3.16.17 \(\int \frac {(d+e x)^4}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [1517]

Optimal. Leaf size=103 \[ \frac {e^4 x}{b^4}-\frac {(b d-a e)^4}{3 b^5 (a+b x)^3}-\frac {2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac {6 e^2 (b d-a e)^2}{b^5 (a+b x)}+\frac {4 e^3 (b d-a e) \log (a+b x)}{b^5} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \begin {gather*} \frac {4 e^3 (b d-a e) \log (a+b x)}{b^5}-\frac {6 e^2 (b d-a e)^2}{b^5 (a+b x)}-\frac {2 e (b d-a e)^3}{b^5 (a+b x)^2}-\frac {(b d-a e)^4}{3 b^5 (a+b x)^3}+\frac {e^4 x}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*x)/b^4 - (b*d - a*e)^4/(3*b^5*(a + b*x)^3) - (2*e*(b*d - a*e)^3)/(b^5*(a + b*x)^2) - (6*e^2*(b*d - a*e)^2
)/(b^5*(a + b*x)) + (4*e^3*(b*d - a*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 45

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

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Mathematica [A]
time = 0.04, size = 166, normalized size = 1.61 \begin {gather*} \frac {-13 a^4 e^4+a^3 b e^3 (22 d-27 e x)-3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a b^3 e \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-b^4 \left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )-12 e^3 (-b d+a e) (a+b x)^3 \log (a+b x)}{3 b^5 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*a^4*e^4 + a^3*b*e^3*(22*d - 27*e*x) - 3*a^2*b^2*e^2*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*b^3*e*(-2*d^3 - 18
*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - b^4*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4*x^4) - 12*e^3*(-(b*d) + a
*e)*(a + b*x)^3*Log[a + b*x])/(3*b^5*(a + b*x)^3)

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Maple [A]
time = 0.63, size = 178, normalized size = 1.73

method	result	size
default	\(\frac {e^{4} x}{b^{4}}-\frac {6 e^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{5} \left (b x +a \right )}+\frac {2 e \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{5} \left (b x +a \right )^{2}}-\frac {4 e^{3} \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{5}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{3 b^{5} \left (b x +a \right )^{3}}\)	\(178\)
norman	\(\frac {\frac {e^{4} x^{4}}{b}-\frac {22 e^{4} a^{4}-22 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}}{3 b^{5}}-\frac {3 \left (4 a^{2} e^{4}-4 a b d \,e^{3}+2 b^{2} d^{2} e^{2}\right ) x^{2}}{b^{3}}-\frac {\left (18 e^{4} a^{3}-18 a^{2} b d \,e^{3}+6 a \,b^{2} d^{2} e^{2}+2 b^{3} d^{3} e \right ) x}{b^{4}}}{\left (b x +a \right )^{3}}-\frac {4 e^{3} \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{5}}\)	\(180\)
risch	\(\frac {e^{4} x}{b^{4}}+\frac {\left (-6 a^{2} b \,e^{4}+12 a \,b^{2} d \,e^{3}-6 d^{2} e^{2} b^{3}\right ) x^{2}-2 e \left (5 e^{3} a^{3}-9 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x -\frac {13 e^{4} a^{4}-22 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}}{3 b}}{b^{4} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}-\frac {4 e^{4} \ln \left (b x +a \right ) a}{b^{5}}+\frac {4 e^{3} \ln \left (b x +a \right ) d}{b^{4}}\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 191, normalized size = 1.85 \begin {gather*} -\frac {b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {x e^{4}}{b^{4}} + \frac {4 \, {\left (b d e^{3} - a 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((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-1/3*(b^4*d^4 + 2*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 22*a^3*b*d*e^3 + 13*a^4*e^4 + 18*(b^4*d^2*e^2 - 2*a*b^3*d*
e^3 + a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x)/(b^8*x^3 + 3*a*b^7
*x^2 + 3*a^2*b^6*x + a^3*b^5) + x*e^4/b^4 + 4*(b*d*e^3 - a*e^4)*log(b*x + a)/b^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (103) = 206\).
time = 2.65, size = 261, normalized size = 2.53 \begin {gather*} -\frac {b^{4} d^{4} - {\left (3 \, b^{4} x^{4} + 9 \, a b^{3} x^{3} - 9 \, a^{2} b^{2} x^{2} - 27 \, a^{3} b x - 13 \, a^{4}\right )} e^{4} - 2 \, {\left (18 \, a b^{3} d x^{2} + 27 \, a^{2} b^{2} d x + 11 \, a^{3} b d\right )} e^{3} + 6 \, {\left (3 \, b^{4} d^{2} x^{2} + 3 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} + 2 \, {\left (3 \, b^{4} d^{3} x + a b^{3} d^{3}\right )} e + 12 \, {\left ({\left (a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 3 \, a^{3} b x + a^{4}\right )} e^{4} - {\left (b^{4} d x^{3} + 3 \, a b^{3} d x^{2} + 3 \, a^{2} b^{2} d x + a^{3} b d\right )} e^{3}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b^4*d^4 - (3*b^4*x^4 + 9*a*b^3*x^3 - 9*a^2*b^2*x^2 - 27*a^3*b*x - 13*a^4)*e^4 - 2*(18*a*b^3*d*x^2 + 27*a
^2*b^2*d*x + 11*a^3*b*d)*e^3 + 6*(3*b^4*d^2*x^2 + 3*a*b^3*d^2*x + a^2*b^2*d^2)*e^2 + 2*(3*b^4*d^3*x + a*b^3*d^
3)*e + 12*((a*b^3*x^3 + 3*a^2*b^2*x^2 + 3*a^3*b*x + a^4)*e^4 - (b^4*d*x^3 + 3*a*b^3*d*x^2 + 3*a^2*b^2*d*x + a^
3*b*d)*e^3)*log(b*x + a))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (94) = 188\).
time = 1.19, size = 209, normalized size = 2.03 \begin {gather*} \frac {- 13 a^{4} e^{4} + 22 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 2 a b^{3} d^{3} e - b^{4} d^{4} + x^{2} \left (- 18 a^{2} b^{2} e^{4} + 36 a b^{3} d e^{3} - 18 b^{4} d^{2} e^{2}\right ) + x \left (- 30 a^{3} b e^{4} + 54 a^{2} b^{2} d e^{3} - 18 a b^{3} d^{2} e^{2} - 6 b^{4} d^{3} e\right )}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac {e^{4} x}{b^{4}} - \frac {4 e^{3} \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-13*a**4*e**4 + 22*a**3*b*d*e**3 - 6*a**2*b**2*d**2*e**2 - 2*a*b**3*d**3*e - b**4*d**4 + x**2*(-18*a**2*b**2*
e**4 + 36*a*b**3*d*e**3 - 18*b**4*d**2*e**2) + x*(-30*a**3*b*e**4 + 54*a**2*b**2*d*e**3 - 18*a*b**3*d**2*e**2
- 6*b**4*d**3*e))/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + e**4*x/b**4 - 4*e**3*(a*e - b*
d)*log(a + b*x)/b**5

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Giac [A]
time = 1.81, size = 167, normalized size = 1.62 \begin {gather*} \frac {x e^{4}}{b^{4}} + \frac {4 \, {\left (b d e^{3} - a e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 22 \, a^{3} b d e^{3} + 13 \, a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^4/b^4 + 4*(b*d*e^3 - a*e^4)*log(abs(b*x + a))/b^5 - 1/3*(b^4*d^4 + 2*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 22*
a^3*b*d*e^3 + 13*a^4*e^4 + 18*(b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 3*a*b^3*d^2*e^2
- 9*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x)/((b*x + a)^3*b^5)

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Mupad [B]
time = 0.66, size = 204, normalized size = 1.98 \begin {gather*} \frac {e^4\,x}{b^4}-\frac {\ln \left (a+b\,x\right )\,\left (4\,a\,e^4-4\,b\,d\,e^3\right )}{b^5}-\frac {\frac {13\,a^4\,e^4-22\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+b^4\,d^4}{3\,b}+x\,\left (10\,a^3\,e^4-18\,a^2\,b\,d\,e^3+6\,a\,b^2\,d^2\,e^2+2\,b^3\,d^3\,e\right )+x^2\,\left (6\,a^2\,b\,e^4-12\,a\,b^2\,d\,e^3+6\,b^3\,d^2\,e^2\right )}{a^3\,b^4+3\,a^2\,b^5\,x+3\,a\,b^6\,x^2+b^7\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^4*x)/b^4 - (log(a + b*x)*(4*a*e^4 - 4*b*d*e^3))/b^5 - ((13*a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 + 2*a*b^3*
d^3*e - 22*a^3*b*d*e^3)/(3*b) + x*(10*a^3*e^4 + 2*b^3*d^3*e + 6*a*b^2*d^2*e^2 - 18*a^2*b*d*e^3) + x^2*(6*a^2*b
*e^4 + 6*b^3*d^2*e^2 - 12*a*b^2*d*e^3))/(a^3*b^4 + b^7*x^3 + 3*a^2*b^5*x + 3*a*b^6*x^2)

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