 + 12*e^4*x^4) + b^5*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5
)) + 60*(b*d - a*e)^6*Log[d + e*x])/(60*e^7)

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Maple [B]  time = 0.043, size = 412, normalized size = 2.8 \begin{align*} -6\,{\frac{\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}}{{e}^{6}}}-{\frac{3\,{b}^{5}{x}^{4}ad}{2\,{e}^{2}}}-5\,{\frac{{b}^{4}{x}^{3}{a}^{2}d}{{e}^{2}}}-20\,{\frac{\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}}{{e}^{4}}}+15\,{\frac{\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}}{{e}^{5}}}+{\frac{\ln \left ( ex+d \right ){a}^{6}}{e}}+{\frac{{b}^{6}{x}^{6}}{6\,e}}+{\frac{6\,{b}^{5}{x}^{5}a}{5\,e}}-{\frac{{b}^{6}{d}^{5}x}{{e}^{6}}}+{\frac{{b}^{6}{x}^{4}{d}^{2}}{4\,{e}^{3}}}+{\frac{20\,{b}^{3}{x}^{3}{a}^{3}}{3\,e}}-{\frac{{b}^{6}{x}^{3}{d}^{3}}{3\,{e}^{4}}}+{\frac{15\,{b}^{2}{x}^{2}{a}^{4}}{2\,e}}+{\frac{{b}^{6}{x}^{2}{d}^{4}}{2\,{e}^{5}}}+6\,{\frac{{a}^{5}bx}{e}}-{\frac{{b}^{6}{x}^{5}d}{5\,{e}^{2}}}+{\frac{15\,{b}^{4}{x}^{4}{a}^{2}}{4\,e}}+{\frac{\ln \left ( ex+d \right ){d}^{6}{b}^{6}}{{e}^{7}}}-6\,{\frac{\ln \left ( ex+d \right ){a}^{5}bd}{{e}^{2}}}+15\,{\frac{\ln \left ( ex+d \right ){d}^{2}{a}^{4}{b}^{2}}{{e}^{3}}}-15\,{\frac{{a}^{2}{b}^{4}{d}^{3}x}{{e}^{4}}}+6\,{\frac{a{b}^{5}{d}^{4}x}{{e}^{5}}}+2\,{\frac{{b}^{5}{x}^{3}a{d}^{2}}{{e}^{3}}}-10\,{\frac{{x}^{2}{a}^{3}{b}^{3}d}{{e}^{2}}}+{\frac{15\,{b}^{4}{x}^{2}{a}^{2}{d}^{2}}{2\,{e}^{3}}}-3\,{\frac{{b}^{5}{x}^{2}a{d}^{3}}{{e}^{4}}}-15\,{\frac{{a}^{4}{b}^{2}dx}{{e}^{2}}}+20\,{\frac{{a}^{3}{b}^{3}{d}^{2}x}{{e}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d),x)

[Out]

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

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Maxima [B]  time = 1.14834, size = 471, normalized size = 3.23 \begin{align*} \frac{10 \, b^{6} e^{5} x^{6} - 12 \,{\left (b^{6} d e^{4} - 6 \, a b^{5} e^{5}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{3} - 6 \, a b^{5} d e^{4} + 15 \, a^{2} b^{4} e^{5}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} - 20 \, a^{3} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e - 6 \, a b^{5} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{2} e^{3} - 20 \, a^{3} b^{3} d e^{4} + 15 \, a^{4} b^{2} e^{5}\right )} x^{2} - 60 \,{\left (b^{6} d^{5} - 6 \, a b^{5} d^{4} e + 15 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} + 15 \, a^{4} b^{2} d e^{4} - 6 \, a^{5} b e^{5}\right )} x}{60 \, e^{6}} + \frac{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.82455, size = 726, normalized size = 4.97 \begin{align*} \frac{10 \, b^{6} e^{6} x^{6} - 12 \,{\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \,{\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.60546, size = 286, normalized size = 1.96 \begin{align*} \frac{b^{6} x^{6}}{6 e} + \frac{x^{5} \left (6 a b^{5} e - b^{6} d\right )}{5 e^{2}} + \frac{x^{4} \left (15 a^{2} b^{4} e^{2} - 6 a b^{5} d e + b^{6} d^{2}\right )}{4 e^{3}} + \frac{x^{3} \left (20 a^{3} b^{3} e^{3} - 15 a^{2} b^{4} d e^{2} + 6 a b^{5} d^{2} e - b^{6} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (15 a^{4} b^{2} e^{4} - 20 a^{3} b^{3} d e^{3} + 15 a^{2} b^{4} d^{2} e^{2} - 6 a b^{5} d^{3} e + b^{6} d^{4}\right )}{2 e^{5}} + \frac{x \left (6 a^{5} b e^{5} - 15 a^{4} b^{2} d e^{4} + 20 a^{3} b^{3} d^{2} e^{3} - 15 a^{2} b^{4} d^{3} e^{2} + 6 a b^{5} d^{4} e - b^{6} d^{5}\right )}{e^{6}} + \frac{\left (a e - b d\right )^{6} \log{\left (d + e x \right )}}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d),x)

[Out]

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

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Giac [B]  time = 1.1816, size = 478, normalized size = 3.27 \begin{align*}{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, b^{6} x^{6} e^{5} - 12 \, b^{6} d x^{5} e^{4} + 15 \, b^{6} d^{2} x^{4} e^{3} - 20 \, b^{6} d^{3} x^{3} e^{2} + 30 \, b^{6} d^{4} x^{2} e - 60 \, b^{6} d^{5} x + 72 \, a b^{5} x^{5} e^{5} - 90 \, a b^{5} d x^{4} e^{4} + 120 \, a b^{5} d^{2} x^{3} e^{3} - 180 \, a b^{5} d^{3} x^{2} e^{2} + 360 \, a b^{5} d^{4} x e + 225 \, a^{2} b^{4} x^{4} e^{5} - 300 \, a^{2} b^{4} d x^{3} e^{4} + 450 \, a^{2} b^{4} d^{2} x^{2} e^{3} - 900 \, a^{2} b^{4} d^{3} x e^{2} + 400 \, a^{3} b^{3} x^{3} e^{5} - 600 \, a^{3} b^{3} d x^{2} e^{4} + 1200 \, a^{3} b^{3} d^{2} x e^{3} + 450 \, a^{4} b^{2} x^{2} e^{5} - 900 \, a^{4} b^{2} d x e^{4} + 360 \, a^{5} b x e^{5}\right )} e^{\left (-6\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)^3/(e*x+d),x, algorithm="giac")

[Out]

(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*
e^6)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*b^6*x^6*e^5 - 12*b^6*d*x^5*e^4 + 15*b^6*d^2*x^4*e^3 - 20*b^6*d^3*x^3*
e^2 + 30*b^6*d^4*x^2*e - 60*b^6*d^5*x + 72*a*b^5*x^5*e^5 - 90*a*b^5*d*x^4*e^4 + 120*a*b^5*d^2*x^3*e^3 - 180*a*
b^5*d^3*x^2*e^2 + 360*a*b^5*d^4*x*e + 225*a^2*b^4*x^4*e^5 - 300*a^2*b^4*d*x^3*e^4 + 450*a^2*b^4*d^2*x^2*e^3 -
900*a^2*b^4*d^3*x*e^2 + 400*a^3*b^3*x^3*e^5 - 600*a^3*b^3*d*x^2*e^4 + 1200*a^3*b^3*d^2*x*e^3 + 450*a^4*b^2*x^2
*e^5 - 900*a^4*b^2*d*x*e^4 + 360*a^5*b*x*e^5)*e^(-6)