3.15.90 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{d+e x} \, dx\) [1490]

Optimal. Leaf size=146 \[ -\frac {b (b d-a e)^5 x}{e^6}+\frac {(b d-a e)^4 (a+b x)^2}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^3}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^4}{4 e^3}-\frac {(b d-a e) (a+b x)^5}{5 e^2}+\frac {(a+b x)^6}{6 e}+\frac {(b d-a e)^6 \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 146, 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 {(b d-a e)^6 \log (d+e x)}{e^7}-\frac {b x (b d-a e)^5}{e^6}+\frac {(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac {(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac {(a+b x)^5 (b d-a e)}{5 e^2}+\frac {(a+b x)^6}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.06, size = 230, normalized size = 1.58 \begin {gather*} \frac {b e x \left (360 a^5 e^5+450 a^4 b e^4 (-2 d+e x)+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+75 a^2 b^3 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+6 a b^4 e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^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),x]

[Out]

(b*e*x*(360*a^5*e^5 + 450*a^4*b*e^4*(-2*d + e*x) + 200*a^3*b^2*e^3*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 75*a^2*b^3*
e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 6*a*b^4*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e
^3*x^3 + 12*e^4*x^4) + b^5*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5
)) + 60*(b*d - a*e)^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. \(658\) vs. \(2(136)=272\).
time = 0.74, size = 659, normalized size = 4.51

method	result	size
norman	\(\frac {b \left (6 a^{5} e^{5}-15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-15 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x}{e^{6}}+\frac {b^{6} x^{6}}{6 e}+\frac {b^{2} \left (15 e^{4} a^{4}-20 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}-6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{2 e^{5}}+\frac {b^{3} \left (20 e^{3} a^{3}-15 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{3}}{3 e^{4}}+\frac {b^{5} \left (6 a e -b d \right ) x^{5}}{5 e^{2}}+\frac {b^{4} \left (15 a^{2} e^{2}-6 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(333\)
risch	\(\frac {b^{6} x^{6}}{6 e}-\frac {6 \ln \left (e x +d \right ) a^{5} b d}{e^{2}}+\frac {15 \ln \left (e x +d \right ) a^{4} b^{2} d^{2}}{e^{3}}-\frac {20 \ln \left (e x +d \right ) a^{3} b^{3} d^{3}}{e^{4}}+\frac {15 \ln \left (e x +d \right ) a^{2} b^{4} d^{4}}{e^{5}}-\frac {6 \ln \left (e x +d \right ) a \,b^{5} d^{5}}{e^{6}}+\frac {6 b^{5} x^{5} a}{5 e}-\frac {b^{6} x^{5} d}{5 e^{2}}+\frac {15 b^{4} x^{4} a^{2}}{4 e}+\frac {b^{6} x^{4} d^{2}}{4 e^{3}}+\frac {20 b^{3} x^{3} a^{3}}{3 e}-\frac {b^{6} x^{3} d^{3}}{3 e^{4}}+\frac {15 b^{2} x^{2} a^{4}}{2 e}+\frac {b^{6} x^{2} d^{4}}{2 e^{5}}+\frac {6 b \,a^{5} x}{e}-\frac {b^{6} d^{5} x}{e^{6}}+\frac {\ln \left (e x +d \right ) a^{6}}{e}-\frac {5 b^{4} x^{3} a^{2} d}{e^{2}}+\frac {2 b^{5} x^{3} a \,d^{2}}{e^{3}}-\frac {10 b^{3} x^{2} a^{3} d}{e^{2}}+\frac {15 b^{4} x^{2} a^{2} d^{2}}{2 e^{3}}-\frac {3 b^{5} x^{2} a \,d^{3}}{e^{4}}-\frac {15 b^{2} a^{4} d x}{e^{2}}+\frac {20 b^{3} a^{3} d^{2} x}{e^{3}}-\frac {15 b^{4} a^{2} d^{3} x}{e^{4}}+\frac {6 b^{5} a \,d^{4} x}{e^{5}}-\frac {3 b^{5} x^{4} a d}{2 e^{2}}+\frac {\ln \left (e x +d \right ) b^{6} d^{6}}{e^{7}}\)	\(412\)
default	\(\frac {b \left (\frac {b^{5} x^{6} e^{5}}{6}+\frac {\left (\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (3 a b \,e^{2}-b^{2} d e \right )\right ) b^{2} e^{2}+b^{3} e^{3} \left (a b \,e^{2}+b^{2} d e \right )\right ) x^{5}}{5}+\frac {\left (\left (\left (2 a e -b d \right ) \left (3 a b \,e^{2}-b^{2} d e \right )+b e \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (3 a b \,e^{2}-b^{2} d e \right )\right ) \left (a b \,e^{2}+b^{2} d e \right )+b^{3} e^{3} \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a e -b d \right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) \left (3 a b \,e^{2}-b^{2} d e \right )+b e \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) \left (a b \,e^{2}+b^{2} d e \right )+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (3 a b \,e^{2}-b^{2} d e \right )\right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (2 a e -b d \right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right ) \left (a b \,e^{2}+b^{2} d e \right )+\left (\left (2 a e -b d \right ) \left (3 a b \,e^{2}-b^{2} d e \right )+b e \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (2 a e -b d \right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) x \right )}{e^{6}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(659\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (136) = 272\).
time = 0.33, size = 328, normalized size = 2.25 \begin {gather*} {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, 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} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{60} \, {\left (10 \, b^{6} x^{6} e^{5} - 12 \, {\left (b^{6} d e^{4} - 6 \, a b^{5} e^{5}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{3} - 6 \, a b^{5} d e^{4} + 15 \, a^{2} b^{4} e^{5}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} - 20 \, a^{3} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e - 6 \, a b^{5} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{2} e^{3} - 20 \, a^{3} b^{3} d e^{4} + 15 \, a^{4} b^{2} e^{5}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} - 6 \, a b^{5} d^{4} e + 15 \, 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} - 6 \, a^{5} b e^{5}\right )} x\right )} e^{\left (-6\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),x, algorithm="maxima")

[Out]

(b^6*d^6 - 6*a*b^5*d^5*e + 15*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 - 6*a^5*b*d*e^5 + a^6*
e^6)*e^(-7)*log(x*e + d) + 1/60*(10*b^6*x^6*e^5 - 12*(b^6*d*e^4 - 6*a*b^5*e^5)*x^5 + 15*(b^6*d^2*e^3 - 6*a*b^5
*d*e^4 + 15*a^2*b^4*e^5)*x^4 - 20*(b^6*d^3*e^2 - 6*a*b^5*d^2*e^3 + 15*a^2*b^4*d*e^4 - 20*a^3*b^3*e^5)*x^3 + 30
*(b^6*d^4*e - 6*a*b^5*d^3*e^2 + 15*a^2*b^4*d^2*e^3 - 20*a^3*b^3*d*e^4 + 15*a^4*b^2*e^5)*x^2 - 60*(b^6*d^5 - 6*
a*b^5*d^4*e + 15*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 - 6*a^5*b*e^5)*x)*e^(-6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (136) = 272\).
time = 3.18, size = 335, normalized size = 2.29 \begin {gather*} -\frac {1}{60} \, {\left (60 \, b^{6} d^{5} x e - {\left (10 \, b^{6} x^{6} + 72 \, a b^{5} x^{5} + 225 \, a^{2} b^{4} x^{4} + 400 \, a^{3} b^{3} x^{3} + 450 \, a^{4} b^{2} x^{2} + 360 \, a^{5} b x\right )} e^{6} + 6 \, {\left (2 \, b^{6} d x^{5} + 15 \, a b^{5} d x^{4} + 50 \, a^{2} b^{4} d x^{3} + 100 \, a^{3} b^{3} d x^{2} + 150 \, a^{4} b^{2} d x\right )} e^{5} - 15 \, {\left (b^{6} d^{2} x^{4} + 8 \, a b^{5} d^{2} x^{3} + 30 \, a^{2} b^{4} d^{2} x^{2} + 80 \, a^{3} b^{3} d^{2} x\right )} e^{4} + 20 \, {\left (b^{6} d^{3} x^{3} + 9 \, a b^{5} d^{3} x^{2} + 45 \, a^{2} b^{4} d^{3} x\right )} e^{3} - 30 \, {\left (b^{6} d^{4} x^{2} + 12 \, a b^{5} d^{4} x\right )} e^{2} - 60 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, 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} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (x e + d\right )\right )} e^{\left (-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),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (124) = 248\).
time = 0.37, size = 296, normalized size = 2.03 \begin {gather*} \frac {b^{6} x^{6}}{6 e} + x^{5} \cdot \left (\frac {6 a b^{5}}{5 e} - \frac {b^{6} d}{5 e^{2}}\right ) + x^{4} \cdot \left (\frac {15 a^{2} b^{4}}{4 e} - \frac {3 a b^{5} d}{2 e^{2}} + \frac {b^{6} d^{2}}{4 e^{3}}\right ) + x^{3} \cdot \left (\frac {20 a^{3} b^{3}}{3 e} - \frac {5 a^{2} b^{4} d}{e^{2}} + \frac {2 a b^{5} d^{2}}{e^{3}} - \frac {b^{6} d^{3}}{3 e^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{4} b^{2}}{2 e} - \frac {10 a^{3} b^{3} d}{e^{2}} + \frac {15 a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 a b^{5} d^{3}}{e^{4}} + \frac {b^{6} d^{4}}{2 e^{5}}\right ) + x \left (\frac {6 a^{5} b}{e} - \frac {15 a^{4} b^{2} d}{e^{2}} + \frac {20 a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 a b^{5} d^{4}}{e^{5}} - \frac {b^{6} d^{5}}{e^{6}}\right ) + \frac {\left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{7}} \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),x)

[Out]

b**6*x**6/(6*e) + x**5*(6*a*b**5/(5*e) - b**6*d/(5*e**2)) + x**4*(15*a**2*b**4/(4*e) - 3*a*b**5*d/(2*e**2) + b
**6*d**2/(4*e**3)) + x**3*(20*a**3*b**3/(3*e) - 5*a**2*b**4*d/e**2 + 2*a*b**5*d**2/e**3 - b**6*d**3/(3*e**4))
+ x**2*(15*a**4*b**2/(2*e) - 10*a**3*b**3*d/e**2 + 15*a**2*b**4*d**2/(2*e**3) - 3*a*b**5*d**3/e**4 + b**6*d**4
/(2*e**5)) + x*(6*a**5*b/e - 15*a**4*b**2*d/e**2 + 20*a**3*b**3*d**2/e**3 - 15*a**2*b**4*d**3/e**4 + 6*a*b**5*
d**4/e**5 - b**6*d**5/e**6) + (a*e - b*d)**6*log(d + e*x)/e**7

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (136) = 272\).
time = 1.82, size = 354, normalized size = 2.42 \begin {gather*} {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, 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} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, b^{6} x^{6} e^{5} - 12 \, b^{6} d x^{5} e^{4} + 15 \, b^{6} d^{2} x^{4} e^{3} - 20 \, b^{6} d^{3} x^{3} e^{2} + 30 \, b^{6} d^{4} x^{2} e - 60 \, b^{6} d^{5} x + 72 \, a b^{5} x^{5} e^{5} - 90 \, a b^{5} d x^{4} e^{4} + 120 \, a b^{5} d^{2} x^{3} e^{3} - 180 \, a b^{5} d^{3} x^{2} e^{2} + 360 \, a b^{5} d^{4} x e + 225 \, a^{2} b^{4} x^{4} e^{5} - 300 \, a^{2} b^{4} d x^{3} e^{4} + 450 \, a^{2} b^{4} d^{2} x^{2} e^{3} - 900 \, a^{2} b^{4} d^{3} x e^{2} + 400 \, a^{3} b^{3} x^{3} e^{5} - 600 \, a^{3} b^{3} d x^{2} e^{4} + 1200 \, a^{3} b^{3} d^{2} x e^{3} + 450 \, a^{4} b^{2} x^{2} e^{5} - 900 \, a^{4} b^{2} d x e^{4} + 360 \, a^{5} b x e^{5}\right )} e^{\left (-6\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),x, algorithm="giac")

[Out]

(b^6*d^6 - 6*a*b^5*d^5*e + 15*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 - 6*a^5*b*d*e^5 + a^6*
e^6)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*b^6*x^6*e^5 - 12*b^6*d*x^5*e^4 + 15*b^6*d^2*x^4*e^3 - 20*b^6*d^3*x^3*
e^2 + 30*b^6*d^4*x^2*e - 60*b^6*d^5*x + 72*a*b^5*x^5*e^5 - 90*a*b^5*d*x^4*e^4 + 120*a*b^5*d^2*x^3*e^3 - 180*a*
b^5*d^3*x^2*e^2 + 360*a*b^5*d^4*x*e + 225*a^2*b^4*x^4*e^5 - 300*a^2*b^4*d*x^3*e^4 + 450*a^2*b^4*d^2*x^2*e^3 -
900*a^2*b^4*d^3*x*e^2 + 400*a^3*b^3*x^3*e^5 - 600*a^3*b^3*d*x^2*e^4 + 1200*a^3*b^3*d^2*x*e^3 + 450*a^4*b^2*x^2
*e^5 - 900*a^4*b^2*d*x*e^4 + 360*a^5*b*x*e^5)*e^(-6)

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Mupad [B]
time = 0.52, size = 385, normalized size = 2.64 \begin {gather*} x^5\,\left (\frac {6\,a\,b^5}{5\,e}-\frac {b^6\,d}{5\,e^2}\right )+x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{3\,e}+\frac {20\,a^3\,b^3}{3\,e}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{e}-\frac {15\,a^4\,b^2}{e}\right )}{e}+\frac {6\,a^5\,b}{e}\right )-x^4\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{4\,e}-\frac {15\,a^2\,b^4}{4\,e}\right )-x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{2\,e}-\frac {15\,a^4\,b^2}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (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\right )}{e^7}+\frac {b^6\,x^6}{6\,e} \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),x)

[Out]

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

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