3.15.80 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{11}} \, dx\) [1480]

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

[Out]

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

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Rubi [A]
time = 0.05, 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, 45} \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]

-1/10*(b*d - a*e)^4/(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 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 {\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 = 0.03, 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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Maple [A]
time = 0.63, size = 186, normalized size = 1.56

method	result	size
risch	\(\frac {-\frac {b^{4} x^{4}}{6 e}-\frac {2 b^{3} \left (6 a e +b d \right ) x^{3}}{21 e^{2}}-\frac {b^{2} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{2}}{28 e^{3}}-\frac {b \left (56 e^{3} a^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{126 e^{4}}-\frac {126 e^{4} a^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5}}}{\left (e x +d \right )^{10}}\)	\(171\)
gosper	\(-\frac {210 b^{4} x^{4} e^{4}+720 a \,b^{3} e^{4} x^{3}+120 b^{4} d \,e^{3} x^{3}+945 a^{2} b^{2} e^{4} x^{2}+270 a \,b^{3} d \,e^{3} x^{2}+45 b^{4} d^{2} e^{2} x^{2}+560 a^{3} b \,e^{4} x +210 a^{2} b^{2} d \,e^{3} x +60 a \,b^{3} d^{2} e^{2} x +10 b^{4} d^{3} e x +126 e^{4} a^{4}+56 a^{3} b d \,e^{3}+21 a^{2} b^{2} d^{2} e^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5} \left (e x +d \right )^{10}}\)	\(185\)
default	\(-\frac {4 b^{3} \left (a e -b d \right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{10 e^{5} \left (e x +d \right )^{10}}-\frac {b^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {4 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{9 e^{5} \left (e x +d \right )^{9}}-\frac {3 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{4 e^{5} \left (e x +d \right )^{8}}\)	\(186\)
norman	\(\frac {-\frac {b^{4} x^{4}}{6 e}-\frac {2 \left (6 a \,b^{3} e^{6}+b^{4} d \,e^{5}\right ) x^{3}}{21 e^{7}}-\frac {\left (21 a^{2} b^{2} e^{7}+6 a \,b^{3} d \,e^{6}+b^{4} d^{2} e^{5}\right ) x^{2}}{28 e^{8}}-\frac {\left (56 a^{3} b \,e^{8}+21 a^{2} b^{2} d \,e^{7}+6 a \,b^{3} d^{2} e^{6}+b^{4} d^{3} e^{5}\right ) x}{126 e^{9}}-\frac {126 a^{4} e^{9}+56 a^{3} b d \,e^{8}+21 a^{2} b^{2} d^{2} e^{7}+6 a \,b^{3} d^{3} e^{6}+b^{4} d^{4} e^{5}}{1260 e^{10}}}{\left (e x +d \right )^{10}}\)	\(197\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (113) = 226\).
time = 0.30, size = 259, normalized size = 2.18 \begin {gather*} -\frac {210 \, b^{4} x^{4} e^{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 (x^{10} e^{15} + 10 \, d x^{9} e^{14} + 45 \, d^{2} x^{8} e^{13} + 120 \, d^{3} x^{7} e^{12} + 210 \, d^{4} x^{6} e^{11} + 252 \, d^{5} x^{5} e^{10} + 210 \, d^{6} x^{4} e^{9} + 120 \, d^{7} x^{3} e^{8} + 45 \, d^{8} x^{2} e^{7} + 10 \, d^{9} x e^{6} + 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*x^4*e^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)/(x^10*e^15 + 10*d*x^9*e^14 + 45*d^2*x^8*e^13 + 120*d^3*x^7*e^12
+ 210*d^4*x^6*e^11 + 252*d^5*x^5*e^10 + 210*d^6*x^4*e^9 + 120*d^7*x^3*e^8 + 45*d^8*x^2*e^7 + 10*d^9*x*e^6 + d
^10*e^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (113) = 226\).
time = 3.06, size = 256, normalized size = 2.15 \begin {gather*} -\frac {b^{4} d^{4} + {\left (210 \, b^{4} x^{4} + 720 \, a b^{3} x^{3} + 945 \, a^{2} b^{2} x^{2} + 560 \, a^{3} b x + 126 \, a^{4}\right )} e^{4} + 2 \, {\left (60 \, b^{4} d x^{3} + 135 \, a b^{3} d x^{2} + 105 \, a^{2} b^{2} d x + 28 \, a^{3} b d\right )} e^{3} + 3 \, {\left (15 \, b^{4} d^{2} x^{2} + 20 \, a b^{3} d^{2} x + 7 \, a^{2} b^{2} d^{2}\right )} e^{2} + 2 \, {\left (5 \, b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e}{1260 \, {\left (x^{10} e^{15} + 10 \, d x^{9} e^{14} + 45 \, d^{2} x^{8} e^{13} + 120 \, d^{3} x^{7} e^{12} + 210 \, d^{4} x^{6} e^{11} + 252 \, d^{5} x^{5} e^{10} + 210 \, d^{6} x^{4} e^{9} + 120 \, d^{7} x^{3} e^{8} + 45 \, d^{8} x^{2} e^{7} + 10 \, d^{9} x e^{6} + 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*(b^4*d^4 + (210*b^4*x^4 + 720*a*b^3*x^3 + 945*a^2*b^2*x^2 + 560*a^3*b*x + 126*a^4)*e^4 + 2*(60*b^4*d*x
^3 + 135*a*b^3*d*x^2 + 105*a^2*b^2*d*x + 28*a^3*b*d)*e^3 + 3*(15*b^4*d^2*x^2 + 20*a*b^3*d^2*x + 7*a^2*b^2*d^2)
*e^2 + 2*(5*b^4*d^3*x + 3*a*b^3*d^3)*e)/(x^10*e^15 + 10*d*x^9*e^14 + 45*d^2*x^8*e^13 + 120*d^3*x^7*e^12 + 210*
d^4*x^6*e^11 + 252*d^5*x^5*e^10 + 210*d^6*x^4*e^9 + 120*d^7*x^3*e^8 + 45*d^8*x^2*e^7 + 10*d^9*x*e^6 + d^10*e^5
)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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Giac [A]
time = 1.54, 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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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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