3.1928 \(\int \frac{(a+b x) (d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=73 \[ \frac{e x (b d-a e)^2}{b^3}+\frac{(d+e x)^2 (b d-a e)}{2 b^2}+\frac{(b d-a e)^3 \log (a+b x)}{b^4}+\frac{(d+e x)^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0286476, antiderivative size = 73, 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} \[ \frac{e x (b d-a e)^2}{b^3}+\frac{(d+e x)^2 (b d-a e)}{2 b^2}+\frac{(b d-a e)^3 \log (a+b x)}{b^4}+\frac{(d+e x)^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0293133, size = 74, normalized size = 1.01 \[ \frac{b e x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (b d-a e)^3 \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 133, normalized size = 1.8 \begin{align*}{\frac{{e}^{3}{x}^{3}}{3\,b}}-{\frac{{e}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,{e}^{2}{x}^{2}d}{2\,b}}+{\frac{{e}^{3}{a}^{2}x}{{b}^{3}}}-3\,{\frac{a{e}^{2}dx}{{b}^{2}}}+3\,{\frac{e{d}^{2}x}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{e}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ){a}^{2}d{e}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) a{d}^{2}e}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959333, size = 154, normalized size = 2.11 \begin{align*} \frac{2 \, b^{2} e^{3} x^{3} + 3 \,{\left (3 \, b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x}{6 \, b^{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 (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.49214, size = 238, normalized size = 3.26 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 3 \,{\left (3 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 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 (b x + a\right )}{6 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.486319, size = 82, normalized size = 1.12 \begin{align*} \frac{e^{3} x^{3}}{3 b} - \frac{x^{2} \left (a e^{3} - 3 b d e^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} e^{3} - 3 a b d e^{2} + 3 b^{2} d^{2} e\right )}{b^{3}} - \frac{\left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08954, size = 149, normalized size = 2.04 \begin{align*} \frac{2 \, b^{2} x^{3} e^{3} + 9 \, b^{2} d x^{2} e^{2} + 18 \, b^{2} d^{2} x e - 3 \, a b x^{2} e^{3} - 18 \, a b d x e^{2} + 6 \, a^{2} x e^{3}}{6 \, b^{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 ({\left | b x + a \right |}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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