3.1901 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)}{d+e x} \, dx\)

Optimal. Leaf size=74 \[ \frac{b x (b d-a e)^2}{e^3}-\frac{(a+b x)^2 (b d-a e)}{2 e^2}-\frac{(b d-a e)^3 \log (d+e x)}{e^4}+\frac{(a+b x)^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.030184, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ \frac{b x (b d-a e)^2}{e^3}-\frac{(a+b x)^2 (b d-a e)}{2 e^2}-\frac{(b d-a e)^3 \log (d+e x)}{e^4}+\frac{(a+b x)^3}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0301018, size = 74, normalized size = 1. \[ \frac{b e x \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (b d-a e)^3 \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(b*d - a*e)^3*Log[d + e*x])
/(6*e^4)

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Maple [A]  time = 0.003, size = 133, normalized size = 1.8 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3\,e}}+{\frac{3\,{b}^{2}{x}^{2}a}{2\,e}}-{\frac{{b}^{3}{x}^{2}d}{2\,{e}^{2}}}+3\,{\frac{b{a}^{2}x}{e}}-3\,{\frac{{b}^{2}dax}{{e}^{2}}}+{\frac{{b}^{3}{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}-3\,{\frac{\ln \left ( ex+d \right ){a}^{2}bd}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{3}}{{e}^{4}}} \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)/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.970727, size = 154, normalized size = 2.08 \begin{align*} \frac{2 \, b^{3} e^{2} x^{3} - 3 \,{\left (b^{3} d e - 3 \, a b^{2} e^{2}\right )} x^{2} + 6 \,{\left (b^{3} d^{2} - 3 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{4}} \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)/(e*x+d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.52299, size = 238, normalized size = 3.22 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} - 3 \,{\left (b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \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)/(e*x+d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.460467, size = 82, normalized size = 1.11 \begin{align*} \frac{b^{3} x^{3}}{3 e} + \frac{x^{2} \left (3 a b^{2} e - b^{3} d\right )}{2 e^{2}} + \frac{x \left (3 a^{2} b e^{2} - 3 a b^{2} d e + b^{3} d^{2}\right )}{e^{3}} + \frac{\left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{4}} \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)/(e*x+d),x)

[Out]

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

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Giac [A]  time = 1.17326, size = 153, normalized size = 2.07 \begin{align*} -{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, b^{3} x^{3} e^{2} - 3 \, b^{3} d x^{2} e + 6 \, b^{3} d^{2} x + 9 \, a b^{2} x^{2} e^{2} - 18 \, a b^{2} d x e + 18 \, a^{2} b x e^{2}\right )} e^{\left (-3\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)/(e*x+d),x, algorithm="giac")

[Out]

-(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*e^(-4)*log(abs(x*e + d)) + 1/6*(2*b^3*x^3*e^2 - 3*b^3*d*x
^2*e + 6*b^3*d^2*x + 9*a*b^2*x^2*e^2 - 18*a*b^2*d*x*e + 18*a^2*b*x*e^2)*e^(-3)