3.16.27 \(\int \frac {(d+e x)^6}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1527]

Optimal. Leaf size=155 \[ \frac {e^6 x}{b^6}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}+\frac {6 e^5 (b d-a e) \log (a+b x)}{b^7} \]

[Out]

e^6*x/b^6-1/5*(-a*e+b*d)^6/b^7/(b*x+a)^5-3/2*e*(-a*e+b*d)^5/b^7/(b*x+a)^4-5*e^2*(-a*e+b*d)^4/b^7/(b*x+a)^3-10*
e^3*(-a*e+b*d)^3/b^7/(b*x+a)^2-15*e^4*(-a*e+b*d)^2/b^7/(b*x+a)+6*e^5*(-a*e+b*d)*ln(b*x+a)/b^7

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Rubi [A]
time = 0.12, antiderivative size = 155, 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 {6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac {e^6 x}{b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(e^6*x)/b^6 - (b*d - a*e)^6/(5*b^7*(a + b*x)^5) - (3*e*(b*d - a*e)^5)/(2*b^7*(a + b*x)^4) - (5*e^2*(b*d - a*e)
^4)/(b^7*(a + b*x)^3) - (10*e^3*(b*d - a*e)^3)/(b^7*(a + b*x)^2) - (15*e^4*(b*d - a*e)^2)/(b^7*(a + b*x)) + (6
*e^5*(b*d - a*e)*Log[a + b*x])/b^7

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

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Mathematica [A]
time = 0.10, size = 300, normalized size = 1.94 \begin {gather*} -\frac {87 a^6 e^6+a^5 b e^5 (-137 d+375 e x)+5 a^4 b^2 e^4 \left (6 d^2-125 d e x+120 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+15 d^2 e x-110 d e^2 x^2+40 e^3 x^3\right )+5 a^2 b^4 e^2 \left (d^4+10 d^3 e x+60 d^2 e^2 x^2-180 d e^3 x^3+10 e^4 x^4\right )+a b^5 e \left (3 d^5+25 d^4 e x+100 d^3 e^2 x^2+300 d^2 e^3 x^3-300 d e^4 x^4-50 e^5 x^5\right )+b^6 \left (2 d^6+15 d^5 e x+50 d^4 e^2 x^2+100 d^3 e^3 x^3+150 d^2 e^4 x^4-10 e^6 x^6\right )+60 e^5 (-b d+a e) (a+b x)^5 \log (a+b x)}{10 b^7 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/10*(87*a^6*e^6 + a^5*b*e^5*(-137*d + 375*e*x) + 5*a^4*b^2*e^4*(6*d^2 - 125*d*e*x + 120*e^2*x^2) + 10*a^3*b^
3*e^3*(d^3 + 15*d^2*e*x - 110*d*e^2*x^2 + 40*e^3*x^3) + 5*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 60*d^2*e^2*x^2 - 180
*d*e^3*x^3 + 10*e^4*x^4) + a*b^5*e*(3*d^5 + 25*d^4*e*x + 100*d^3*e^2*x^2 + 300*d^2*e^3*x^3 - 300*d*e^4*x^4 - 5
0*e^5*x^5) + b^6*(2*d^6 + 15*d^5*e*x + 50*d^4*e^2*x^2 + 100*d^3*e^3*x^3 + 150*d^2*e^4*x^4 - 10*e^6*x^6) + 60*e
^5*(-(b*d) + a*e)*(a + b*x)^5*Log[a + b*x])/(b^7*(a + b*x)^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs. \(2(151)=302\).
time = 0.64, size = 349, normalized size = 2.25

method	result	size
default	\(\frac {e^{6} x}{b^{6}}-\frac {15 e^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{7} \left (b x +a \right )}+\frac {10 e^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{7} \left (b x +a \right )^{2}}-\frac {6 e^{5} \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{7}}-\frac {5 e^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{b^{7} \left (b x +a \right )^{3}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5 b^{7} \left (b x +a \right )^{5}}+\frac {3 e \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{2 b^{7} \left (b x +a \right )^{4}}\)	\(349\)
norman	\(\frac {\frac {e^{6} x^{6}}{b}-\frac {137 a^{6} e^{6}-137 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+5 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +2 b^{6} d^{6}}{10 b^{7}}-\frac {5 \left (6 e^{6} a^{2}-6 a b d \,e^{5}+3 b^{2} d^{2} e^{4}\right ) x^{4}}{b^{3}}-\frac {10 \left (9 e^{6} a^{3}-9 a^{2} b d \,e^{5}+3 a \,b^{2} d^{2} e^{4}+d^{3} e^{3} b^{3}\right ) x^{3}}{b^{4}}-\frac {5 \left (22 a^{4} e^{6}-22 a^{3} b d \,e^{5}+6 a^{2} b^{2} d^{2} e^{4}+2 a \,b^{3} d^{3} e^{3}+b^{4} d^{4} e^{2}\right ) x^{2}}{b^{5}}-\frac {\left (125 e^{6} a^{5}-125 a^{4} b d \,e^{5}+30 a^{3} b^{2} d^{2} e^{4}+10 a^{2} b^{3} d^{3} e^{3}+5 a \,b^{4} d^{4} e^{2}+3 d^{5} e \,b^{5}\right ) x}{2 b^{6}}}{\left (b x +a \right )^{5}}-\frac {6 e^{5} \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{7}}\)	\(349\)
risch	\(\frac {e^{6} x}{b^{6}}+\frac {\left (-15 a^{2} b^{3} e^{6}+30 a \,b^{4} d \,e^{5}-15 d^{2} e^{4} b^{5}\right ) x^{4}-10 b^{2} e^{3} \left (5 e^{3} a^{3}-9 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}-5 b \,e^{2} \left (13 e^{4} a^{4}-22 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}-\frac {e \left (77 a^{5} e^{5}-125 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +3 b^{5} d^{5}\right ) x}{2}-\frac {87 a^{6} e^{6}-137 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+5 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +2 b^{6} d^{6}}{10 b}}{b^{6} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}-\frac {6 e^{6} \ln \left (b x +a \right ) a}{b^{7}}+\frac {6 e^{5} \ln \left (b x +a \right ) d}{b^{6}}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

e^6*x/b^6-15/b^7*e^4*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(b*x+a)+10/b^7*e^3*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d
^3)/(b*x+a)^2-6/b^7*e^5*(a*e-b*d)*ln(b*x+a)-5/b^7*e^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)/(b*x+a)^3-1/5/b^7*(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)/(b*x+a)^5+3/2/b^7*e*(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)/(b*x+a)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (153) = 306\).
time = 0.30, size = 374, normalized size = 2.41 \begin {gather*} -\frac {2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {x e^{6}}{b^{6}} + \frac {6 \, {\left (b d e^{5} - a e^{6}\right )} \log \left (b x + a\right )}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/10*(2*b^6*d^6 + 3*a*b^5*d^5*e + 5*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 30*a^4*b^2*d^2*e^4 - 137*a^5*b*d*e
^5 + 87*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 100*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 - 9
*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 50*(b^6*d^4*e^2 + 2*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 22*a^3*b^3*d*e^5
+ 13*a^4*b^2*e^6)*x^2 + 5*(3*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 10*a^2*b^4*d^3*e^3 + 30*a^3*b^3*d^2*e^4 - 125*a^4*
b^2*d*e^5 + 77*a^5*b*e^6)*x)/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b
^7) + x*e^6/b^6 + 6*(b*d*e^5 - a*e^6)*log(b*x + a)/b^7

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (153) = 306\).
time = 3.26, size = 490, normalized size = 3.16 \begin {gather*} -\frac {2 \, b^{6} d^{6} - {\left (10 \, b^{6} x^{6} + 50 \, a b^{5} x^{5} - 50 \, a^{2} b^{4} x^{4} - 400 \, a^{3} b^{3} x^{3} - 600 \, a^{4} b^{2} x^{2} - 375 \, a^{5} b x - 87 \, a^{6}\right )} e^{6} - {\left (300 \, a b^{5} d x^{4} + 900 \, a^{2} b^{4} d x^{3} + 1100 \, a^{3} b^{3} d x^{2} + 625 \, a^{4} b^{2} d x + 137 \, a^{5} b d\right )} e^{5} + 30 \, {\left (5 \, b^{6} d^{2} x^{4} + 10 \, a b^{5} d^{2} x^{3} + 10 \, a^{2} b^{4} d^{2} x^{2} + 5 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\right )} e^{4} + 10 \, {\left (10 \, b^{6} d^{3} x^{3} + 10 \, a b^{5} d^{3} x^{2} + 5 \, a^{2} b^{4} d^{3} x + a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (10 \, b^{6} d^{4} x^{2} + 5 \, a b^{5} d^{4} x + a^{2} b^{4} d^{4}\right )} e^{2} + 3 \, {\left (5 \, b^{6} d^{5} x + a b^{5} d^{5}\right )} e + 60 \, {\left ({\left (a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{3} + 10 \, a^{4} b^{2} x^{2} + 5 \, a^{5} b x + a^{6}\right )} e^{6} - {\left (b^{6} d x^{5} + 5 \, a b^{5} d x^{4} + 10 \, a^{2} b^{4} d x^{3} + 10 \, a^{3} b^{3} d x^{2} + 5 \, a^{4} b^{2} d x + a^{5} b d\right )} e^{5}\right )} \log \left (b x + a\right )}{10 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/10*(2*b^6*d^6 - (10*b^6*x^6 + 50*a*b^5*x^5 - 50*a^2*b^4*x^4 - 400*a^3*b^3*x^3 - 600*a^4*b^2*x^2 - 375*a^5*b
*x - 87*a^6)*e^6 - (300*a*b^5*d*x^4 + 900*a^2*b^4*d*x^3 + 1100*a^3*b^3*d*x^2 + 625*a^4*b^2*d*x + 137*a^5*b*d)*
e^5 + 30*(5*b^6*d^2*x^4 + 10*a*b^5*d^2*x^3 + 10*a^2*b^4*d^2*x^2 + 5*a^3*b^3*d^2*x + a^4*b^2*d^2)*e^4 + 10*(10*
b^6*d^3*x^3 + 10*a*b^5*d^3*x^2 + 5*a^2*b^4*d^3*x + a^3*b^3*d^3)*e^3 + 5*(10*b^6*d^4*x^2 + 5*a*b^5*d^4*x + a^2*
b^4*d^4)*e^2 + 3*(5*b^6*d^5*x + a*b^5*d^5)*e + 60*((a*b^5*x^5 + 5*a^2*b^4*x^4 + 10*a^3*b^3*x^3 + 10*a^4*b^2*x^
2 + 5*a^5*b*x + a^6)*e^6 - (b^6*d*x^5 + 5*a*b^5*d*x^4 + 10*a^2*b^4*d*x^3 + 10*a^3*b^3*d*x^2 + 5*a^4*b^2*d*x +
a^5*b*d)*e^5)*log(b*x + a))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^
7)

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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((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (153) = 306\).
time = 2.01, size = 328, normalized size = 2.12 \begin {gather*} \frac {x e^{6}}{b^{6}} + \frac {6 \, {\left (b d e^{5} - a e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \, {\left (b x + a\right )}^{5} b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

x*e^6/b^6 + 6*(b*d*e^5 - a*e^6)*log(abs(b*x + a))/b^7 - 1/10*(2*b^6*d^6 + 3*a*b^5*d^5*e + 5*a^2*b^4*d^4*e^2 +
10*a^3*b^3*d^3*e^3 + 30*a^4*b^2*d^2*e^4 - 137*a^5*b*d*e^5 + 87*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^
2*b^4*e^6)*x^4 + 100*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 - 9*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 50*(b^6*d^4*e^2 +
2*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 22*a^3*b^3*d*e^5 + 13*a^4*b^2*e^6)*x^2 + 5*(3*b^6*d^5*e + 5*a*b^5*d^4*e
^2 + 10*a^2*b^4*d^3*e^3 + 30*a^3*b^3*d^2*e^4 - 125*a^4*b^2*d*e^5 + 77*a^5*b*e^6)*x)/((b*x + a)^5*b^7)

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Mupad [B]
time = 0.64, size = 399, normalized size = 2.57 \begin {gather*} \frac {e^6\,x}{b^6}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a\,e^6-6\,b\,d\,e^5\right )}{b^7}-\frac {x^2\,\left (65\,a^4\,b\,e^6-110\,a^3\,b^2\,d\,e^5+30\,a^2\,b^3\,d^2\,e^4+10\,a\,b^4\,d^3\,e^3+5\,b^5\,d^4\,e^2\right )+x^4\,\left (15\,a^2\,b^3\,e^6-30\,a\,b^4\,d\,e^5+15\,b^5\,d^2\,e^4\right )+\frac {87\,a^6\,e^6-137\,a^5\,b\,d\,e^5+30\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+5\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+2\,b^6\,d^6}{10\,b}+x\,\left (\frac {77\,a^5\,e^6}{2}-\frac {125\,a^4\,b\,d\,e^5}{2}+15\,a^3\,b^2\,d^2\,e^4+5\,a^2\,b^3\,d^3\,e^3+\frac {5\,a\,b^4\,d^4\,e^2}{2}+\frac {3\,b^5\,d^5\,e}{2}\right )+x^3\,\left (50\,a^3\,b^2\,e^6-90\,a^2\,b^3\,d\,e^5+30\,a\,b^4\,d^2\,e^4+10\,b^5\,d^3\,e^3\right )}{a^5\,b^6+5\,a^4\,b^7\,x+10\,a^3\,b^8\,x^2+10\,a^2\,b^9\,x^3+5\,a\,b^{10}\,x^4+b^{11}\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^6/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(e^6*x)/b^6 - (log(a + b*x)*(6*a*e^6 - 6*b*d*e^5))/b^7 - (x^2*(65*a^4*b*e^6 + 5*b^5*d^4*e^2 + 10*a*b^4*d^3*e^3
- 110*a^3*b^2*d*e^5 + 30*a^2*b^3*d^2*e^4) + x^4*(15*a^2*b^3*e^6 + 15*b^5*d^2*e^4 - 30*a*b^4*d*e^5) + (87*a^6*
e^6 + 2*b^6*d^6 + 5*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 30*a^4*b^2*d^2*e^4 + 3*a*b^5*d^5*e - 137*a^5*b*d*e^
5)/(10*b) + x*((77*a^5*e^6)/2 + (3*b^5*d^5*e)/2 + (5*a*b^4*d^4*e^2)/2 + 5*a^2*b^3*d^3*e^3 + 15*a^3*b^2*d^2*e^4
- (125*a^4*b*d*e^5)/2) + x^3*(50*a^3*b^2*e^6 + 10*b^5*d^3*e^3 + 30*a*b^4*d^2*e^4 - 90*a^2*b^3*d*e^5))/(a^5*b^
6 + b^11*x^5 + 5*a^4*b^7*x + 5*a*b^10*x^4 + 10*a^3*b^8*x^2 + 10*a^2*b^9*x^3)

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