3.1504 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{15}} \, dx\)

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

[Out]

-(b*d - a*e)^6/(14*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)

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Rubi [A]  time = 0.115182, antiderivative size = 173, 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} \[ \frac{2 b^5 (b d-a e)}{3 e^7 (d+e x)^9}-\frac{3 b^4 (b d-a e)^2}{2 e^7 (d+e x)^{10}}+\frac{20 b^3 (b d-a e)^3}{11 e^7 (d+e x)^{11}}-\frac{5 b^2 (b d-a e)^4}{4 e^7 (d+e x)^{12}}+\frac{6 b (b d-a e)^5}{13 e^7 (d+e x)^{13}}-\frac{(b d-a e)^6}{14 e^7 (d+e x)^{14}}-\frac{b^6}{8 e^7 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d - a*e)^6/(14*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 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 )^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 = 0.094877, size = 277, normalized size = 1.6 \[ -\frac{36 a^2 b^4 e^2 \left (91 d^2 e^2 x^2+14 d^3 e x+d^4+364 d e^3 x^3+1001 e^4 x^4\right )+120 a^3 b^3 e^3 \left (14 d^2 e x+d^3+91 d e^2 x^2+364 e^3 x^3\right )+330 a^4 b^2 e^4 \left (d^2+14 d e x+91 e^2 x^2\right )+792 a^5 b e^5 (d+14 e x)+1716 a^6 e^6+8 a b^5 e \left (91 d^3 e^2 x^2+364 d^2 e^3 x^3+14 d^4 e x+d^5+1001 d e^4 x^4+2002 e^5 x^5\right )+b^6 \left (91 d^4 e^2 x^2+364 d^3 e^3 x^3+1001 d^2 e^4 x^4+14 d^5 e x+d^6+2002 d e^5 x^5+3003 e^6 x^6\right )}{24024 e^7 (d+e x)^{14}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 + 300
3*e^6*x^6))/(24024*e^7*(d + e*x)^14)

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Maple [B]  time = 0.048, size = 357, normalized size = 2.1 \begin{align*} -{\frac{3\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{2\,{b}^{5} \left ( ae-bd \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{5\,{b}^{2} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{12}}}-{\frac{{e}^{6}{a}^{6}-6\,{a}^{5}bd{e}^{5}+15\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-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+{d}^{6}{b}^{6}}{14\,{e}^{7} \left ( ex+d \right ) ^{14}}}-{\frac{20\,{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{11\,{e}^{7} \left ( ex+d \right ) ^{11}}}-{\frac{6\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{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 ( ex+d \right ) ^{13}}}-{\frac{{b}^{6}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}} \end{align*}

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)

[Out]

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

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Maxima [B]  time = 1.24489, size = 670, normalized size = 3.87 \begin{align*} -\frac{3003 \, b^{6} e^{6} x^{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 (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\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)^3/(e*x+d)^15,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.7311, size = 1104, normalized size = 6.38 \begin{align*} -\frac{3003 \, b^{6} e^{6} x^{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 (e^{21} x^{14} + 14 \, d e^{20} x^{13} + 91 \, d^{2} e^{19} x^{12} + 364 \, d^{3} e^{18} x^{11} + 1001 \, d^{4} e^{17} x^{10} + 2002 \, d^{5} e^{16} x^{9} + 3003 \, d^{6} e^{15} x^{8} + 3432 \, d^{7} e^{14} x^{7} + 3003 \, d^{8} e^{13} x^{6} + 2002 \, d^{9} e^{12} x^{5} + 1001 \, d^{10} e^{11} x^{4} + 364 \, d^{11} e^{10} x^{3} + 91 \, d^{12} e^{9} x^{2} + 14 \, d^{13} e^{8} x + d^{14} e^{7}\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)^3/(e*x+d)^15,x, algorithm="fricas")

[Out]

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

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

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

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Giac [B]  time = 1.15519, size = 475, normalized size = 2.75 \begin{align*} -\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{align*}

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