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

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 111, 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 {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} \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)^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 = 0.06, size = 120, normalized size = 1.08 \begin {gather*} \frac {e^4 \log (a+b x)}{b^5}-\frac {(b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )}{12 b^5 (a+b x)^4} \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)^3,x]

[Out]

-1/12*((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)))/(b^5*(a + b*x)^4) + (e^4*Log[a + b*x])/b^5

________________________________________________________________________________________

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 )^3} \, 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)^3,x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 267, normalized size = 2.41 \begin {gather*} -\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 {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)^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)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 174, normalized size = 1.57 \begin {gather*} \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 {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)^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)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 260, normalized size = 2.34 \begin {gather*} -\frac {a^{4} e^{4}}{4 \left (b x +a \right )^{4} b^{5}}+\frac {a^{3} d \,e^{3}}{\left (b x +a \right )^{4} b^{4}}-\frac {3 a^{2} d^{2} e^{2}}{2 \left (b x +a \right )^{4} b^{3}}+\frac {a \,d^{3} e}{\left (b x +a \right )^{4} b^{2}}-\frac {d^{4}}{4 \left (b x +a \right )^{4} b}+\frac {4 a^{3} e^{4}}{3 \left (b x +a \right )^{3} b^{5}}-\frac {4 a^{2} d \,e^{3}}{\left (b x +a \right )^{3} b^{4}}+\frac {4 a \,d^{2} e^{2}}{\left (b x +a \right )^{3} b^{3}}-\frac {4 d^{3} e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {3 a^{2} e^{4}}{\left (b x +a \right )^{2} b^{5}}+\frac {6 a d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {3 d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}+\frac {4 a \,e^{4}}{\left (b x +a \right ) b^{5}}-\frac {4 d \,e^{3}}{\left (b x +a \right ) b^{4}}+\frac {e^{4} \ln \left (b x +a \right )}{b^{5}} \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)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.59, size = 219, normalized size = 1.97 \begin {gather*} -\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 {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)^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

________________________________________________________________________________________

mupad [B]  time = 2.09, size = 213, normalized size = 1.92 \begin {gather*} \frac {e^4\,\ln \left (a+b\,x\right )}{b^5}-\frac {\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}{12\,b^5}+\frac {3\,x^2\,\left (-3\,a^2\,e^4+2\,a\,b\,d\,e^3+b^2\,d^2\,e^2\right )}{b^3}+\frac {2\,x\,\left (-11\,a^3\,e^4+6\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2+2\,b^3\,d^3\,e\right )}{3\,b^4}-\frac {4\,e^3\,x^3\,\left (a\,e-b\,d\right )}{b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \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)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 3.34, size = 230, normalized size = 2.07 \begin {gather*} \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 {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)**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

________________________________________________________________________________________