3.1945 \(\int \frac{(a+b x) (d+e x)^4}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac{3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac{4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac{(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac{e^4 \log (a+b x)}{b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.099135, antiderivative size = 111, 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} \[ -\frac{4 e^3 (b d-a e)}{b^5 (a+b x)}-\frac{3 e^2 (b d-a e)^2}{b^5 (a+b x)^2}-\frac{4 e (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac{(b d-a e)^4}{4 b^5 (a+b x)^4}+\frac{e^4 \log (a+b x)}{b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.064781, size = 120, normalized size = 1.08 \[ \frac{e^4 \log (a+b x)}{b^5}-\frac{(b d-a e) \left (a^2 b e^2 (13 d+88 e x)+25 a^3 e^3+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (16 d^2 e x+3 d^3+36 d e^2 x^2+48 e^3 x^3\right )\right )}{12 b^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d - a*e)*(25*a^3*e^3 + a^2*b*e^2*(13*d + 88*e*x) + a*b^2*e*(7*d^2 + 40*d*e*x + 108*e^2*x^2) + b^3*(3*d^3
+ 16*d^2*e*x + 36*d*e^2*x^2 + 48*e^3*x^3)))/(12*b^5*(a + b*x)^4) + (e^4*Log[a + b*x])/b^5

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 260, normalized size = 2.3 \begin{align*} 4\,{\frac{{e}^{4}a}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{d{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+{\frac{4\,{e}^{4}{a}^{3}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}-4\,{\frac{d{e}^{3}{a}^{2}}{{b}^{4} \left ( bx+a \right ) ^{3}}}+4\,{\frac{a{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{4\,{d}^{3}e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-3\,{\frac{{e}^{4}{a}^{2}}{{b}^{5} \left ( bx+a \right ) ^{2}}}+6\,{\frac{d{e}^{3}a}{{b}^{4} \left ( bx+a \right ) ^{2}}}-3\,{\frac{{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{4}\ln \left ( bx+a \right ) }{{b}^{5}}}-{\frac{{a}^{4}{e}^{4}}{4\,{b}^{5} \left ( bx+a \right ) ^{4}}}+{\frac{d{e}^{3}{a}^{3}}{{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{3\,{d}^{2}{e}^{2}{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{a{d}^{3}e}{{b}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{4}}{4\,b \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)^4/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.990153, size = 296, normalized size = 2.67 \begin{align*} -\frac{3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac{e^{4} \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)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/12*(3*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 - 25*a^4*e^4 + 48*(b^4*d*e^3 - a*b^3*e^4
)*x^3 + 36*(b^4*d^2*e^2 + 2*a*b^3*d*e^3 - 3*a^2*b^2*e^4)*x^2 + 8*(2*b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*
e^3 - 11*a^3*b*e^4)*x)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + e^4*log(b*x + a)/b^5

________________________________________________________________________________________

Fricas [B]  time = 1.47541, size = 545, normalized size = 4.91 \begin{align*} -\frac{3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} - 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (2 \, b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 11 \, a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/12*(3*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 - 25*a^4*e^4 + 48*(b^4*d*e^3 - a*b^3*e^4
)*x^3 + 36*(b^4*d^2*e^2 + 2*a*b^3*d*e^3 - 3*a^2*b^2*e^4)*x^2 + 8*(2*b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*
e^3 - 11*a^3*b*e^4)*x - 12*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*log(b
*x + a))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)

________________________________________________________________________________________

Sympy [B]  time = 3.86187, size = 230, normalized size = 2.07 \begin{align*} \frac{25 a^{4} e^{4} - 12 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - 3 b^{4} d^{4} + x^{3} \left (48 a b^{3} e^{4} - 48 b^{4} d e^{3}\right ) + x^{2} \left (108 a^{2} b^{2} e^{4} - 72 a b^{3} d e^{3} - 36 b^{4} d^{2} e^{2}\right ) + x \left (88 a^{3} b e^{4} - 48 a^{2} b^{2} d e^{3} - 24 a b^{3} d^{2} e^{2} - 16 b^{4} d^{3} e\right )}{12 a^{4} b^{5} + 48 a^{3} b^{6} x + 72 a^{2} b^{7} x^{2} + 48 a b^{8} x^{3} + 12 b^{9} x^{4}} + \frac{e^{4} \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)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(25*a**4*e**4 - 12*a**3*b*d*e**3 - 6*a**2*b**2*d**2*e**2 - 4*a*b**3*d**3*e - 3*b**4*d**4 + x**3*(48*a*b**3*e**
4 - 48*b**4*d*e**3) + x**2*(108*a**2*b**2*e**4 - 72*a*b**3*d*e**3 - 36*b**4*d**2*e**2) + x*(88*a**3*b*e**4 - 4
8*a**2*b**2*d*e**3 - 24*a*b**3*d**2*e**2 - 16*b**4*d**3*e))/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2
+ 48*a*b**8*x**3 + 12*b**9*x**4) + e**4*log(a + b*x)/b**5

________________________________________________________________________________________

Giac [A]  time = 1.1178, size = 235, normalized size = 2.12 \begin{align*} \frac{e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{48 \,{\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \,{\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \,{\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac{3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \,{\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)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

e^4*log(abs(b*x + a))/b^5 - 1/12*(48*(b^3*d*e^3 - a*b^2*e^4)*x^3 + 36*(b^3*d^2*e^2 + 2*a*b^2*d*e^3 - 3*a^2*b*e
^4)*x^2 + 8*(2*b^3*d^3*e + 3*a*b^2*d^2*e^2 + 6*a^2*b*d*e^3 - 11*a^3*e^4)*x + (3*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^
2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 - 25*a^4*e^4)/b)/((b*x + a)^4*b^4)