3.1694 \(\int \frac{(A+B x) (d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=145 \[ \frac{e^2 (a+b x)^2 (-4 a B e+A b e+3 b B d)}{2 b^5}-\frac{(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac{3 e x (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac{(b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{b^5}+\frac{B e^3 (a+b x)^3}{3 b^5} \]

[Out]

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

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Rubi [A]  time = 0.176026, antiderivative size = 145, 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, 77} \[ \frac{e^2 (a+b x)^2 (-4 a B e+A b e+3 b B d)}{2 b^5}-\frac{(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac{3 e x (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac{(b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{b^5}+\frac{B e^3 (a+b x)^3}{3 b^5} \]

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]

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

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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

Mathematica [A]  time = 0.127935, size = 250, normalized size = 1.72 \[ \frac{-3 A b \left (2 a^2 b e^2 (3 d+2 e x)-2 a^3 e^3+3 a b^2 e \left (-2 d^2-2 d e x+e^2 x^2\right )+b^3 \left (2 d^3-6 d e^2 x^2-e^3 x^3\right )\right )+B \left (6 a^2 b^2 e \left (-3 d^2-6 d e x+2 e^2 x^2\right )+18 a^3 b e^2 (d+e x)-6 a^4 e^3+a b^3 \left (18 d^2 e x+6 d^3-27 d e^2 x^2-4 e^3 x^3\right )+b^4 e x^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (a+b x) (b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{6 b^5 (a+b x)} \]

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*(-6*a^4*e^3 + 18*a^3*b*e^2*(d + e*x) + 6*a^2*b^2*e*(-3*d^2 - 6*d*e*x + 2*e^2*x^2) + b^4*e*x^2*(18*d^2 + 9*d
*e*x + 2*e^2*x^2) + a*b^3*(6*d^3 + 18*d^2*e*x - 27*d*e^2*x^2 - 4*e^3*x^3)) - 3*A*b*(-2*a^3*e^3 + 2*a^2*b*e^2*(
3*d + 2*e*x) + 3*a*b^2*e*(-2*d^2 - 2*d*e*x + e^2*x^2) + b^3*(2*d^3 - 6*d*e^2*x^2 - e^3*x^3)) + 6*(b*d - a*e)^2
*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)*Log[a + b*x])/(6*b^5*(a + b*x))

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Maple [B]  time = 0.01, size = 376, normalized size = 2.6 \begin{align*}{\frac{B{e}^{3}{x}^{3}}{3\,{b}^{2}}}+{\frac{{e}^{3}A{x}^{2}}{2\,{b}^{2}}}-{\frac{B{e}^{3}{x}^{2}a}{{b}^{3}}}+{\frac{3\,{e}^{2}B{x}^{2}d}{2\,{b}^{2}}}-2\,{\frac{aA{e}^{3}x}{{b}^{3}}}+3\,{\frac{A{e}^{2}dx}{{b}^{2}}}+3\,{\frac{{a}^{2}B{e}^{3}x}{{b}^{4}}}-6\,{\frac{aBd{e}^{2}x}{{b}^{3}}}+3\,{\frac{B{d}^{2}ex}{{b}^{2}}}+{\frac{A{a}^{3}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{A{a}^{2}d{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{A{d}^{2}ae}{{b}^{2} \left ( bx+a \right ) }}-{\frac{A{d}^{3}}{b \left ( bx+a \right ) }}-{\frac{B{e}^{3}{a}^{4}}{{b}^{5} \left ( bx+a \right ) }}+3\,{\frac{B{a}^{3}d{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{B{a}^{2}{d}^{2}e}{{b}^{3} \left ( bx+a \right ) }}+{\frac{aB{d}^{3}}{{b}^{2} \left ( bx+a \right ) }}+3\,{\frac{\ln \left ( bx+a \right ) A{a}^{2}{e}^{3}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a \right ) Aad{e}^{2}}{{b}^{3}}}+3\,{\frac{\ln \left ( bx+a \right ) A{d}^{2}e}{{b}^{2}}}-4\,{\frac{\ln \left ( bx+a \right ) B{e}^{3}{a}^{3}}{{b}^{5}}}+9\,{\frac{B\ln \left ( bx+a \right ){a}^{2}d{e}^{2}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a \right ) Ba{d}^{2}e}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) B{d}^{3}}{{b}^{2}}} \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^2*B*x^3+1/2*e^3/b^2*A*x^2-e^3/b^3*B*x^2*a+3/2*e^2/b^2*B*x^2*d-2*e^3/b^3*A*a*x+3*e^2/b^2*A*d*x+3*e^3/
b^4*a^2*B*x-6*e^2/b^3*B*a*d*x+3*e/b^2*B*d^2*x+1/b^4/(b*x+a)*A*a^3*e^3-3/b^3/(b*x+a)*A*a^2*d*e^2+3/b^2/(b*x+a)*
A*d^2*a*e-1/b/(b*x+a)*A*d^3-1/b^5/(b*x+a)*B*e^3*a^4+3/b^4/(b*x+a)*B*a^3*d*e^2-3/b^3/(b*x+a)*B*a^2*d^2*e+1/b^2/
(b*x+a)*B*a*d^3+3/b^4*ln(b*x+a)*A*a^2*e^3-6/b^3*ln(b*x+a)*A*a*d*e^2+3/b^2*ln(b*x+a)*A*d^2*e-4/b^5*ln(b*x+a)*B*
e^3*a^3+9/b^4*ln(b*x+a)*B*a^2*d*e^2-6/b^3*ln(b*x+a)*B*a*d^2*e+1/b^2*ln(b*x+a)*B*d^3

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Maxima [A]  time = 1.06651, size = 369, normalized size = 2.54 \begin{align*} \frac{{\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} -{\left (B a^{4} - A a^{3} b\right )} e^{3}}{b^{6} x + a b^{5}} + \frac{2 \, B b^{2} e^{3} x^{3} + 3 \,{\left (3 \, B b^{2} d e^{2} -{\left (2 \, B a b - A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B b^{2} d^{2} e - 3 \,{\left (2 \, B a b - A b^{2}\right )} d e^{2} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} x}{6 \, b^{4}} + \frac{{\left (B b^{3} d^{3} - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{2} -{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \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]

((B*a*b^3 - A*b^4)*d^3 - 3*(B*a^2*b^2 - A*a*b^3)*d^2*e + 3*(B*a^3*b - A*a^2*b^2)*d*e^2 - (B*a^4 - A*a^3*b)*e^3
)/(b^6*x + a*b^5) + 1/6*(2*B*b^2*e^3*x^3 + 3*(3*B*b^2*d*e^2 - (2*B*a*b - A*b^2)*e^3)*x^2 + 6*(3*B*b^2*d^2*e -
3*(2*B*a*b - A*b^2)*d*e^2 + (3*B*a^2 - 2*A*a*b)*e^3)*x)/b^4 + (B*b^3*d^3 - 3*(2*B*a*b^2 - A*b^3)*d^2*e + 3*(3*
B*a^2*b - 2*A*a*b^2)*d*e^2 - (4*B*a^3 - 3*A*a^2*b)*e^3)*log(b*x + a)/b^5

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Fricas [B]  time = 1.61226, size = 848, normalized size = 5.85 \begin{align*} \frac{2 \, B b^{4} e^{3} x^{4} + 6 \,{\left (B a b^{3} - A b^{4}\right )} d^{3} - 18 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 18 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 6 \,{\left (B a^{4} - A a^{3} b\right )} e^{3} +{\left (9 \, B b^{4} d e^{2} -{\left (4 \, B a b^{3} - 3 \, A b^{4}\right )} e^{3}\right )} x^{3} + 3 \,{\left (6 \, B b^{4} d^{2} e - 3 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d e^{2} +{\left (4 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B a b^{3} d^{2} e - 3 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} +{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \,{\left (B a b^{3} d^{3} - 3 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{2} -{\left (4 \, B a^{4} - 3 \, A a^{3} b\right )} e^{3} +{\left (B b^{4} d^{3} - 3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} -{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{6} x + a b^{5}\right )}} \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*b^4*e^3*x^4 + 6*(B*a*b^3 - A*b^4)*d^3 - 18*(B*a^2*b^2 - A*a*b^3)*d^2*e + 18*(B*a^3*b - A*a^2*b^2)*d*e
^2 - 6*(B*a^4 - A*a^3*b)*e^3 + (9*B*b^4*d*e^2 - (4*B*a*b^3 - 3*A*b^4)*e^3)*x^3 + 3*(6*B*b^4*d^2*e - 3*(3*B*a*b
^3 - 2*A*b^4)*d*e^2 + (4*B*a^2*b^2 - 3*A*a*b^3)*e^3)*x^2 + 6*(3*B*a*b^3*d^2*e - 3*(2*B*a^2*b^2 - A*a*b^3)*d*e^
2 + (3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x + 6*(B*a*b^3*d^3 - 3*(2*B*a^2*b^2 - A*a*b^3)*d^2*e + 3*(3*B*a^3*b - 2*A*a
^2*b^2)*d*e^2 - (4*B*a^4 - 3*A*a^3*b)*e^3 + (B*b^4*d^3 - 3*(2*B*a*b^3 - A*b^4)*d^2*e + 3*(3*B*a^2*b^2 - 2*A*a*
b^3)*d*e^2 - (4*B*a^3*b - 3*A*a^2*b^2)*e^3)*x)*log(b*x + a))/(b^6*x + a*b^5)

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Sympy [A]  time = 1.9666, size = 250, normalized size = 1.72 \begin{align*} \frac{B e^{3} x^{3}}{3 b^{2}} - \frac{- A a^{3} b e^{3} + 3 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e + A b^{4} d^{3} + B a^{4} e^{3} - 3 B a^{3} b d e^{2} + 3 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3}}{a b^{5} + b^{6} x} - \frac{x^{2} \left (- A b e^{3} + 2 B a e^{3} - 3 B b d e^{2}\right )}{2 b^{3}} + \frac{x \left (- 2 A a b e^{3} + 3 A b^{2} d e^{2} + 3 B a^{2} e^{3} - 6 B a b d e^{2} + 3 B b^{2} d^{2} e\right )}{b^{4}} - \frac{\left (a e - b d\right )^{2} \left (- 3 A b e + 4 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{5}} \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]

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

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Giac [B]  time = 1.1246, size = 381, normalized size = 2.63 \begin{align*} \frac{{\left (B b^{3} d^{3} - 6 \, B a b^{2} d^{2} e + 3 \, A b^{3} d^{2} e + 9 \, B a^{2} b d e^{2} - 6 \, A a b^{2} d e^{2} - 4 \, B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{2 \, B b^{4} x^{3} e^{3} + 9 \, B b^{4} d x^{2} e^{2} + 18 \, B b^{4} d^{2} x e - 6 \, B a b^{3} x^{2} e^{3} + 3 \, A b^{4} x^{2} e^{3} - 36 \, B a b^{3} d x e^{2} + 18 \, A b^{4} d x e^{2} + 18 \, B a^{2} b^{2} x e^{3} - 12 \, A a b^{3} x e^{3}}{6 \, b^{6}} + \frac{B a b^{3} d^{3} - A b^{4} d^{3} - 3 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} - 3 \, A a^{2} b^{2} d e^{2} - B a^{4} e^{3} + A a^{3} b e^{3}}{{\left (b x + a\right )} b^{5}} \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]

(B*b^3*d^3 - 6*B*a*b^2*d^2*e + 3*A*b^3*d^2*e + 9*B*a^2*b*d*e^2 - 6*A*a*b^2*d*e^2 - 4*B*a^3*e^3 + 3*A*a^2*b*e^3
)*log(abs(b*x + a))/b^5 + 1/6*(2*B*b^4*x^3*e^3 + 9*B*b^4*d*x^2*e^2 + 18*B*b^4*d^2*x*e - 6*B*a*b^3*x^2*e^3 + 3*
A*b^4*x^2*e^3 - 36*B*a*b^3*d*x*e^2 + 18*A*b^4*d*x*e^2 + 18*B*a^2*b^2*x*e^3 - 12*A*a*b^3*x*e^3)/b^6 + (B*a*b^3*
d^3 - A*b^4*d^3 - 3*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e + 3*B*a^3*b*d*e^2 - 3*A*a^2*b^2*d*e^2 - B*a^4*e^3 + A*a^
3*b*e^3)/((b*x + a)*b^5)