3.16.4 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{15}} \, dx\) [1504]

Optimal. Leaf size=173 \[ -\frac {(b d-a e)^6}{14 e^7 (d+e x)^{14}}+\frac {6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac {5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac {20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac {3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac {2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac {b^6}{8 e^7 (d+e x)^8} \]

[Out]

-1/14*(-a*e+b*d)^6/e^7/(e*x+d)^14+6/13*b*(-a*e+b*d)^5/e^7/(e*x+d)^13-5/4*b^2*(-a*e+b*d)^4/e^7/(e*x+d)^12+20/11
*b^3*(-a*e+b*d)^3/e^7/(e*x+d)^11-3/2*b^4*(-a*e+b*d)^2/e^7/(e*x+d)^10+2/3*b^5*(-a*e+b*d)/e^7/(e*x+d)^9-1/8*b^6/
e^7/(e*x+d)^8

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 173, 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 {2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac {3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac {5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac {(b d-a e)^6}{14 e^7 (d+e x)^{14}}-\frac {b^6}{8 e^7 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^15,x]

[Out]

-1/14*(b*d - a*e)^6/(e^7*(d + e*x)^14) + (6*b*(b*d - a*e)^5)/(13*e^7*(d + e*x)^13) - (5*b^2*(b*d - a*e)^4)/(4*
e^7*(d + e*x)^12) + (20*b^3*(b*d - a*e)^3)/(11*e^7*(d + e*x)^11) - (3*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^10)
+ (2*b^5*(b*d - a*e))/(3*e^7*(d + e*x)^9) - b^6/(8*e^7*(d + e*x)^8)

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 )^3}{(d+e x)^{15}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{15}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{15}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{14}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{13}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{12}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{11}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{10}}+\frac {b^6}{e^6 (d+e x)^9}\right ) \, dx\\ &=-\frac {(b d-a e)^6}{14 e^7 (d+e x)^{14}}+\frac {6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac {5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac {20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac {3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac {2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac {b^6}{8 e^7 (d+e x)^8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 277, normalized size = 1.60 \begin {gather*} -\frac {1716 a^6 e^6+792 a^5 b e^5 (d+14 e x)+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+120 a^3 b^3 e^3 \left (d^3+14 d^2 e x+91 d e^2 x^2+364 e^3 x^3\right )+36 a^2 b^4 e^2 \left (d^4+14 d^3 e x+91 d^2 e^2 x^2+364 d e^3 x^3+1001 e^4 x^4\right )+8 a b^5 e \left (d^5+14 d^4 e x+91 d^3 e^2 x^2+364 d^2 e^3 x^3+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (d^6+14 d^5 e x+91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+2002 d e^5 x^5+3003 e^6 x^6\right )}{24024 e^7 (d+e x)^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^15,x]

[Out]

-1/24024*(1716*a^6*e^6 + 792*a^5*b*e^5*(d + 14*e*x) + 330*a^4*b^2*e^4*(d^2 + 14*d*e*x + 91*e^2*x^2) + 120*a^3*
b^3*e^3*(d^3 + 14*d^2*e*x + 91*d*e^2*x^2 + 364*e^3*x^3) + 36*a^2*b^4*e^2*(d^4 + 14*d^3*e*x + 91*d^2*e^2*x^2 +
364*d*e^3*x^3 + 1001*e^4*x^4) + 8*a*b^5*e*(d^5 + 14*d^4*e*x + 91*d^3*e^2*x^2 + 364*d^2*e^3*x^3 + 1001*d*e^4*x^
4 + 2002*e^5*x^5) + b^6*(d^6 + 14*d^5*e*x + 91*d^4*e^2*x^2 + 364*d^3*e^3*x^3 + 1001*d^2*e^4*x^4 + 2002*d*e^5*x
^5 + 3003*e^6*x^6))/(e^7*(d + e*x)^14)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(159)=318\).
time = 0.62, size = 357, normalized size = 2.06

method	result	size
risch	\(\frac {-\frac {b^{6} x^{6}}{8 e}-\frac {b^{5} \left (8 a e +b d \right ) x^{5}}{12 e^{2}}-\frac {b^{4} \left (36 a^{2} e^{2}+8 a b d e +b^{2} d^{2}\right ) x^{4}}{24 e^{3}}-\frac {b^{3} \left (120 e^{3} a^{3}+36 a^{2} b d \,e^{2}+8 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{66 e^{4}}-\frac {b^{2} \left (330 e^{4} a^{4}+120 a^{3} b d \,e^{3}+36 a^{2} b^{2} d^{2} e^{2}+8 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{264 e^{5}}-\frac {b \left (792 a^{5} e^{5}+330 a^{4} b d \,e^{4}+120 a^{3} b^{2} d^{2} e^{3}+36 a^{2} b^{3} d^{3} e^{2}+8 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{1716 e^{6}}-\frac {1716 a^{6} e^{6}+792 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}+120 a^{3} b^{3} d^{3} e^{3}+36 a^{2} b^{4} d^{4} e^{2}+8 a \,b^{5} d^{5} e +b^{6} d^{6}}{24024 e^{7}}}{\left (e x +d \right )^{14}}\)	\(335\)
default	\(-\frac {5 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{12}}-\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{11 e^{7} \left (e x +d \right )^{11}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{14 e^{7} \left (e x +d \right )^{14}}-\frac {2 b^{5} \left (a e -b d \right )}{3 e^{7} \left (e x +d \right )^{9}}-\frac {3 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{10}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{13 e^{7} \left (e x +d \right )^{13}}-\frac {b^{6}}{8 e^{7} \left (e x +d \right )^{8}}\)	\(357\)
norman	\(\frac {-\frac {1716 a^{6} e^{13}+792 a^{5} b d \,e^{12}+330 a^{4} b^{2} d^{2} e^{11}+120 a^{3} b^{3} d^{3} e^{10}+36 a^{2} b^{4} d^{4} e^{9}+8 a \,b^{5} d^{5} e^{8}+b^{6} d^{6} e^{7}}{24024 e^{14}}-\frac {\left (792 a^{5} b \,e^{12}+330 a^{4} b^{2} d \,e^{11}+120 a^{3} b^{3} d^{2} e^{10}+36 a^{2} b^{4} d^{3} e^{9}+8 a \,b^{5} d^{4} e^{8}+b^{6} d^{5} e^{7}\right ) x}{1716 e^{13}}-\frac {\left (330 a^{4} b^{2} e^{11}+120 a^{3} b^{3} d \,e^{10}+36 a^{2} b^{4} d^{2} e^{9}+8 a \,b^{5} d^{3} e^{8}+b^{6} d^{4} e^{7}\right ) x^{2}}{264 e^{12}}-\frac {\left (120 a^{3} b^{3} e^{10}+36 a^{2} b^{4} d \,e^{9}+8 a \,b^{5} d^{2} e^{8}+b^{6} d^{3} e^{7}\right ) x^{3}}{66 e^{11}}-\frac {\left (36 a^{2} b^{4} e^{9}+8 a \,b^{5} d \,e^{8}+b^{6} d^{2} e^{7}\right ) x^{4}}{24 e^{10}}-\frac {\left (8 a \,b^{5} e^{8}+b^{6} d \,e^{7}\right ) x^{5}}{12 e^{9}}-\frac {b^{6} x^{6}}{8 e}}{\left (e x +d \right )^{14}}\)	\(375\)
gosper	\(-\frac {3003 b^{6} x^{6} e^{6}+16016 a \,b^{5} e^{6} x^{5}+2002 b^{6} d \,e^{5} x^{5}+36036 a^{2} b^{4} e^{6} x^{4}+8008 a \,b^{5} d \,e^{5} x^{4}+1001 b^{6} d^{2} e^{4} x^{4}+43680 a^{3} b^{3} e^{6} x^{3}+13104 a^{2} b^{4} d \,e^{5} x^{3}+2912 a \,b^{5} d^{2} e^{4} x^{3}+364 b^{6} d^{3} e^{3} x^{3}+30030 a^{4} b^{2} e^{6} x^{2}+10920 a^{3} b^{3} d \,e^{5} x^{2}+3276 a^{2} b^{4} d^{2} e^{4} x^{2}+728 a \,b^{5} d^{3} e^{3} x^{2}+91 b^{6} d^{4} e^{2} x^{2}+11088 a^{5} b \,e^{6} x +4620 a^{4} b^{2} d \,e^{5} x +1680 a^{3} b^{3} d^{2} e^{4} x +504 a^{2} b^{4} d^{3} e^{3} x +112 a \,b^{5} d^{4} e^{2} x +14 b^{6} d^{5} e x +1716 a^{6} e^{6}+792 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}+120 a^{3} b^{3} d^{3} e^{3}+36 a^{2} b^{4} d^{4} e^{2}+8 a \,b^{5} d^{5} e +b^{6} d^{6}}{24024 e^{7} \left (e x +d \right )^{14}}\)	\(376\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x,method=_RETURNVERBOSE)

[Out]

-5/4*b^2/e^7*(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*x+d)^12-20/11*b^3/e^7*(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)^11-1/14*(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3
*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)/e^7/(e*x+d)^14-2/3*b^5/e^7*(a*e-b*d)/(e*x+d)^9-3/2*b^4/e^7*(a^2
*e^2-2*a*b*d*e+b^2*d^2)/(e*x+d)^10-6/13*b/e^7*(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a
*b^4*d^4*e-b^5*d^5)/(e*x+d)^13-1/8*b^6/e^7/(e*x+d)^8

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (165) = 330\).
time = 0.30, size = 458, normalized size = 2.65 \begin {gather*} -\frac {3003 \, b^{6} x^{6} e^{6} + b^{6} d^{6} + 8 \, a b^{5} d^{5} e + 36 \, a^{2} b^{4} d^{4} e^{2} + 120 \, a^{3} b^{3} d^{3} e^{3} + 330 \, a^{4} b^{2} d^{2} e^{4} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6} + 2002 \, {\left (b^{6} d e^{5} + 8 \, a b^{5} e^{6}\right )} x^{5} + 1001 \, {\left (b^{6} d^{2} e^{4} + 8 \, a b^{5} d e^{5} + 36 \, a^{2} b^{4} e^{6}\right )} x^{4} + 364 \, {\left (b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 36 \, a^{2} b^{4} d e^{5} + 120 \, a^{3} b^{3} e^{6}\right )} x^{3} + 91 \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 36 \, a^{2} b^{4} d^{2} e^{4} + 120 \, a^{3} b^{3} d e^{5} + 330 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (b^{6} d^{5} e + 8 \, a b^{5} d^{4} e^{2} + 36 \, a^{2} b^{4} d^{3} e^{3} + 120 \, a^{3} b^{3} d^{2} e^{4} + 330 \, a^{4} b^{2} d e^{5} + 792 \, a^{5} b e^{6}\right )} x}{24024 \, {\left (x^{14} e^{21} + 14 \, d x^{13} e^{20} + 91 \, d^{2} x^{12} e^{19} + 364 \, d^{3} x^{11} e^{18} + 1001 \, d^{4} x^{10} e^{17} + 2002 \, d^{5} x^{9} e^{16} + 3003 \, d^{6} x^{8} e^{15} + 3432 \, d^{7} x^{7} e^{14} + 3003 \, d^{8} x^{6} e^{13} + 2002 \, d^{9} x^{5} e^{12} + 1001 \, d^{10} x^{4} e^{11} + 364 \, d^{11} x^{3} e^{10} + 91 \, d^{12} x^{2} e^{9} + 14 \, d^{13} x e^{8} + d^{14} e^{7}\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)^3/(e*x+d)^15,x, algorithm="maxima")

[Out]

-1/24024*(3003*b^6*x^6*e^6 + b^6*d^6 + 8*a*b^5*d^5*e + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*
d^2*e^4 + 792*a^5*b*d*e^5 + 1716*a^6*e^6 + 2002*(b^6*d*e^5 + 8*a*b^5*e^6)*x^5 + 1001*(b^6*d^2*e^4 + 8*a*b^5*d*
e^5 + 36*a^2*b^4*e^6)*x^4 + 364*(b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 36*a^2*b^4*d*e^5 + 120*a^3*b^3*e^6)*x^3 + 91*
(b^6*d^4*e^2 + 8*a*b^5*d^3*e^3 + 36*a^2*b^4*d^2*e^4 + 120*a^3*b^3*d*e^5 + 330*a^4*b^2*e^6)*x^2 + 14*(b^6*d^5*e
+ 8*a*b^5*d^4*e^2 + 36*a^2*b^4*d^3*e^3 + 120*a^3*b^3*d^2*e^4 + 330*a^4*b^2*d*e^5 + 792*a^5*b*e^6)*x)/(x^14*e^
21 + 14*d*x^13*e^20 + 91*d^2*x^12*e^19 + 364*d^3*x^11*e^18 + 1001*d^4*x^10*e^17 + 2002*d^5*x^9*e^16 + 3003*d^6
*x^8*e^15 + 3432*d^7*x^7*e^14 + 3003*d^8*x^6*e^13 + 2002*d^9*x^5*e^12 + 1001*d^10*x^4*e^11 + 364*d^11*x^3*e^10
+ 91*d^12*x^2*e^9 + 14*d^13*x*e^8 + d^14*e^7)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (165) = 330\).
time = 2.09, size = 456, normalized size = 2.64 \begin {gather*} -\frac {b^{6} d^{6} + {\left (3003 \, b^{6} x^{6} + 16016 \, a b^{5} x^{5} + 36036 \, a^{2} b^{4} x^{4} + 43680 \, a^{3} b^{3} x^{3} + 30030 \, a^{4} b^{2} x^{2} + 11088 \, a^{5} b x + 1716 \, a^{6}\right )} e^{6} + 2 \, {\left (1001 \, b^{6} d x^{5} + 4004 \, a b^{5} d x^{4} + 6552 \, a^{2} b^{4} d x^{3} + 5460 \, a^{3} b^{3} d x^{2} + 2310 \, a^{4} b^{2} d x + 396 \, a^{5} b d\right )} e^{5} + {\left (1001 \, b^{6} d^{2} x^{4} + 2912 \, a b^{5} d^{2} x^{3} + 3276 \, a^{2} b^{4} d^{2} x^{2} + 1680 \, a^{3} b^{3} d^{2} x + 330 \, a^{4} b^{2} d^{2}\right )} e^{4} + 4 \, {\left (91 \, b^{6} d^{3} x^{3} + 182 \, a b^{5} d^{3} x^{2} + 126 \, a^{2} b^{4} d^{3} x + 30 \, a^{3} b^{3} d^{3}\right )} e^{3} + {\left (91 \, b^{6} d^{4} x^{2} + 112 \, a b^{5} d^{4} x + 36 \, a^{2} b^{4} d^{4}\right )} e^{2} + 2 \, {\left (7 \, b^{6} d^{5} x + 4 \, a b^{5} d^{5}\right )} e}{24024 \, {\left (x^{14} e^{21} + 14 \, d x^{13} e^{20} + 91 \, d^{2} x^{12} e^{19} + 364 \, d^{3} x^{11} e^{18} + 1001 \, d^{4} x^{10} e^{17} + 2002 \, d^{5} x^{9} e^{16} + 3003 \, d^{6} x^{8} e^{15} + 3432 \, d^{7} x^{7} e^{14} + 3003 \, d^{8} x^{6} e^{13} + 2002 \, d^{9} x^{5} e^{12} + 1001 \, d^{10} x^{4} e^{11} + 364 \, d^{11} x^{3} e^{10} + 91 \, d^{12} x^{2} e^{9} + 14 \, d^{13} x e^{8} + d^{14} e^{7}\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)^3/(e*x+d)^15,x, algorithm="fricas")

[Out]

-1/24024*(b^6*d^6 + (3003*b^6*x^6 + 16016*a*b^5*x^5 + 36036*a^2*b^4*x^4 + 43680*a^3*b^3*x^3 + 30030*a^4*b^2*x^
2 + 11088*a^5*b*x + 1716*a^6)*e^6 + 2*(1001*b^6*d*x^5 + 4004*a*b^5*d*x^4 + 6552*a^2*b^4*d*x^3 + 5460*a^3*b^3*d
*x^2 + 2310*a^4*b^2*d*x + 396*a^5*b*d)*e^5 + (1001*b^6*d^2*x^4 + 2912*a*b^5*d^2*x^3 + 3276*a^2*b^4*d^2*x^2 + 1
680*a^3*b^3*d^2*x + 330*a^4*b^2*d^2)*e^4 + 4*(91*b^6*d^3*x^3 + 182*a*b^5*d^3*x^2 + 126*a^2*b^4*d^3*x + 30*a^3*
b^3*d^3)*e^3 + (91*b^6*d^4*x^2 + 112*a*b^5*d^4*x + 36*a^2*b^4*d^4)*e^2 + 2*(7*b^6*d^5*x + 4*a*b^5*d^5)*e)/(x^1
4*e^21 + 14*d*x^13*e^20 + 91*d^2*x^12*e^19 + 364*d^3*x^11*e^18 + 1001*d^4*x^10*e^17 + 2002*d^5*x^9*e^16 + 3003
*d^6*x^8*e^15 + 3432*d^7*x^7*e^14 + 3003*d^8*x^6*e^13 + 2002*d^9*x^5*e^12 + 1001*d^10*x^4*e^11 + 364*d^11*x^3*
e^10 + 91*d^12*x^2*e^9 + 14*d^13*x*e^8 + d^14*e^7)

________________________________________________________________________________________

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)**3/(e*x+d)**15,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (165) = 330\).
time = 1.90, size = 352, normalized size = 2.03 \begin {gather*} -\frac {{\left (3003 \, b^{6} x^{6} e^{6} + 2002 \, b^{6} d x^{5} e^{5} + 1001 \, b^{6} d^{2} x^{4} e^{4} + 364 \, b^{6} d^{3} x^{3} e^{3} + 91 \, b^{6} d^{4} x^{2} e^{2} + 14 \, b^{6} d^{5} x e + b^{6} d^{6} + 16016 \, a b^{5} x^{5} e^{6} + 8008 \, a b^{5} d x^{4} e^{5} + 2912 \, a b^{5} d^{2} x^{3} e^{4} + 728 \, a b^{5} d^{3} x^{2} e^{3} + 112 \, a b^{5} d^{4} x e^{2} + 8 \, a b^{5} d^{5} e + 36036 \, a^{2} b^{4} x^{4} e^{6} + 13104 \, a^{2} b^{4} d x^{3} e^{5} + 3276 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 504 \, a^{2} b^{4} d^{3} x e^{3} + 36 \, a^{2} b^{4} d^{4} e^{2} + 43680 \, a^{3} b^{3} x^{3} e^{6} + 10920 \, a^{3} b^{3} d x^{2} e^{5} + 1680 \, a^{3} b^{3} d^{2} x e^{4} + 120 \, a^{3} b^{3} d^{3} e^{3} + 30030 \, a^{4} b^{2} x^{2} e^{6} + 4620 \, a^{4} b^{2} d x e^{5} + 330 \, a^{4} b^{2} d^{2} e^{4} + 11088 \, a^{5} b x e^{6} + 792 \, a^{5} b d e^{5} + 1716 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{24024 \, {\left (x e + d\right )}^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^15,x, algorithm="giac")

[Out]

-1/24024*(3003*b^6*x^6*e^6 + 2002*b^6*d*x^5*e^5 + 1001*b^6*d^2*x^4*e^4 + 364*b^6*d^3*x^3*e^3 + 91*b^6*d^4*x^2*
e^2 + 14*b^6*d^5*x*e + b^6*d^6 + 16016*a*b^5*x^5*e^6 + 8008*a*b^5*d*x^4*e^5 + 2912*a*b^5*d^2*x^3*e^4 + 728*a*b
^5*d^3*x^2*e^3 + 112*a*b^5*d^4*x*e^2 + 8*a*b^5*d^5*e + 36036*a^2*b^4*x^4*e^6 + 13104*a^2*b^4*d*x^3*e^5 + 3276*
a^2*b^4*d^2*x^2*e^4 + 504*a^2*b^4*d^3*x*e^3 + 36*a^2*b^4*d^4*e^2 + 43680*a^3*b^3*x^3*e^6 + 10920*a^3*b^3*d*x^2
*e^5 + 1680*a^3*b^3*d^2*x*e^4 + 120*a^3*b^3*d^3*e^3 + 30030*a^4*b^2*x^2*e^6 + 4620*a^4*b^2*d*x*e^5 + 330*a^4*b
^2*d^2*e^4 + 11088*a^5*b*x*e^6 + 792*a^5*b*d*e^5 + 1716*a^6*e^6)*e^(-7)/(x*e + d)^14

________________________________________________________________________________________

Mupad [B]
time = 1.59, size = 478, normalized size = 2.76 \begin {gather*} -\frac {\frac {1716\,a^6\,e^6+792\,a^5\,b\,d\,e^5+330\,a^4\,b^2\,d^2\,e^4+120\,a^3\,b^3\,d^3\,e^3+36\,a^2\,b^4\,d^4\,e^2+8\,a\,b^5\,d^5\,e+b^6\,d^6}{24024\,e^7}+\frac {b^6\,x^6}{8\,e}+\frac {b^3\,x^3\,\left (120\,a^3\,e^3+36\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{66\,e^4}+\frac {b\,x\,\left (792\,a^5\,e^5+330\,a^4\,b\,d\,e^4+120\,a^3\,b^2\,d^2\,e^3+36\,a^2\,b^3\,d^3\,e^2+8\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{1716\,e^6}+\frac {b^5\,x^5\,\left (8\,a\,e+b\,d\right )}{12\,e^2}+\frac {b^2\,x^2\,\left (330\,a^4\,e^4+120\,a^3\,b\,d\,e^3+36\,a^2\,b^2\,d^2\,e^2+8\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{264\,e^5}+\frac {b^4\,x^4\,\left (36\,a^2\,e^2+8\,a\,b\,d\,e+b^2\,d^2\right )}{24\,e^3}}{d^{14}+14\,d^{13}\,e\,x+91\,d^{12}\,e^2\,x^2+364\,d^{11}\,e^3\,x^3+1001\,d^{10}\,e^4\,x^4+2002\,d^9\,e^5\,x^5+3003\,d^8\,e^6\,x^6+3432\,d^7\,e^7\,x^7+3003\,d^6\,e^8\,x^8+2002\,d^5\,e^9\,x^9+1001\,d^4\,e^{10}\,x^{10}+364\,d^3\,e^{11}\,x^{11}+91\,d^2\,e^{12}\,x^{12}+14\,d\,e^{13}\,x^{13}+e^{14}\,x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^15,x)

[Out]

-((1716*a^6*e^6 + b^6*d^6 + 36*a^2*b^4*d^4*e^2 + 120*a^3*b^3*d^3*e^3 + 330*a^4*b^2*d^2*e^4 + 8*a*b^5*d^5*e + 7
92*a^5*b*d*e^5)/(24024*e^7) + (b^6*x^6)/(8*e) + (b^3*x^3*(120*a^3*e^3 + b^3*d^3 + 8*a*b^2*d^2*e + 36*a^2*b*d*e
^2))/(66*e^4) + (b*x*(792*a^5*e^5 + b^5*d^5 + 36*a^2*b^3*d^3*e^2 + 120*a^3*b^2*d^2*e^3 + 8*a*b^4*d^4*e + 330*a
^4*b*d*e^4))/(1716*e^6) + (b^5*x^5*(8*a*e + b*d))/(12*e^2) + (b^2*x^2*(330*a^4*e^4 + b^4*d^4 + 36*a^2*b^2*d^2*
e^2 + 8*a*b^3*d^3*e + 120*a^3*b*d*e^3))/(264*e^5) + (b^4*x^4*(36*a^2*e^2 + b^2*d^2 + 8*a*b*d*e))/(24*e^3))/(d^
14 + e^14*x^14 + 14*d*e^13*x^13 + 91*d^12*e^2*x^2 + 364*d^11*e^3*x^3 + 1001*d^10*e^4*x^4 + 2002*d^9*e^5*x^5 +
3003*d^8*e^6*x^6 + 3432*d^7*e^7*x^7 + 3003*d^6*e^8*x^8 + 2002*d^5*e^9*x^9 + 1001*d^4*e^10*x^10 + 364*d^3*e^11*
x^11 + 91*d^2*e^12*x^12 + 14*d^13*e*x)

________________________________________________________________________________________