3.15.96 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^7} \, dx\) [1496]

Optimal. Leaf size=167 \[ -\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}+\frac {b^6 \log (d+e x)}{e^7} \]

[Out]

-1/6*(-a*e+b*d)^6/e^7/(e*x+d)^6+6/5*b*(-a*e+b*d)^5/e^7/(e*x+d)^5-15/4*b^2*(-a*e+b*d)^4/e^7/(e*x+d)^4+20/3*b^3*
(-a*e+b*d)^3/e^7/(e*x+d)^3-15/2*b^4*(-a*e+b*d)^2/e^7/(e*x+d)^2+6*b^5*(-a*e+b*d)/e^7/(e*x+d)+b^6*ln(e*x+d)/e^7

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Rubi [A]
time = 0.09, antiderivative size = 167, 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 b^5 (b d-a e)}{e^7 (d+e x)}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {b^6 \log (d+e x)}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]

[Out]

-1/6*(b*d - a*e)^6/(e^7*(d + e*x)^6) + (6*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^5) - (15*b^2*(b*d - a*e)^4)/(4*e^7
*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3)/(3*e^7*(d + e*x)^3) - (15*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^2) + (6*b
^5*(b*d - a*e))/(e^7*(d + e*x)) + (b^6*Log[d + e*x])/e^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 {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^7} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}+\frac {b^6 \log (d+e x)}{e^7}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 233, normalized size = 1.40 \begin {gather*} \frac {\frac {(b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )}{(d+e x)^6}+60 b^6 \log (d+e x)}{60 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]

[Out]

(((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x + 225*e^2*x^2) + a^2
*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^
2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*
e^4*x^4 + 360*e^5*x^5)))/(d + e*x)^6 + 60*b^6*Log[d + e*x])/(60*e^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(354\) vs. \(2(157)=314\).
time = 0.62, size = 355, normalized size = 2.13

method	result	size
risch	\(\frac {-\frac {6 b^{5} \left (a e -b d \right ) x^{5}}{e^{2}}-\frac {15 b^{4} \left (a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {10 b^{3} \left (2 e^{3} a^{3}+3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -11 b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {5 b^{2} \left (3 e^{4} a^{4}+4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+12 a \,b^{3} d^{3} e -25 b^{4} d^{4}\right ) x^{2}}{4 e^{5}}-\frac {b \left (12 a^{5} e^{5}+15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+30 a^{2} b^{3} d^{3} e^{2}+60 a \,b^{4} d^{4} e -137 b^{5} d^{5}\right ) x}{10 e^{6}}-\frac {10 a^{6} e^{6}+12 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}+60 a \,b^{5} d^{5} e -147 b^{6} d^{6}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}}\)	\(342\)
norman	\(\frac {-\frac {10 a^{6} e^{6}+12 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}+60 a \,b^{5} d^{5} e -147 b^{6} d^{6}}{60 e^{7}}-\frac {6 \left (e a \,b^{5}-d \,b^{6}\right ) x^{5}}{e^{2}}-\frac {15 \left (e^{2} a^{2} b^{4}+2 d e a \,b^{5}-3 b^{6} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {10 \left (2 e^{3} a^{3} b^{3}+3 d \,e^{2} a^{2} b^{4}+6 d^{2} e a \,b^{5}-11 d^{3} b^{6}\right ) x^{3}}{3 e^{4}}-\frac {5 \left (3 e^{4} a^{4} b^{2}+4 d \,e^{3} a^{3} b^{3}+6 d^{2} e^{2} a^{2} b^{4}+12 d^{3} e a \,b^{5}-25 d^{4} b^{6}\right ) x^{2}}{4 e^{5}}-\frac {\left (12 a^{5} b \,e^{5}+15 d \,e^{4} a^{4} b^{2}+20 d^{2} e^{3} a^{3} b^{3}+30 d^{3} e^{2} a^{2} b^{4}+60 d^{4} e a \,b^{5}-137 d^{5} b^{6}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}}\)	\(352\)
default	\(-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {20 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{7} \left (e x +d \right )^{3}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {6 b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{4}}\)	\(355\)

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)^7,x,method=_RETURNVERBOSE)

[Out]

-15/2*b^4/e^7*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(e*x+d)^2-20/3*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)^3-6/5*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)^5-6*b^5/e^7*(a*e-b*d)/(e*x+d)+b^6*ln(e*x+d)/e^7-1/6*(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)^6-15/4*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)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (163) = 326\).
time = 0.29, size = 387, normalized size = 2.32 \begin {gather*} b^{6} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x}{60 \, {\left (x^{6} e^{13} + 6 \, d x^{5} e^{12} + 15 \, d^{2} x^{4} e^{11} + 20 \, d^{3} x^{3} e^{10} + 15 \, d^{4} x^{2} e^{9} + 6 \, d^{5} x e^{8} + d^{6} 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)^7,x, algorithm="maxima")

[Out]

b^6*e^(-7)*log(x*e + d) + 1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^
4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d
*e^5 - a^2*b^4*e^6)*x^4 + 200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(2
5*b^6*d^4*e^2 - 12*a*b^5*d^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e
- 60*a*b^5*d^4*e^2 - 30*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x)/(x^6*e^13
+ 6*d*x^5*e^12 + 15*d^2*x^4*e^11 + 20*d^3*x^3*e^10 + 15*d^4*x^2*e^9 + 6*d^5*x*e^8 + d^6*e^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (163) = 326\).
time = 2.59, size = 459, normalized size = 2.75 \begin {gather*} \frac {147 \, b^{6} d^{6} - {\left (360 \, a b^{5} x^{5} + 450 \, a^{2} b^{4} x^{4} + 400 \, a^{3} b^{3} x^{3} + 225 \, a^{4} b^{2} x^{2} + 72 \, a^{5} b x + 10 \, a^{6}\right )} e^{6} + 6 \, {\left (60 \, b^{6} d x^{5} - 150 \, a b^{5} d x^{4} - 100 \, a^{2} b^{4} d x^{3} - 50 \, a^{3} b^{3} d x^{2} - 15 \, a^{4} b^{2} d x - 2 \, a^{5} b d\right )} e^{5} + 15 \, {\left (90 \, b^{6} d^{2} x^{4} - 80 \, a b^{5} d^{2} x^{3} - 30 \, a^{2} b^{4} d^{2} x^{2} - 8 \, a^{3} b^{3} d^{2} x - a^{4} b^{2} d^{2}\right )} e^{4} + 20 \, {\left (110 \, b^{6} d^{3} x^{3} - 45 \, a b^{5} d^{3} x^{2} - 9 \, a^{2} b^{4} d^{3} x - a^{3} b^{3} d^{3}\right )} e^{3} + 15 \, {\left (125 \, b^{6} d^{4} x^{2} - 24 \, a b^{5} d^{4} x - 2 \, a^{2} b^{4} d^{4}\right )} e^{2} + 6 \, {\left (137 \, b^{6} d^{5} x - 10 \, a b^{5} d^{5}\right )} e + 60 \, {\left (b^{6} x^{6} e^{6} + 6 \, b^{6} d x^{5} e^{5} + 15 \, b^{6} d^{2} x^{4} e^{4} + 20 \, b^{6} d^{3} x^{3} e^{3} + 15 \, b^{6} d^{4} x^{2} e^{2} + 6 \, b^{6} d^{5} x e + b^{6} d^{6}\right )} \log \left (x e + d\right )}{60 \, {\left (x^{6} e^{13} + 6 \, d x^{5} e^{12} + 15 \, d^{2} x^{4} e^{11} + 20 \, d^{3} x^{3} e^{10} + 15 \, d^{4} x^{2} e^{9} + 6 \, d^{5} x e^{8} + d^{6} 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)^7,x, algorithm="fricas")

[Out]

1/60*(147*b^6*d^6 - (360*a*b^5*x^5 + 450*a^2*b^4*x^4 + 400*a^3*b^3*x^3 + 225*a^4*b^2*x^2 + 72*a^5*b*x + 10*a^6
)*e^6 + 6*(60*b^6*d*x^5 - 150*a*b^5*d*x^4 - 100*a^2*b^4*d*x^3 - 50*a^3*b^3*d*x^2 - 15*a^4*b^2*d*x - 2*a^5*b*d)
*e^5 + 15*(90*b^6*d^2*x^4 - 80*a*b^5*d^2*x^3 - 30*a^2*b^4*d^2*x^2 - 8*a^3*b^3*d^2*x - a^4*b^2*d^2)*e^4 + 20*(1
10*b^6*d^3*x^3 - 45*a*b^5*d^3*x^2 - 9*a^2*b^4*d^3*x - a^3*b^3*d^3)*e^3 + 15*(125*b^6*d^4*x^2 - 24*a*b^5*d^4*x
- 2*a^2*b^4*d^4)*e^2 + 6*(137*b^6*d^5*x - 10*a*b^5*d^5)*e + 60*(b^6*x^6*e^6 + 6*b^6*d*x^5*e^5 + 15*b^6*d^2*x^4
*e^4 + 20*b^6*d^3*x^3*e^3 + 15*b^6*d^4*x^2*e^2 + 6*b^6*d^5*x*e + b^6*d^6)*log(x*e + d))/(x^6*e^13 + 6*d*x^5*e^
12 + 15*d^2*x^4*e^11 + 20*d^3*x^3*e^10 + 15*d^4*x^2*e^9 + 6*d^5*x*e^8 + d^6*e^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((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**7,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (163) = 326\).
time = 1.54, size = 339, normalized size = 2.03 \begin {gather*} b^{6} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (360 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{3} - 2 \, a b^{5} d e^{4} - a^{2} b^{4} e^{5}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} - 3 \, a^{2} b^{4} d e^{4} - 2 \, a^{3} b^{3} e^{5}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e - 12 \, a b^{5} d^{3} e^{2} - 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} - 3 \, a^{4} b^{2} e^{5}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} - 60 \, a b^{5} d^{4} e - 30 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} - 15 \, a^{4} b^{2} d e^{4} - 12 \, a^{5} b e^{5}\right )} x + {\left (147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

b^6*e^(-7)*log(abs(x*e + d)) + 1/60*(360*(b^6*d*e^4 - a*b^5*e^5)*x^5 + 450*(3*b^6*d^2*e^3 - 2*a*b^5*d*e^4 - a^
2*b^4*e^5)*x^4 + 200*(11*b^6*d^3*e^2 - 6*a*b^5*d^2*e^3 - 3*a^2*b^4*d*e^4 - 2*a^3*b^3*e^5)*x^3 + 75*(25*b^6*d^4
*e - 12*a*b^5*d^3*e^2 - 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 - 3*a^4*b^2*e^5)*x^2 + 6*(137*b^6*d^5 - 60*a*b^5*d
^4*e - 30*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 15*a^4*b^2*d*e^4 - 12*a^5*b*e^5)*x + (147*b^6*d^6 - 60*a*b^5*
d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6)*e^(-1))*e^
(-6)/(x*e + d)^6

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Mupad [B]
time = 0.64, size = 353, normalized size = 2.11 \begin {gather*} \frac {b^6\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (6\,a\,b^5\,e^6-6\,b^6\,d\,e^5\right )+x^2\,\left (\frac {15\,a^4\,b^2\,e^6}{4}+5\,a^3\,b^3\,d\,e^5+\frac {15\,a^2\,b^4\,d^2\,e^4}{2}+15\,a\,b^5\,d^3\,e^3-\frac {125\,b^6\,d^4\,e^2}{4}\right )+x^4\,\left (\frac {15\,a^2\,b^4\,e^6}{2}+15\,a\,b^5\,d\,e^5-\frac {45\,b^6\,d^2\,e^4}{2}\right )+x\,\left (\frac {6\,a^5\,b\,e^6}{5}+\frac {3\,a^4\,b^2\,d\,e^5}{2}+2\,a^3\,b^3\,d^2\,e^4+3\,a^2\,b^4\,d^3\,e^3+6\,a\,b^5\,d^4\,e^2-\frac {137\,b^6\,d^5\,e}{10}\right )+\frac {a^6\,e^6}{6}-\frac {49\,b^6\,d^6}{20}+x^3\,\left (\frac {20\,a^3\,b^3\,e^6}{3}+10\,a^2\,b^4\,d\,e^5+20\,a\,b^5\,d^2\,e^4-\frac {110\,b^6\,d^3\,e^3}{3}\right )+\frac {a^2\,b^4\,d^4\,e^2}{2}+\frac {a^3\,b^3\,d^3\,e^3}{3}+\frac {a^4\,b^2\,d^2\,e^4}{4}+a\,b^5\,d^5\,e+\frac {a^5\,b\,d\,e^5}{5}}{e^7\,{\left (d+e\,x\right )}^6} \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)^7,x)

[Out]

(b^6*log(d + e*x))/e^7 - (x^5*(6*a*b^5*e^6 - 6*b^6*d*e^5) + x^2*((15*a^4*b^2*e^6)/4 - (125*b^6*d^4*e^2)/4 + 15
*a*b^5*d^3*e^3 + 5*a^3*b^3*d*e^5 + (15*a^2*b^4*d^2*e^4)/2) + x^4*((15*a^2*b^4*e^6)/2 - (45*b^6*d^2*e^4)/2 + 15
*a*b^5*d*e^5) + x*((6*a^5*b*e^6)/5 - (137*b^6*d^5*e)/10 + 6*a*b^5*d^4*e^2 + (3*a^4*b^2*d*e^5)/2 + 3*a^2*b^4*d^
3*e^3 + 2*a^3*b^3*d^2*e^4) + (a^6*e^6)/6 - (49*b^6*d^6)/20 + x^3*((20*a^3*b^3*e^6)/3 - (110*b^6*d^3*e^3)/3 + 2
0*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5) + (a^2*b^4*d^4*e^2)/2 + (a^3*b^3*d^3*e^3)/3 + (a^4*b^2*d^2*e^4)/4 + a*b^5*
d^5*e + (a^5*b*d*e^5)/5)/(e^7*(d + e*x)^6)

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