3.1478 \(\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^9} \, dx\)

Optimal. Leaf size=117 \[ \frac{4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac{b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac{4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac{(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac{b^4}{4 e^5 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0713198, antiderivative size = 117, normalized size of antiderivative = 1., 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, 43} \[ \frac{4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac{b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac{4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac{(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac{b^4}{4 e^5 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^9,x]

[Out]

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

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{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^9} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^9}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac{b^4}{e^4 (d+e x)^5}\right ) \, dx\\ &=-\frac{(b d-a e)^4}{8 e^5 (d+e x)^8}+\frac{4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac{b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac{4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac{b^4}{4 e^5 (d+e x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0466586, size = 144, normalized size = 1.23 \[ -\frac{10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+20 a^3 b e^3 (d+8 e x)+35 a^4 e^4+4 a b^3 e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )}{280 e^5 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^9,x]

[Out]

-(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2
*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4))/(280*e
^5*(d + e*x)^8)

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Maple [A]  time = 0.046, size = 186, normalized size = 1.6 \begin{align*} -{\frac{{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{8\,{e}^{5} \left ( ex+d \right ) ^{8}}}-{\frac{4\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{4\,{b}^{3} \left ( ae-bd \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x)

[Out]

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

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Maxima [B]  time = 1.15478, size = 348, normalized size = 2.97 \begin{align*} -\frac{70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \,{\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \,{\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x, algorithm="maxima")

[Out]

-1/280*(70*b^4*e^4*x^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*
d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*
e^2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(e^13*x^8 + 8*d*e^12*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^
4*e^9*x^4 + 56*d^5*e^8*x^3 + 28*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

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Fricas [B]  time = 1.65301, size = 540, normalized size = 4.62 \begin{align*} -\frac{70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \,{\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \,{\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/280*(70*b^4*e^4*x^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*
d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*
e^2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(e^13*x^8 + 8*d*e^12*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^
4*e^9*x^4 + 56*d^5*e^8*x^3 + 28*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

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Sympy [B]  time = 23.1461, size = 275, normalized size = 2.35 \begin{align*} - \frac{35 a^{4} e^{4} + 20 a^{3} b d e^{3} + 10 a^{2} b^{2} d^{2} e^{2} + 4 a b^{3} d^{3} e + b^{4} d^{4} + 70 b^{4} e^{4} x^{4} + x^{3} \left (224 a b^{3} e^{4} + 56 b^{4} d e^{3}\right ) + x^{2} \left (280 a^{2} b^{2} e^{4} + 112 a b^{3} d e^{3} + 28 b^{4} d^{2} e^{2}\right ) + x \left (160 a^{3} b e^{4} + 80 a^{2} b^{2} d e^{3} + 32 a b^{3} d^{2} e^{2} + 8 b^{4} d^{3} e\right )}{280 d^{8} e^{5} + 2240 d^{7} e^{6} x + 7840 d^{6} e^{7} x^{2} + 15680 d^{5} e^{8} x^{3} + 19600 d^{4} e^{9} x^{4} + 15680 d^{3} e^{10} x^{5} + 7840 d^{2} e^{11} x^{6} + 2240 d e^{12} x^{7} + 280 e^{13} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**9,x)

[Out]

-(35*a**4*e**4 + 20*a**3*b*d*e**3 + 10*a**2*b**2*d**2*e**2 + 4*a*b**3*d**3*e + b**4*d**4 + 70*b**4*e**4*x**4 +
x**3*(224*a*b**3*e**4 + 56*b**4*d*e**3) + x**2*(280*a**2*b**2*e**4 + 112*a*b**3*d*e**3 + 28*b**4*d**2*e**2) +
x*(160*a**3*b*e**4 + 80*a**2*b**2*d*e**3 + 32*a*b**3*d**2*e**2 + 8*b**4*d**3*e))/(280*d**8*e**5 + 2240*d**7*e
**6*x + 7840*d**6*e**7*x**2 + 15680*d**5*e**8*x**3 + 19600*d**4*e**9*x**4 + 15680*d**3*e**10*x**5 + 7840*d**2*
e**11*x**6 + 2240*d*e**12*x**7 + 280*e**13*x**8)

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Giac [A]  time = 1.09208, size = 235, normalized size = 2.01 \begin{align*} -\frac{{\left (70 \, b^{4} x^{4} e^{4} + 56 \, b^{4} d x^{3} e^{3} + 28 \, b^{4} d^{2} x^{2} e^{2} + 8 \, b^{4} d^{3} x e + b^{4} d^{4} + 224 \, a b^{3} x^{3} e^{4} + 112 \, a b^{3} d x^{2} e^{3} + 32 \, a b^{3} d^{2} x e^{2} + 4 \, a b^{3} d^{3} e + 280 \, a^{2} b^{2} x^{2} e^{4} + 80 \, a^{2} b^{2} d x e^{3} + 10 \, a^{2} b^{2} d^{2} e^{2} + 160 \, a^{3} b x e^{4} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{280 \,{\left (x e + d\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(70*b^4*x^4*e^4 + 56*b^4*d*x^3*e^3 + 28*b^4*d^2*x^2*e^2 + 8*b^4*d^3*x*e + b^4*d^4 + 224*a*b^3*x^3*e^4 +
112*a*b^3*d*x^2*e^3 + 32*a*b^3*d^2*x*e^2 + 4*a*b^3*d^3*e + 280*a^2*b^2*x^2*e^4 + 80*a^2*b^2*d*x*e^3 + 10*a^2*
b^2*d^2*e^2 + 160*a^3*b*x*e^4 + 20*a^3*b*d*e^3 + 35*a^4*e^4)*e^(-5)/(x*e + d)^8