3.12.82 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=119 \[ \frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac {b^4}{6 e^5 (d+e x)^6} \]

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Rubi [A]  time = 0.07, antiderivative size = 119, 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, 43} \begin {gather*} \frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac {b^4}{6 e^5 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11,x]

[Out]

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

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)^{11}} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^{11}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{11}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{10}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^9}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^8}+\frac {b^4}{e^4 (d+e x)^7}\right ) \, dx\\ &=-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {b^4}{6 e^5 (d+e x)^6}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 144, normalized size = 1.21 \begin {gather*} -\frac {126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )}{1260 e^5 (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11,x]

[Out]

-1/1260*(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 + 10*d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d
^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e
^4*x^4))/(e^5*(d + e*x)^10)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11, x]

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fricas [B]  time = 0.38, size = 280, normalized size = 2.35 \begin {gather*} -\frac {210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \, {\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \, {\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \, {\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \end {gather*}

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)^11,x, algorithm="fricas")

[Out]

-1/1260*(210*b^4*e^4*x^4 + b^4*d^4 + 6*a*b^3*d^3*e + 21*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 126*a^4*e^4 + 120*(
b^4*d*e^3 + 6*a*b^3*e^4)*x^3 + 45*(b^4*d^2*e^2 + 6*a*b^3*d*e^3 + 21*a^2*b^2*e^4)*x^2 + 10*(b^4*d^3*e + 6*a*b^3
*d^2*e^2 + 21*a^2*b^2*d*e^3 + 56*a^3*b*e^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e^13*x^8 + 120*d^3*e^12*x^7
+ 210*d^4*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^2 + 10*d^9*e^6*x + d
^10*e^5)

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giac [A]  time = 0.16, size = 174, normalized size = 1.46 \begin {gather*} -\frac {{\left (210 \, b^{4} x^{4} e^{4} + 120 \, b^{4} d x^{3} e^{3} + 45 \, b^{4} d^{2} x^{2} e^{2} + 10 \, b^{4} d^{3} x e + b^{4} d^{4} + 720 \, a b^{3} x^{3} e^{4} + 270 \, a b^{3} d x^{2} e^{3} + 60 \, a b^{3} d^{2} x e^{2} + 6 \, a b^{3} d^{3} e + 945 \, a^{2} b^{2} x^{2} e^{4} + 210 \, a^{2} b^{2} d x e^{3} + 21 \, a^{2} b^{2} d^{2} e^{2} + 560 \, a^{3} b x e^{4} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{1260 \, {\left (x e + d\right )}^{10}} \end {gather*}

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)^11,x, algorithm="giac")

[Out]

-1/1260*(210*b^4*x^4*e^4 + 120*b^4*d*x^3*e^3 + 45*b^4*d^2*x^2*e^2 + 10*b^4*d^3*x*e + b^4*d^4 + 720*a*b^3*x^3*e
^4 + 270*a*b^3*d*x^2*e^3 + 60*a*b^3*d^2*x*e^2 + 6*a*b^3*d^3*e + 945*a^2*b^2*x^2*e^4 + 210*a^2*b^2*d*x*e^3 + 21
*a^2*b^2*d^2*e^2 + 560*a^3*b*x*e^4 + 56*a^3*b*d*e^3 + 126*a^4*e^4)*e^(-5)/(x*e + d)^10

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maple [A]  time = 0.05, size = 186, normalized size = 1.56 \begin {gather*} -\frac {b^{4}}{6 \left (e x +d \right )^{6} e^{5}}-\frac {4 \left (a e -b d \right ) b^{3}}{7 \left (e x +d \right )^{7} e^{5}}-\frac {3 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}}{4 \left (e x +d \right )^{8} e^{5}}-\frac {4 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b}{9 \left (e x +d \right )^{9} e^{5}}-\frac {e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 d^{2} e^{2} b^{2} a^{2}-4 d^{3} a \,b^{3} e +b^{4} d^{4}}{10 \left (e x +d \right )^{10} e^{5}} \end {gather*}

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)^11,x)

[Out]

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

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maxima [B]  time = 1.62, size = 280, normalized size = 2.35 \begin {gather*} -\frac {210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \, {\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \, {\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \, {\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \end {gather*}

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)^11,x, algorithm="maxima")

[Out]

-1/1260*(210*b^4*e^4*x^4 + b^4*d^4 + 6*a*b^3*d^3*e + 21*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 126*a^4*e^4 + 120*(
b^4*d*e^3 + 6*a*b^3*e^4)*x^3 + 45*(b^4*d^2*e^2 + 6*a*b^3*d*e^3 + 21*a^2*b^2*e^4)*x^2 + 10*(b^4*d^3*e + 6*a*b^3
*d^2*e^2 + 21*a^2*b^2*d*e^3 + 56*a^3*b*e^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e^13*x^8 + 120*d^3*e^12*x^7
+ 210*d^4*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^2 + 10*d^9*e^6*x + d
^10*e^5)

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mupad [B]  time = 0.24, size = 270, normalized size = 2.27 \begin {gather*} -\frac {\frac {126\,a^4\,e^4+56\,a^3\,b\,d\,e^3+21\,a^2\,b^2\,d^2\,e^2+6\,a\,b^3\,d^3\,e+b^4\,d^4}{1260\,e^5}+\frac {b^4\,x^4}{6\,e}+\frac {2\,b^3\,x^3\,\left (6\,a\,e+b\,d\right )}{21\,e^2}+\frac {b\,x\,\left (56\,a^3\,e^3+21\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{126\,e^4}+\frac {b^2\,x^2\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{28\,e^3}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^11,x)

[Out]

-((126*a^4*e^4 + b^4*d^4 + 21*a^2*b^2*d^2*e^2 + 6*a*b^3*d^3*e + 56*a^3*b*d*e^3)/(1260*e^5) + (b^4*x^4)/(6*e) +
(2*b^3*x^3*(6*a*e + b*d))/(21*e^2) + (b*x*(56*a^3*e^3 + b^3*d^3 + 6*a*b^2*d^2*e + 21*a^2*b*d*e^2))/(126*e^4)
+ (b^2*x^2*(21*a^2*e^2 + b^2*d^2 + 6*a*b*d*e))/(28*e^3))/(d^10 + e^10*x^10 + 10*d*e^9*x^9 + 45*d^8*e^2*x^2 + 1
20*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d^4*e^6*x^6 + 120*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d
^9*e*x)

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sympy [B]  time = 58.38, size = 303, normalized size = 2.55 \begin {gather*} \frac {- 126 a^{4} e^{4} - 56 a^{3} b d e^{3} - 21 a^{2} b^{2} d^{2} e^{2} - 6 a b^{3} d^{3} e - b^{4} d^{4} - 210 b^{4} e^{4} x^{4} + x^{3} \left (- 720 a b^{3} e^{4} - 120 b^{4} d e^{3}\right ) + x^{2} \left (- 945 a^{2} b^{2} e^{4} - 270 a b^{3} d e^{3} - 45 b^{4} d^{2} e^{2}\right ) + x \left (- 560 a^{3} b e^{4} - 210 a^{2} b^{2} d e^{3} - 60 a b^{3} d^{2} e^{2} - 10 b^{4} d^{3} e\right )}{1260 d^{10} e^{5} + 12600 d^{9} e^{6} x + 56700 d^{8} e^{7} x^{2} + 151200 d^{7} e^{8} x^{3} + 264600 d^{6} e^{9} x^{4} + 317520 d^{5} e^{10} x^{5} + 264600 d^{4} e^{11} x^{6} + 151200 d^{3} e^{12} x^{7} + 56700 d^{2} e^{13} x^{8} + 12600 d e^{14} x^{9} + 1260 e^{15} x^{10}} \end {gather*}

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)**11,x)

[Out]

(-126*a**4*e**4 - 56*a**3*b*d*e**3 - 21*a**2*b**2*d**2*e**2 - 6*a*b**3*d**3*e - b**4*d**4 - 210*b**4*e**4*x**4
+ x**3*(-720*a*b**3*e**4 - 120*b**4*d*e**3) + x**2*(-945*a**2*b**2*e**4 - 270*a*b**3*d*e**3 - 45*b**4*d**2*e*
*2) + x*(-560*a**3*b*e**4 - 210*a**2*b**2*d*e**3 - 60*a*b**3*d**2*e**2 - 10*b**4*d**3*e))/(1260*d**10*e**5 + 1
2600*d**9*e**6*x + 56700*d**8*e**7*x**2 + 151200*d**7*e**8*x**3 + 264600*d**6*e**9*x**4 + 317520*d**5*e**10*x*
*5 + 264600*d**4*e**11*x**6 + 151200*d**3*e**12*x**7 + 56700*d**2*e**13*x**8 + 12600*d*e**14*x**9 + 1260*e**15
*x**10)

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