3.1912 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{d+e x} \, dx\)

Optimal. Leaf size=122 \[ \frac{b x (b d-a e)^4}{e^5}-\frac{(a+b x)^2 (b d-a e)^3}{2 e^4}+\frac{(a+b x)^3 (b d-a e)^2}{3 e^3}-\frac{(a+b x)^4 (b d-a e)}{4 e^2}-\frac{(b d-a e)^5 \log (d+e x)}{e^6}+\frac{(a+b x)^5}{5 e} \]

[Out]

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

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Rubi [A]  time = 0.055885, antiderivative size = 122, 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} \[ \frac{b x (b d-a e)^4}{e^5}-\frac{(a+b x)^2 (b d-a e)^3}{2 e^4}+\frac{(a+b x)^3 (b d-a e)^2}{3 e^3}-\frac{(a+b x)^4 (b d-a e)}{4 e^2}-\frac{(b d-a e)^5 \log (d+e x)}{e^6}+\frac{(a+b x)^5}{5 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0676299, size = 167, normalized size = 1.37 \[ \frac{b e x \left (100 a^2 b^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+300 a^3 b e^3 (e x-2 d)+300 a^4 e^4+25 a b^3 e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (b d-a e)^5 \log (d+e x)}{60 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x*(300*a^4*e^4 + 300*a^3*b*e^3*(-2*d + e*x) + 100*a^2*b^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 25*a*b^3*e*
(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 1
2*e^4*x^4)) - 60*(b*d - a*e)^5*Log[d + e*x])/(60*e^6)

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Maple [B]  time = 0.002, size = 302, normalized size = 2.5 \begin{align*}{\frac{{b}^{5}{x}^{5}}{5\,e}}+{\frac{5\,{b}^{4}{x}^{4}a}{4\,e}}-{\frac{{b}^{5}{x}^{4}d}{4\,{e}^{2}}}+{\frac{10\,{b}^{3}{x}^{3}{a}^{2}}{3\,e}}-{\frac{5\,{b}^{4}{x}^{3}ad}{3\,{e}^{2}}}+{\frac{{b}^{5}{x}^{3}{d}^{2}}{3\,{e}^{3}}}+5\,{\frac{{b}^{2}{x}^{2}{a}^{3}}{e}}-5\,{\frac{{b}^{3}{x}^{2}{a}^{2}d}{{e}^{2}}}+{\frac{5\,{b}^{4}{x}^{2}a{d}^{2}}{2\,{e}^{3}}}-{\frac{{b}^{5}{x}^{2}{d}^{3}}{2\,{e}^{4}}}+5\,{\frac{b{a}^{4}x}{e}}-10\,{\frac{{a}^{3}d{b}^{2}x}{{e}^{2}}}+10\,{\frac{{a}^{2}{d}^{2}{b}^{3}x}{{e}^{3}}}-5\,{\frac{a{d}^{3}{b}^{4}x}{{e}^{4}}}+{\frac{{b}^{5}{d}^{4}x}{{e}^{5}}}+{\frac{\ln \left ( ex+d \right ){a}^{5}}{e}}-5\,{\frac{\ln \left ( ex+d \right ){a}^{4}bd}{{e}^{2}}}+10\,{\frac{\ln \left ( ex+d \right ){a}^{3}{b}^{2}{d}^{2}}{{e}^{3}}}-10\,{\frac{\ln \left ( ex+d \right ){a}^{2}{b}^{3}{d}^{3}}{{e}^{4}}}+5\,{\frac{\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}}{{e}^{5}}}-{\frac{\ln \left ( ex+d \right ){b}^{5}{d}^{5}}{{e}^{6}}} \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),x)

[Out]

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

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Maxima [B]  time = 1.10115, size = 348, normalized size = 2.85 \begin{align*} \frac{12 \, b^{5} e^{4} x^{5} - 15 \,{\left (b^{5} d e^{3} - 5 \, a b^{4} e^{4}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{2} - 5 \, a b^{4} d e^{3} + 10 \, a^{2} b^{3} e^{4}\right )} x^{3} - 30 \,{\left (b^{5} d^{3} e - 5 \, a b^{4} d^{2} e^{2} + 10 \, a^{2} b^{3} d e^{3} - 10 \, a^{3} b^{2} e^{4}\right )} x^{2} + 60 \,{\left (b^{5} d^{4} - 5 \, a b^{4} d^{3} e + 10 \, a^{2} b^{3} d^{2} e^{2} - 10 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x}{60 \, e^{5}} - \frac{{\left (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}\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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.57047, size = 537, normalized size = 4.4 \begin{align*} \frac{12 \, b^{5} e^{5} x^{5} - 15 \,{\left (b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 20 \,{\left (b^{5} d^{2} e^{3} - 5 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \,{\left (b^{5} d^{3} e^{2} - 5 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} - 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 60 \,{\left (b^{5} d^{4} e - 5 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \,{\left (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}\right )} \log \left (e x + d\right )}{60 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.691321, size = 202, normalized size = 1.66 \begin{align*} \frac{b^{5} x^{5}}{5 e} + \frac{x^{4} \left (5 a b^{4} e - b^{5} d\right )}{4 e^{2}} + \frac{x^{3} \left (10 a^{2} b^{3} e^{2} - 5 a b^{4} d e + b^{5} d^{2}\right )}{3 e^{3}} + \frac{x^{2} \left (10 a^{3} b^{2} e^{3} - 10 a^{2} b^{3} d e^{2} + 5 a b^{4} d^{2} e - b^{5} d^{3}\right )}{2 e^{4}} + \frac{x \left (5 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 10 a^{2} b^{3} d^{2} e^{2} - 5 a b^{4} d^{3} e + b^{5} d^{4}\right )}{e^{5}} + \frac{\left (a e - b d\right )^{5} \log{\left (d + e x \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),x)

[Out]

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

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Giac [B]  time = 1.10889, size = 350, normalized size = 2.87 \begin{align*} -{\left (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}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (12 \, b^{5} x^{5} e^{4} - 15 \, b^{5} d x^{4} e^{3} + 20 \, b^{5} d^{2} x^{3} e^{2} - 30 \, b^{5} d^{3} x^{2} e + 60 \, b^{5} d^{4} x + 75 \, a b^{4} x^{4} e^{4} - 100 \, a b^{4} d x^{3} e^{3} + 150 \, a b^{4} d^{2} x^{2} e^{2} - 300 \, a b^{4} d^{3} x e + 200 \, a^{2} b^{3} x^{3} e^{4} - 300 \, a^{2} b^{3} d x^{2} e^{3} + 600 \, a^{2} b^{3} d^{2} x e^{2} + 300 \, a^{3} b^{2} x^{2} e^{4} - 600 \, a^{3} b^{2} d x e^{3} + 300 \, a^{4} b x e^{4}\right )} e^{\left (-5\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),x, algorithm="giac")

[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^(-6)*log(abs(
x*e + d)) + 1/60*(12*b^5*x^5*e^4 - 15*b^5*d*x^4*e^3 + 20*b^5*d^2*x^3*e^2 - 30*b^5*d^3*x^2*e + 60*b^5*d^4*x + 7
5*a*b^4*x^4*e^4 - 100*a*b^4*d*x^3*e^3 + 150*a*b^4*d^2*x^2*e^2 - 300*a*b^4*d^3*x*e + 200*a^2*b^3*x^3*e^4 - 300*
a^2*b^3*d*x^2*e^3 + 600*a^2*b^3*d^2*x*e^2 + 300*a^3*b^2*x^2*e^4 - 600*a^3*b^2*d*x*e^3 + 300*a^4*b*x*e^4)*e^(-5
)