3.15.64 \(\int (d+e x)^5 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1464]

Optimal. Leaf size=119 \[ \frac {(b d-a e)^4 (d+e x)^6}{6 e^5}-\frac {4 b (b d-a e)^3 (d+e x)^7}{7 e^5}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8}{4 e^5}-\frac {4 b^3 (b d-a e) (d+e x)^9}{9 e^5}+\frac {b^4 (d+e x)^{10}}{10 e^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 119, 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 {4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac {3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac {4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac {b^4 (d+e x)^{10}}{10 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

((b*d - a*e)^4*(d + e*x)^6)/(6*e^5) - (4*b*(b*d - a*e)^3*(d + e*x)^7)/(7*e^5) + (3*b^2*(b*d - a*e)^2*(d + e*x)
^8)/(4*e^5) - (4*b^3*(b*d - a*e)*(d + e*x)^9)/(9*e^5) + (b^4*(d + e*x)^10)/(10*e^5)

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(350\) vs. \(2(119)=238\).
time = 0.03, size = 350, normalized size = 2.94 \begin {gather*} a^4 d^5 x+\frac {1}{2} a^3 d^4 (4 b d+5 a e) x^2+\frac {2}{3} a^2 d^3 \left (3 b^2 d^2+10 a b d e+5 a^2 e^2\right ) x^3+\frac {1}{2} a d^2 \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right ) x^4+\frac {1}{5} d \left (b^4 d^4+20 a b^3 d^3 e+60 a^2 b^2 d^2 e^2+40 a^3 b d e^3+5 a^4 e^4\right ) x^5+\frac {1}{6} e \left (5 b^4 d^4+40 a b^3 d^3 e+60 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4\right ) x^6+\frac {2}{7} b e^2 \left (5 b^3 d^3+20 a b^2 d^2 e+15 a^2 b d e^2+2 a^3 e^3\right ) x^7+\frac {1}{4} b^2 e^3 \left (5 b^2 d^2+10 a b d e+3 a^2 e^2\right ) x^8+\frac {1}{9} b^3 e^4 (5 b d+4 a e) x^9+\frac {1}{10} b^4 e^5 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*d^5*x + (a^3*d^4*(4*b*d + 5*a*e)*x^2)/2 + (2*a^2*d^3*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2)*x^3)/3 + (a*d^2*
(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3)*x^4)/2 + (d*(b^4*d^4 + 20*a*b^3*d^3*e + 60*a^2*b^2*d
^2*e^2 + 40*a^3*b*d*e^3 + 5*a^4*e^4)*x^5)/5 + (e*(5*b^4*d^4 + 40*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 20*a^3*b*d
*e^3 + a^4*e^4)*x^6)/6 + (2*b*e^2*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 2*a^3*e^3)*x^7)/7 + (b^2*e^3*
(5*b^2*d^2 + 10*a*b*d*e + 3*a^2*e^2)*x^8)/4 + (b^3*e^4*(5*b*d + 4*a*e)*x^9)/9 + (b^4*e^5*x^10)/10

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(109)=218\).
time = 0.68, size = 361, normalized size = 3.03

method	result	size
norman	\(\frac {b^{4} e^{5} x^{10}}{10}+\left (\frac {4}{9} a \,b^{3} e^{5}+\frac {5}{9} b^{4} d \,e^{4}\right ) x^{9}+\left (\frac {3}{4} a^{2} b^{2} e^{5}+\frac {5}{2} a \,b^{3} d \,e^{4}+\frac {5}{4} b^{4} d^{2} e^{3}\right ) x^{8}+\left (\frac {4}{7} a^{3} b \,e^{5}+\frac {30}{7} a^{2} b^{2} d \,e^{4}+\frac {40}{7} a \,b^{3} d^{2} e^{3}+\frac {10}{7} b^{4} d^{3} e^{2}\right ) x^{7}+\left (\frac {1}{6} a^{4} e^{5}+\frac {10}{3} a^{3} b d \,e^{4}+10 a^{2} b^{2} d^{2} e^{3}+\frac {20}{3} a \,b^{3} d^{3} e^{2}+\frac {5}{6} b^{4} d^{4} e \right ) x^{6}+\left (a^{4} d \,e^{4}+8 a^{3} b \,d^{2} e^{3}+12 a^{2} b^{2} d^{3} e^{2}+4 a \,b^{3} d^{4} e +\frac {1}{5} b^{4} d^{5}\right ) x^{5}+\left (\frac {5}{2} a^{4} d^{2} e^{3}+10 a^{3} b \,d^{3} e^{2}+\frac {15}{2} a^{2} b^{2} d^{4} e +a \,b^{3} d^{5}\right ) x^{4}+\left (\frac {10}{3} a^{4} d^{3} e^{2}+\frac {20}{3} a^{3} b \,d^{4} e +2 a^{2} b^{2} d^{5}\right ) x^{3}+\left (\frac {5}{2} a^{4} d^{4} e +2 a^{3} b \,d^{5}\right ) x^{2}+a^{4} d^{5} x\)	\(353\)
default	\(\frac {b^{4} e^{5} x^{10}}{10}+\frac {\left (4 a \,b^{3} e^{5}+5 b^{4} d \,e^{4}\right ) x^{9}}{9}+\frac {\left (6 a^{2} b^{2} e^{5}+20 a \,b^{3} d \,e^{4}+10 b^{4} d^{2} e^{3}\right ) x^{8}}{8}+\frac {\left (4 a^{3} b \,e^{5}+30 a^{2} b^{2} d \,e^{4}+40 a \,b^{3} d^{2} e^{3}+10 b^{4} d^{3} e^{2}\right ) x^{7}}{7}+\frac {\left (a^{4} e^{5}+20 a^{3} b d \,e^{4}+60 a^{2} b^{2} d^{2} e^{3}+40 a \,b^{3} d^{3} e^{2}+5 b^{4} d^{4} e \right ) x^{6}}{6}+\frac {\left (5 a^{4} d \,e^{4}+40 a^{3} b \,d^{2} e^{3}+60 a^{2} b^{2} d^{3} e^{2}+20 a \,b^{3} d^{4} e +b^{4} d^{5}\right ) x^{5}}{5}+\frac {\left (10 a^{4} d^{2} e^{3}+40 a^{3} b \,d^{3} e^{2}+30 a^{2} b^{2} d^{4} e +4 a \,b^{3} d^{5}\right ) x^{4}}{4}+\frac {\left (10 a^{4} d^{3} e^{2}+20 a^{3} b \,d^{4} e +6 a^{2} b^{2} d^{5}\right ) x^{3}}{3}+\frac {\left (5 a^{4} d^{4} e +4 a^{3} b \,d^{5}\right ) x^{2}}{2}+a^{4} d^{5} x\)	\(361\)
risch	\(a^{4} d^{5} x +\frac {5}{2} x^{4} a^{4} d^{2} e^{3}+x^{4} a \,b^{3} d^{5}+\frac {10}{3} x^{3} a^{4} d^{3} e^{2}+2 x^{3} a^{2} b^{2} d^{5}+2 x^{2} a^{3} b \,d^{5}+\frac {1}{10} b^{4} e^{5} x^{10}+\frac {5}{2} x^{8} a \,b^{3} d \,e^{4}+\frac {30}{7} x^{7} a^{2} b^{2} d \,e^{4}+\frac {4}{9} x^{9} a \,b^{3} e^{5}+\frac {5}{9} x^{9} b^{4} d \,e^{4}+\frac {3}{4} x^{8} a^{2} b^{2} e^{5}+\frac {5}{4} x^{8} b^{4} d^{2} e^{3}+\frac {4}{7} x^{7} a^{3} b \,e^{5}+\frac {10}{7} x^{7} b^{4} d^{3} e^{2}+\frac {5}{6} x^{6} b^{4} d^{4} e +x^{5} a^{4} d \,e^{4}+\frac {1}{6} x^{6} a^{4} e^{5}+\frac {1}{5} x^{5} b^{4} d^{5}+\frac {5}{2} d^{4} e \,a^{4} x^{2}+\frac {15}{2} x^{4} a^{2} b^{2} d^{4} e +\frac {20}{3} x^{3} a^{3} b \,d^{4} e +\frac {40}{7} x^{7} a \,b^{3} d^{2} e^{3}+\frac {10}{3} x^{6} a^{3} b d \,e^{4}+10 x^{6} a^{2} b^{2} d^{2} e^{3}+\frac {20}{3} x^{6} a \,b^{3} d^{3} e^{2}+8 x^{5} a^{3} b \,d^{2} e^{3}+12 x^{5} a^{2} b^{2} d^{3} e^{2}+4 x^{5} a \,b^{3} d^{4} e +10 x^{4} a^{3} b \,d^{3} e^{2}\)	\(397\)
gosper	\(\frac {x \left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} b^{4} d^{2} e^{3}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right )}{1260}\)	\(398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/10*b^4*e^5*x^10+1/9*(4*a*b^3*e^5+5*b^4*d*e^4)*x^9+1/8*(6*a^2*b^2*e^5+20*a*b^3*d*e^4+10*b^4*d^2*e^3)*x^8+1/7*
(4*a^3*b*e^5+30*a^2*b^2*d*e^4+40*a*b^3*d^2*e^3+10*b^4*d^3*e^2)*x^7+1/6*(a^4*e^5+20*a^3*b*d*e^4+60*a^2*b^2*d^2*
e^3+40*a*b^3*d^3*e^2+5*b^4*d^4*e)*x^6+1/5*(5*a^4*d*e^4+40*a^3*b*d^2*e^3+60*a^2*b^2*d^3*e^2+20*a*b^3*d^4*e+b^4*
d^5)*x^5+1/4*(10*a^4*d^2*e^3+40*a^3*b*d^3*e^2+30*a^2*b^2*d^4*e+4*a*b^3*d^5)*x^4+1/3*(10*a^4*d^3*e^2+20*a^3*b*d
^4*e+6*a^2*b^2*d^5)*x^3+1/2*(5*a^4*d^4*e+4*a^3*b*d^5)*x^2+a^4*d^5*x

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (113) = 226\).
time = 0.27, size = 345, normalized size = 2.90 \begin {gather*} \frac {1}{10} \, b^{4} x^{10} e^{5} + a^{4} d^{5} x + \frac {1}{9} \, {\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac {1}{4} \, {\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac {2}{7} \, {\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/10*b^4*x^10*e^5 + a^4*d^5*x + 1/9*(5*b^4*d*e^4 + 4*a*b^3*e^5)*x^9 + 1/4*(5*b^4*d^2*e^3 + 10*a*b^3*d*e^4 + 3*
a^2*b^2*e^5)*x^8 + 2/7*(5*b^4*d^3*e^2 + 20*a*b^3*d^2*e^3 + 15*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^7 + 1/6*(5*b^4*d^
4*e + 40*a*b^3*d^3*e^2 + 60*a^2*b^2*d^2*e^3 + 20*a^3*b*d*e^4 + a^4*e^5)*x^6 + 1/5*(b^4*d^5 + 20*a*b^3*d^4*e +
60*a^2*b^2*d^3*e^2 + 40*a^3*b*d^2*e^3 + 5*a^4*d*e^4)*x^5 + 1/2*(2*a*b^3*d^5 + 15*a^2*b^2*d^4*e + 20*a^3*b*d^3*
e^2 + 5*a^4*d^2*e^3)*x^4 + 2/3*(3*a^2*b^2*d^5 + 10*a^3*b*d^4*e + 5*a^4*d^3*e^2)*x^3 + 1/2*(4*a^3*b*d^5 + 5*a^4
*d^4*e)*x^2

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (113) = 226\).
time = 1.92, size = 357, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, b^{4} d^{5} x^{5} + a b^{3} d^{5} x^{4} + 2 \, a^{2} b^{2} d^{5} x^{3} + 2 \, a^{3} b d^{5} x^{2} + a^{4} d^{5} x + \frac {1}{1260} \, {\left (126 \, b^{4} x^{10} + 560 \, a b^{3} x^{9} + 945 \, a^{2} b^{2} x^{8} + 720 \, a^{3} b x^{7} + 210 \, a^{4} x^{6}\right )} e^{5} + \frac {1}{126} \, {\left (70 \, b^{4} d x^{9} + 315 \, a b^{3} d x^{8} + 540 \, a^{2} b^{2} d x^{7} + 420 \, a^{3} b d x^{6} + 126 \, a^{4} d x^{5}\right )} e^{4} + \frac {1}{28} \, {\left (35 \, b^{4} d^{2} x^{8} + 160 \, a b^{3} d^{2} x^{7} + 280 \, a^{2} b^{2} d^{2} x^{6} + 224 \, a^{3} b d^{2} x^{5} + 70 \, a^{4} d^{2} x^{4}\right )} e^{3} + \frac {2}{21} \, {\left (15 \, b^{4} d^{3} x^{7} + 70 \, a b^{3} d^{3} x^{6} + 126 \, a^{2} b^{2} d^{3} x^{5} + 105 \, a^{3} b d^{3} x^{4} + 35 \, a^{4} d^{3} x^{3}\right )} e^{2} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} x^{6} + 24 \, a b^{3} d^{4} x^{5} + 45 \, a^{2} b^{2} d^{4} x^{4} + 40 \, a^{3} b d^{4} x^{3} + 15 \, a^{4} d^{4} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/5*b^4*d^5*x^5 + a*b^3*d^5*x^4 + 2*a^2*b^2*d^5*x^3 + 2*a^3*b*d^5*x^2 + a^4*d^5*x + 1/1260*(126*b^4*x^10 + 560
*a*b^3*x^9 + 945*a^2*b^2*x^8 + 720*a^3*b*x^7 + 210*a^4*x^6)*e^5 + 1/126*(70*b^4*d*x^9 + 315*a*b^3*d*x^8 + 540*
a^2*b^2*d*x^7 + 420*a^3*b*d*x^6 + 126*a^4*d*x^5)*e^4 + 1/28*(35*b^4*d^2*x^8 + 160*a*b^3*d^2*x^7 + 280*a^2*b^2*
d^2*x^6 + 224*a^3*b*d^2*x^5 + 70*a^4*d^2*x^4)*e^3 + 2/21*(15*b^4*d^3*x^7 + 70*a*b^3*d^3*x^6 + 126*a^2*b^2*d^3*
x^5 + 105*a^3*b*d^3*x^4 + 35*a^4*d^3*x^3)*e^2 + 1/6*(5*b^4*d^4*x^6 + 24*a*b^3*d^4*x^5 + 45*a^2*b^2*d^4*x^4 + 4
0*a^3*b*d^4*x^3 + 15*a^4*d^4*x^2)*e

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (107) = 214\).
time = 0.04, size = 401, normalized size = 3.37 \begin {gather*} a^{4} d^{5} x + \frac {b^{4} e^{5} x^{10}}{10} + x^{9} \cdot \left (\frac {4 a b^{3} e^{5}}{9} + \frac {5 b^{4} d e^{4}}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} b^{2} e^{5}}{4} + \frac {5 a b^{3} d e^{4}}{2} + \frac {5 b^{4} d^{2} e^{3}}{4}\right ) + x^{7} \cdot \left (\frac {4 a^{3} b e^{5}}{7} + \frac {30 a^{2} b^{2} d e^{4}}{7} + \frac {40 a b^{3} d^{2} e^{3}}{7} + \frac {10 b^{4} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac {a^{4} e^{5}}{6} + \frac {10 a^{3} b d e^{4}}{3} + 10 a^{2} b^{2} d^{2} e^{3} + \frac {20 a b^{3} d^{3} e^{2}}{3} + \frac {5 b^{4} d^{4} e}{6}\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} b d^{2} e^{3} + 12 a^{2} b^{2} d^{3} e^{2} + 4 a b^{3} d^{4} e + \frac {b^{4} d^{5}}{5}\right ) + x^{4} \cdot \left (\frac {5 a^{4} d^{2} e^{3}}{2} + 10 a^{3} b d^{3} e^{2} + \frac {15 a^{2} b^{2} d^{4} e}{2} + a b^{3} d^{5}\right ) + x^{3} \cdot \left (\frac {10 a^{4} d^{3} e^{2}}{3} + \frac {20 a^{3} b d^{4} e}{3} + 2 a^{2} b^{2} d^{5}\right ) + x^{2} \cdot \left (\frac {5 a^{4} d^{4} e}{2} + 2 a^{3} b d^{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**4*d**5*x + b**4*e**5*x**10/10 + x**9*(4*a*b**3*e**5/9 + 5*b**4*d*e**4/9) + x**8*(3*a**2*b**2*e**5/4 + 5*a*b
**3*d*e**4/2 + 5*b**4*d**2*e**3/4) + x**7*(4*a**3*b*e**5/7 + 30*a**2*b**2*d*e**4/7 + 40*a*b**3*d**2*e**3/7 + 1
0*b**4*d**3*e**2/7) + x**6*(a**4*e**5/6 + 10*a**3*b*d*e**4/3 + 10*a**2*b**2*d**2*e**3 + 20*a*b**3*d**3*e**2/3
+ 5*b**4*d**4*e/6) + x**5*(a**4*d*e**4 + 8*a**3*b*d**2*e**3 + 12*a**2*b**2*d**3*e**2 + 4*a*b**3*d**4*e + b**4*
d**5/5) + x**4*(5*a**4*d**2*e**3/2 + 10*a**3*b*d**3*e**2 + 15*a**2*b**2*d**4*e/2 + a*b**3*d**5) + x**3*(10*a**
4*d**3*e**2/3 + 20*a**3*b*d**4*e/3 + 2*a**2*b**2*d**5) + x**2*(5*a**4*d**4*e/2 + 2*a**3*b*d**5)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (113) = 226\).
time = 1.95, size = 381, normalized size = 3.20 \begin {gather*} \frac {1}{10} \, b^{4} x^{10} e^{5} + \frac {5}{9} \, b^{4} d x^{9} e^{4} + \frac {5}{4} \, b^{4} d^{2} x^{8} e^{3} + \frac {10}{7} \, b^{4} d^{3} x^{7} e^{2} + \frac {5}{6} \, b^{4} d^{4} x^{6} e + \frac {1}{5} \, b^{4} d^{5} x^{5} + \frac {4}{9} \, a b^{3} x^{9} e^{5} + \frac {5}{2} \, a b^{3} d x^{8} e^{4} + \frac {40}{7} \, a b^{3} d^{2} x^{7} e^{3} + \frac {20}{3} \, a b^{3} d^{3} x^{6} e^{2} + 4 \, a b^{3} d^{4} x^{5} e + a b^{3} d^{5} x^{4} + \frac {3}{4} \, a^{2} b^{2} x^{8} e^{5} + \frac {30}{7} \, a^{2} b^{2} d x^{7} e^{4} + 10 \, a^{2} b^{2} d^{2} x^{6} e^{3} + 12 \, a^{2} b^{2} d^{3} x^{5} e^{2} + \frac {15}{2} \, a^{2} b^{2} d^{4} x^{4} e + 2 \, a^{2} b^{2} d^{5} x^{3} + \frac {4}{7} \, a^{3} b x^{7} e^{5} + \frac {10}{3} \, a^{3} b d x^{6} e^{4} + 8 \, a^{3} b d^{2} x^{5} e^{3} + 10 \, a^{3} b d^{3} x^{4} e^{2} + \frac {20}{3} \, a^{3} b d^{4} x^{3} e + 2 \, a^{3} b d^{5} x^{2} + \frac {1}{6} \, a^{4} x^{6} e^{5} + a^{4} d x^{5} e^{4} + \frac {5}{2} \, a^{4} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{4} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{4} d^{4} x^{2} e + a^{4} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/10*b^4*x^10*e^5 + 5/9*b^4*d*x^9*e^4 + 5/4*b^4*d^2*x^8*e^3 + 10/7*b^4*d^3*x^7*e^2 + 5/6*b^4*d^4*x^6*e + 1/5*b
^4*d^5*x^5 + 4/9*a*b^3*x^9*e^5 + 5/2*a*b^3*d*x^8*e^4 + 40/7*a*b^3*d^2*x^7*e^3 + 20/3*a*b^3*d^3*x^6*e^2 + 4*a*b
^3*d^4*x^5*e + a*b^3*d^5*x^4 + 3/4*a^2*b^2*x^8*e^5 + 30/7*a^2*b^2*d*x^7*e^4 + 10*a^2*b^2*d^2*x^6*e^3 + 12*a^2*
b^2*d^3*x^5*e^2 + 15/2*a^2*b^2*d^4*x^4*e + 2*a^2*b^2*d^5*x^3 + 4/7*a^3*b*x^7*e^5 + 10/3*a^3*b*d*x^6*e^4 + 8*a^
3*b*d^2*x^5*e^3 + 10*a^3*b*d^3*x^4*e^2 + 20/3*a^3*b*d^4*x^3*e + 2*a^3*b*d^5*x^2 + 1/6*a^4*x^6*e^5 + a^4*d*x^5*
e^4 + 5/2*a^4*d^2*x^4*e^3 + 10/3*a^4*d^3*x^3*e^2 + 5/2*a^4*d^4*x^2*e + a^4*d^5*x

________________________________________________________________________________________

Mupad [B]
time = 0.13, size = 340, normalized size = 2.86 \begin {gather*} x^4\,\left (\frac {5\,a^4\,d^2\,e^3}{2}+10\,a^3\,b\,d^3\,e^2+\frac {15\,a^2\,b^2\,d^4\,e}{2}+a\,b^3\,d^5\right )+x^7\,\left (\frac {4\,a^3\,b\,e^5}{7}+\frac {30\,a^2\,b^2\,d\,e^4}{7}+\frac {40\,a\,b^3\,d^2\,e^3}{7}+\frac {10\,b^4\,d^3\,e^2}{7}\right )+x^5\,\left (a^4\,d\,e^4+8\,a^3\,b\,d^2\,e^3+12\,a^2\,b^2\,d^3\,e^2+4\,a\,b^3\,d^4\,e+\frac {b^4\,d^5}{5}\right )+x^6\,\left (\frac {a^4\,e^5}{6}+\frac {10\,a^3\,b\,d\,e^4}{3}+10\,a^2\,b^2\,d^2\,e^3+\frac {20\,a\,b^3\,d^3\,e^2}{3}+\frac {5\,b^4\,d^4\,e}{6}\right )+a^4\,d^5\,x+\frac {b^4\,e^5\,x^{10}}{10}+\frac {a^3\,d^4\,x^2\,\left (5\,a\,e+4\,b\,d\right )}{2}+\frac {b^3\,e^4\,x^9\,\left (4\,a\,e+5\,b\,d\right )}{9}+\frac {2\,a^2\,d^3\,x^3\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b^2\,e^3\,x^8\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^4*(a*b^3*d^5 + (5*a^4*d^2*e^3)/2 + (15*a^2*b^2*d^4*e)/2 + 10*a^3*b*d^3*e^2) + x^7*((4*a^3*b*e^5)/7 + (10*b^4
*d^3*e^2)/7 + (40*a*b^3*d^2*e^3)/7 + (30*a^2*b^2*d*e^4)/7) + x^5*((b^4*d^5)/5 + a^4*d*e^4 + 8*a^3*b*d^2*e^3 +
12*a^2*b^2*d^3*e^2 + 4*a*b^3*d^4*e) + x^6*((a^4*e^5)/6 + (5*b^4*d^4*e)/6 + (20*a*b^3*d^3*e^2)/3 + 10*a^2*b^2*d
^2*e^3 + (10*a^3*b*d*e^4)/3) + a^4*d^5*x + (b^4*e^5*x^10)/10 + (a^3*d^4*x^2*(5*a*e + 4*b*d))/2 + (b^3*e^4*x^9*
(4*a*e + 5*b*d))/9 + (2*a^2*d^3*x^3*(5*a^2*e^2 + 3*b^2*d^2 + 10*a*b*d*e))/3 + (b^2*e^3*x^8*(3*a^2*e^2 + 5*b^2*
d^2 + 10*a*b*d*e))/4

________________________________________________________________________________________