3.16.29 \(\int \frac {(d+e x)^4}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1529]

Optimal. Leaf size=28 \[ -\frac {(d+e x)^5}{5 (b d-a e) (a+b x)^5} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 37} \begin {gather*} -\frac {(d+e x)^5}{5 (a+b x)^5 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(d + e*x)^5/((b*d - a*e)*(a + b*x)^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 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 {(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)^6} \, dx\\ &=-\frac {(d+e x)^5}{5 (b d-a e) (a+b x)^5}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(28)=56\).
time = 0.04, size = 140, normalized size = 5.00 \begin {gather*} -\frac {a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a b^3 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{5 b^5 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + a*b^3*e*(d^3 + 5*d^2*e*x +
10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(b^5*(a + b*x)
^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(26)=52\).
time = 0.72, size = 185, normalized size = 6.61

method	result	size
risch	\(\frac {-\frac {e^{4} x^{4}}{b}-\frac {2 e^{3} \left (a e +b d \right ) x^{3}}{b^{2}}-\frac {2 e^{2} \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x^{2}}{b^{3}}-\frac {e \left (e^{3} a^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{b^{4}}-\frac {e^{4} a^{4}+a^{3} b d \,e^{3}+a^{2} b^{2} d^{2} e^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 b^{5}}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}\)	\(179\)
norman	\(\frac {-\frac {e^{4} x^{4}}{b}+\frac {2 \left (-e^{4} a -b d \,e^{3}\right ) x^{3}}{b^{2}}+\frac {2 \left (-a^{2} e^{4}-a b d \,e^{3}-b^{2} d^{2} e^{2}\right ) x^{2}}{b^{3}}+\frac {\left (-e^{4} a^{3}-d \,e^{3} a^{2} b -a \,d^{2} e^{2} b^{2}-b^{3} d^{3} e \right ) x}{b^{4}}+\frac {-e^{4} a^{4}-a^{3} b d \,e^{3}-a^{2} b^{2} d^{2} e^{2}-a \,b^{3} d^{3} e -b^{4} d^{4}}{5 b^{5}}}{\left (b x +a \right )^{5}}\)	\(180\)
default	\(-\frac {e^{4}}{b^{5} \left (b x +a \right )}+\frac {2 e^{3} \left (a e -b d \right )}{b^{5} \left (b x +a \right )^{2}}-\frac {2 e^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{5} \left (b x +a \right )^{3}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{5 b^{5} \left (b x +a \right )^{5}}+\frac {e \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{5} \left (b x +a \right )^{4}}\)	\(185\)
gosper	\(-\frac {5 e^{4} x^{4} b^{4}+10 a \,b^{3} e^{4} x^{3}+10 b^{4} d \,e^{3} x^{3}+10 a^{2} b^{2} e^{4} x^{2}+10 a \,b^{3} d \,e^{3} x^{2}+10 b^{4} d^{2} e^{2} x^{2}+5 a^{3} b \,e^{4} x +5 a^{2} b^{2} d \,e^{3} x +5 a \,b^{3} d^{2} e^{2} x +5 b^{4} d^{3} e x +e^{4} a^{4}+a^{3} b d \,e^{3}+a^{2} b^{2} d^{2} e^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}}{5 \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} b^{5}}\)	\(199\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (28) = 56\).
time = 0.28, size = 205, normalized size = 7.32 \begin {gather*} -\frac {5 \, b^{4} x^{4} e^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \, {\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} 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)^3,x, algorithm="maxima")

[Out]

-1/5*(5*b^4*x^4*e^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^4*e^4 + 10*(b^4*d*e^3 + a*b^3*
e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a
^3*b*e^4)*x)/(b^10*x^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (28) = 56\).
time = 2.83, size = 204, normalized size = 7.29 \begin {gather*} -\frac {b^{4} d^{4} + {\left (5 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x + a^{4}\right )} e^{4} + {\left (10 \, b^{4} d x^{3} + 10 \, a b^{3} d x^{2} + 5 \, a^{2} b^{2} d x + a^{3} b d\right )} e^{3} + {\left (10 \, b^{4} d^{2} x^{2} + 5 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} + {\left (5 \, b^{4} d^{3} x + a b^{3} d^{3}\right )} e}{5 \, {\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} 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)^3,x, algorithm="fricas")

[Out]

-1/5*(b^4*d^4 + (5*b^4*x^4 + 10*a*b^3*x^3 + 10*a^2*b^2*x^2 + 5*a^3*b*x + a^4)*e^4 + (10*b^4*d*x^3 + 10*a*b^3*d
*x^2 + 5*a^2*b^2*d*x + a^3*b*d)*e^3 + (10*b^4*d^2*x^2 + 5*a*b^3*d^2*x + a^2*b^2*d^2)*e^2 + (5*b^4*d^3*x + a*b^
3*d^3)*e)/(b^10*x^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (22) = 44\).
time = 5.12, size = 236, normalized size = 8.43 \begin {gather*} \frac {- a^{4} e^{4} - a^{3} b d e^{3} - a^{2} b^{2} d^{2} e^{2} - a b^{3} d^{3} e - b^{4} d^{4} - 5 b^{4} e^{4} x^{4} + x^{3} \left (- 10 a b^{3} e^{4} - 10 b^{4} d e^{3}\right ) + x^{2} \left (- 10 a^{2} b^{2} e^{4} - 10 a b^{3} d e^{3} - 10 b^{4} d^{2} e^{2}\right ) + x \left (- 5 a^{3} b e^{4} - 5 a^{2} b^{2} d e^{3} - 5 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{5 a^{5} b^{5} + 25 a^{4} b^{6} x + 50 a^{3} b^{7} x^{2} + 50 a^{2} b^{8} x^{3} + 25 a b^{9} x^{4} + 5 b^{10} x^{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)**3,x)

[Out]

(-a**4*e**4 - a**3*b*d*e**3 - a**2*b**2*d**2*e**2 - a*b**3*d**3*e - b**4*d**4 - 5*b**4*e**4*x**4 + x**3*(-10*a
*b**3*e**4 - 10*b**4*d*e**3) + x**2*(-10*a**2*b**2*e**4 - 10*a*b**3*d*e**3 - 10*b**4*d**2*e**2) + x*(-5*a**3*b
*e**4 - 5*a**2*b**2*d*e**3 - 5*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(5*a**5*b**5 + 25*a**4*b**6*x + 50*a**3*b**7
*x**2 + 50*a**2*b**8*x**3 + 25*a*b**9*x**4 + 5*b**10*x**5)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (28) = 56\).
time = 1.56, size = 170, normalized size = 6.07 \begin {gather*} -\frac {5 \, b^{4} x^{4} e^{4} + 10 \, b^{4} d x^{3} e^{3} + 10 \, b^{4} d^{2} x^{2} e^{2} + 5 \, b^{4} d^{3} x e + b^{4} d^{4} + 10 \, a b^{3} x^{3} e^{4} + 10 \, a b^{3} d x^{2} e^{3} + 5 \, a b^{3} d^{2} x e^{2} + a b^{3} d^{3} e + 10 \, a^{2} b^{2} x^{2} e^{4} + 5 \, a^{2} b^{2} d x e^{3} + a^{2} b^{2} d^{2} e^{2} + 5 \, a^{3} b x e^{4} + a^{3} b d e^{3} + a^{4} e^{4}}{5 \, {\left (b x + a\right )}^{5} 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)^3,x, algorithm="giac")

[Out]

-1/5*(5*b^4*x^4*e^4 + 10*b^4*d*x^3*e^3 + 10*b^4*d^2*x^2*e^2 + 5*b^4*d^3*x*e + b^4*d^4 + 10*a*b^3*x^3*e^4 + 10*
a*b^3*d*x^2*e^3 + 5*a*b^3*d^2*x*e^2 + a*b^3*d^3*e + 10*a^2*b^2*x^2*e^4 + 5*a^2*b^2*d*x*e^3 + a^2*b^2*d^2*e^2 +
5*a^3*b*x*e^4 + a^3*b*d*e^3 + a^4*e^4)/((b*x + a)^5*b^5)

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Mupad [B]
time = 0.08, size = 203, normalized size = 7.25 \begin {gather*} -\frac {\frac {a^4\,e^4+a^3\,b\,d\,e^3+a^2\,b^2\,d^2\,e^2+a\,b^3\,d^3\,e+b^4\,d^4}{5\,b^5}+\frac {e^4\,x^4}{b}+\frac {2\,e^3\,x^3\,\left (a\,e+b\,d\right )}{b^2}+\frac {e\,x\,\left (a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3\right )}{b^4}+\frac {2\,e^2\,x^2\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \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)^3,x)

[Out]

-((a^4*e^4 + b^4*d^4 + a^2*b^2*d^2*e^2 + a*b^3*d^3*e + a^3*b*d*e^3)/(5*b^5) + (e^4*x^4)/b + (2*e^3*x^3*(a*e +
b*d))/b^2 + (e*x*(a^3*e^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2))/b^4 + (2*e^2*x^2*(a^2*e^2 + b^2*d^2 + a*b*d*
e))/b^3)/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)

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