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

Optimal. Leaf size=102 \[ \frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}+\frac {e^3 x (4 b d-3 a e)}{b^4}+\frac {e^4 x^2}{2 b^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 102, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {e^3 x (4 b d-3 a e)}{b^4}+\frac {6 e^2 (b d-a e)^2 \log (a+b x)}{b^5}-\frac {4 e (b d-a e)^3}{b^5 (a+b x)}-\frac {(b d-a e)^4}{2 b^5 (a+b x)^2}+\frac {e^4 x^2}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 163, normalized size = 1.60 \begin {gather*} \frac {7 a^4 e^4+2 a^3 b e^3 (e x-10 d)+a^2 b^2 e^2 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+12 e^2 (a+b x)^2 (b d-a e)^2 \log (a+b x)+b^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )}{2 b^5 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^4*e^4 + 2*a^3*b*e^3*(-10*d + e*x) + a^2*b^2*e^2*(18*d^2 - 16*d*e*x - 11*e^2*x^2) - 4*a*b^3*e*(d^3 - 6*d^2
*e*x - 4*d*e^2*x^2 + e^3*x^3) + b^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(b*d - a*e)^2*(a + b*x
)^2*Log[a + b*x])/(2*b^5*(a + b*x)^2)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 292, normalized size = 2.86 \begin {gather*} \frac {b^{4} e^{4} x^{4} - b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 20 \, a^{3} b d e^{3} + 7 \, a^{4} e^{4} + 4 \, {\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + {\left (16 \, a b^{3} d e^{3} - 11 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (4 \, b^{4} d^{3} e - 12 \, a b^{3} d^{2} e^{2} + 8 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

1/2*(b^4*e^4*x^4 - b^4*d^4 - 4*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 20*a^3*b*d*e^3 + 7*a^4*e^4 + 4*(2*b^4*d*e^3
- a*b^3*e^4)*x^3 + (16*a*b^3*d*e^3 - 11*a^2*b^2*e^4)*x^2 - 2*(4*b^4*d^3*e - 12*a*b^3*d^2*e^2 + 8*a^2*b^2*d*e^3
- a^3*b*e^4)*x + 12*(a^2*b^2*d^2*e^2 - 2*a^3*b*d*e^3 + a^4*e^4 + (b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*
x^2 + 2*(a*b^3*d^2*e^2 - 2*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(b*x + a))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 172, normalized size = 1.69 \begin {gather*} \frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{2} e^{4} + 8 \, b^{3} d x e^{3} - 6 \, a b^{2} x e^{4}}{2 \, b^{6}} - \frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

6*(b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*log(abs(b*x + a))/b^5 + 1/2*(b^3*x^2*e^4 + 8*b^3*d*x*e^3 - 6*a*b^2*x*e
^4)/b^6 - 1/2*(b^4*d^4 + 4*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 7*a^4*e^4 + 8*(b^4*d^3*e - 3*a*
b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x)/((b*x + a)^2*b^5)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 245, normalized size = 2.40 \begin {gather*} -\frac {a^{4} e^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {2 a^{3} d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2} d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}+\frac {2 a \,d^{3} e}{\left (b x +a \right )^{2} b^{2}}-\frac {d^{4}}{2 \left (b x +a \right )^{2} b}+\frac {e^{4} x^{2}}{2 b^{3}}+\frac {4 a^{3} e^{4}}{\left (b x +a \right ) b^{5}}-\frac {12 a^{2} d \,e^{3}}{\left (b x +a \right ) b^{4}}+\frac {6 a^{2} e^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {12 a \,d^{2} e^{2}}{\left (b x +a \right ) b^{3}}-\frac {12 a d \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 a \,e^{4} x}{b^{4}}-\frac {4 d^{3} e}{\left (b x +a \right ) b^{2}}+\frac {6 d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {4 d \,e^{3} x}{b^{3}} \end {gather*}

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)^2,x)

[Out]

1/2*e^4*x^2/b^3-3*e^4/b^4*a*x+4*e^3/b^3*x*d-1/2/b^5/(b*x+a)^2*a^4*e^4+2/b^4/(b*x+a)^2*a^3*d*e^3-3/b^3/(b*x+a)^
2*a^2*d^2*e^2+2/b^2/(b*x+a)^2*a*d^3*e-1/2/b/(b*x+a)^2*d^4+6/b^5*e^4*ln(b*x+a)*a^2-12/b^4*e^3*ln(b*x+a)*a*d+6/b
^3*e^2*ln(b*x+a)*d^2+4/b^5*e^4/(b*x+a)*a^3-12/b^4*e^3/(b*x+a)*a^2*d+12/b^3*e^2/(b*x+a)*a*d^2-4/b^2*e/(b*x+a)*d
^3

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 190, normalized size = 1.86 \begin {gather*} -\frac {b^{4} d^{4} + 4 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 7 \, a^{4} e^{4} + 8 \, {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {b e^{4} x^{2} + 2 \, {\left (4 \, b d e^{3} - 3 \, a e^{4}\right )} x}{2 \, b^{4}} + \frac {6 \, {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

-1/2*(b^4*d^4 + 4*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 7*a^4*e^4 + 8*(b^4*d^3*e - 3*a*b^3*d^2*e
^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x)/(b^7*x^2 + 2*a*b^6*x + a^2*b^5) + 1/2*(b*e^4*x^2 + 2*(4*b*d*e^3 - 3*a*e^4
)*x)/b^4 + 6*(b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*log(b*x + a)/b^5

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 197, normalized size = 1.93 \begin {gather*} \frac {e^4\,x^2}{2\,b^3}-x\,\left (\frac {3\,a\,e^4}{b^4}-\frac {4\,d\,e^3}{b^3}\right )-\frac {\frac {-7\,a^4\,e^4+20\,a^3\,b\,d\,e^3-18\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4}{2\,b}-x\,\left (4\,a^3\,e^4-12\,a^2\,b\,d\,e^3+12\,a\,b^2\,d^2\,e^2-4\,b^3\,d^3\,e\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\frac {\ln \left (a+b\,x\right )\,\left (6\,a^2\,e^4-12\,a\,b\,d\,e^3+6\,b^2\,d^2\,e^2\right )}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(d + e*x)^4)/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

(e^4*x^2)/(2*b^3) - x*((3*a*e^4)/b^4 - (4*d*e^3)/b^3) - ((b^4*d^4 - 7*a^4*e^4 - 18*a^2*b^2*d^2*e^2 + 4*a*b^3*d
^3*e + 20*a^3*b*d*e^3)/(2*b) - x*(4*a^3*e^4 - 4*b^3*d^3*e + 12*a*b^2*d^2*e^2 - 12*a^2*b*d*e^3))/(a^2*b^4 + b^6
*x^2 + 2*a*b^5*x) + (log(a + b*x)*(6*a^2*e^4 + 6*b^2*d^2*e^2 - 12*a*b*d*e^3))/b^5

________________________________________________________________________________________

sympy [A]  time = 1.24, size = 185, normalized size = 1.81 \begin {gather*} x \left (- \frac {3 a e^{4}}{b^{4}} + \frac {4 d e^{3}}{b^{3}}\right ) + \frac {7 a^{4} e^{4} - 20 a^{3} b d e^{3} + 18 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e - b^{4} d^{4} + x \left (8 a^{3} b e^{4} - 24 a^{2} b^{2} d e^{3} + 24 a b^{3} d^{2} e^{2} - 8 b^{4} d^{3} e\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {e^{4} x^{2}}{2 b^{3}} + \frac {6 e^{2} \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}

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)**2,x)

[Out]

x*(-3*a*e**4/b**4 + 4*d*e**3/b**3) + (7*a**4*e**4 - 20*a**3*b*d*e**3 + 18*a**2*b**2*d**2*e**2 - 4*a*b**3*d**3*
e - b**4*d**4 + x*(8*a**3*b*e**4 - 24*a**2*b**2*d*e**3 + 24*a*b**3*d**2*e**2 - 8*b**4*d**3*e))/(2*a**2*b**5 +
4*a*b**6*x + 2*b**7*x**2) + e**4*x**2/(2*b**3) + 6*e**2*(a*e - b*d)**2*log(a + b*x)/b**5

________________________________________________________________________________________