3.1946 \(\int \frac{(a+b x) (d+e x)^3}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(d+e x)^4}{4 (a+b x)^4 (b d-a e)} \]

[Out]

-(d + e*x)^4/(4*(b*d - a*e)*(a + b*x)^4)

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Rubi [A]  time = 0.0045802, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 37} \[ -\frac{(d+e x)^4}{4 (a+b x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d + e*x)^4/(4*(b*d - a*e)*(a + b*x)^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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [B]  time = 0.0310323, size = 91, normalized size = 3.25 \[ -\frac{a^2 b e^2 (d+4 e x)+a^3 e^3+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{4 b^4 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.005, size = 122, normalized size = 4.4 \begin{align*} -{\frac{{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-{\frac{e \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{3\,{e}^{2} \left ( ae-bd \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{-{e}^{3}{a}^{3}+3\,d{e}^{2}{a}^{2}b-3\,a{d}^{2}e{b}^{2}+{d}^{3}{b}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}} \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)^3,x)

[Out]

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

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Maxima [B]  time = 1.00468, size = 193, normalized size = 6.89 \begin{align*} -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\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)^3,x, algorithm="maxima")

[Out]

-1/4*(4*b^3*e^3*x^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2 + a^3*e^3 + 6*(b^3*d*e^2 + a*b^2*e^3)*x^2 + 4*(b^3*d
^2*e + a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 + 4*a^3*b^5*x + a^4*b^4)

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Fricas [B]  time = 1.43656, size = 284, normalized size = 10.14 \begin{align*} -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\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)^3,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*e^3*x^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2 + a^3*e^3 + 6*(b^3*d*e^2 + a*b^2*e^3)*x^2 + 4*(b^3*d
^2*e + a*b^2*d*e^2 + a^2*b*e^3)*x)/(b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 + 4*a^3*b^5*x + a^4*b^4)

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Sympy [B]  time = 1.87146, size = 153, normalized size = 5.46 \begin{align*} - \frac{a^{3} e^{3} + a^{2} b d e^{2} + a b^{2} d^{2} e + b^{3} d^{3} + 4 b^{3} e^{3} x^{3} + x^{2} \left (6 a b^{2} e^{3} + 6 b^{3} d e^{2}\right ) + x \left (4 a^{2} b e^{3} + 4 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{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)**3,x)

[Out]

-(a**3*e**3 + a**2*b*d*e**2 + a*b**2*d**2*e + b**3*d**3 + 4*b**3*e**3*x**3 + x**2*(6*a*b**2*e**3 + 6*b**3*d*e*
*2) + x*(4*a**2*b*e**3 + 4*a*b**2*d*e**2 + 4*b**3*d**2*e))/(4*a**4*b**4 + 16*a**3*b**5*x + 24*a**2*b**6*x**2 +
16*a*b**7*x**3 + 4*b**8*x**4)

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Giac [B]  time = 1.11751, size = 143, normalized size = 5.11 \begin{align*} -\frac{4 \, b^{3} x^{3} e^{3} + 6 \, b^{3} d x^{2} e^{2} + 4 \, b^{3} d^{2} x e + b^{3} d^{3} + 6 \, a b^{2} x^{2} e^{3} + 4 \, a b^{2} d x e^{2} + a b^{2} d^{2} e + 4 \, a^{2} b x e^{3} + a^{2} b d e^{2} + a^{3} e^{3}}{4 \,{\left (b x + a\right )}^{4} 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)^3,x, algorithm="giac")

[Out]

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