3.15.85 \(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1485]

Optimal. Leaf size=119 \[ \frac {(b d-a e)^4 (a+b x)^7}{7 b^5}+\frac {e (b d-a e)^3 (a+b x)^8}{2 b^5}+\frac {2 e^2 (b d-a e)^2 (a+b x)^9}{3 b^5}+\frac {2 e^3 (b d-a e) (a+b x)^{10}}{5 b^5}+\frac {e^4 (a+b x)^{11}}{11 b^5} \]

[Out]

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

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Rubi [A]
time = 0.18, 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 {2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac {2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac {e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac {(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac {e^4 (a+b x)^{11}}{11 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^4*(a + b*x)^7)/(7*b^5) + (e*(b*d - a*e)^3*(a + b*x)^8)/(2*b^5) + (2*e^2*(b*d - a*e)^2*(a + b*x)^9
)/(3*b^5) + (2*e^3*(b*d - a*e)*(a + b*x)^10)/(5*b^5) + (e^4*(a + b*x)^11)/(11*b^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)^4 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^4 \, dx\\ &=\int \left (\frac {(b d-a e)^4 (a+b x)^6}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^7}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^8}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^9}{b^4}+\frac {e^4 (a+b x)^{10}}{b^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (a+b x)^7}{7 b^5}+\frac {e (b d-a e)^3 (a+b x)^8}{2 b^5}+\frac {2 e^2 (b d-a e)^2 (a+b x)^9}{3 b^5}+\frac {2 e^3 (b d-a e) (a+b x)^{10}}{5 b^5}+\frac {e^4 (a+b x)^{11}}{11 b^5}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(119)=238\).
time = 0.04, size = 398, normalized size = 3.34 \begin {gather*} a^6 d^4 x+a^5 d^3 (3 b d+2 a e) x^2+a^4 d^2 \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right ) x^3+a^3 d \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {1}{5} a^2 \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right ) x^5+a b \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right ) x^6+\frac {1}{7} b^2 \left (b^4 d^4+24 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4\right ) x^7+\frac {1}{2} b^3 e \left (b^3 d^3+9 a b^2 d^2 e+15 a^2 b d e^2+5 a^3 e^3\right ) x^8+\frac {1}{3} b^4 e^2 \left (2 b^2 d^2+8 a b d e+5 a^2 e^2\right ) x^9+\frac {1}{5} b^5 e^3 (2 b d+3 a e) x^{10}+\frac {1}{11} b^6 e^4 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

a^6*d^4*x + a^5*d^3*(3*b*d + 2*a*e)*x^2 + a^4*d^2*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^3 + a^3*d*(5*b^3*d^3 +
15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^4 + (a^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a
^3*b*d*e^3 + a^4*e^4)*x^5)/5 + a*b*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*
x^6 + (b^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4)*x^7)/7 + (b^3*e*(b^3*
d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^8)/2 + (b^4*e^2*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^9)/3
+ (b^5*e^3*(2*b*d + 3*a*e)*x^10)/5 + (b^6*e^4*x^11)/11

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(109)=218\).
time = 0.66, size = 427, normalized size = 3.59

method	result	size
norman	\(\frac {e^{4} b^{6} x^{11}}{11}+\left (\frac {3}{5} e^{4} a \,b^{5}+\frac {2}{5} d \,e^{3} b^{6}\right ) x^{10}+\left (\frac {5}{3} e^{4} a^{2} b^{4}+\frac {8}{3} d \,e^{3} a \,b^{5}+\frac {2}{3} d^{2} e^{2} b^{6}\right ) x^{9}+\left (\frac {5}{2} e^{4} a^{3} b^{3}+\frac {15}{2} d \,e^{3} a^{2} b^{4}+\frac {9}{2} d^{2} e^{2} a \,b^{5}+\frac {1}{2} d^{3} e \,b^{6}\right ) x^{8}+\left (\frac {15}{7} e^{4} a^{4} b^{2}+\frac {80}{7} d \,e^{3} a^{3} b^{3}+\frac {90}{7} d^{2} e^{2} a^{2} b^{4}+\frac {24}{7} d^{3} e a \,b^{5}+\frac {1}{7} d^{4} b^{6}\right ) x^{7}+\left (e^{4} a^{5} b +10 d \,e^{3} a^{4} b^{2}+20 d^{2} e^{2} a^{3} b^{3}+10 d^{3} e \,a^{2} b^{4}+d^{4} a \,b^{5}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{6}+\frac {24}{5} d \,e^{3} a^{5} b +18 d^{2} e^{2} a^{4} b^{2}+16 d^{3} e \,a^{3} b^{3}+3 b^{4} d^{4} a^{2}\right ) x^{5}+\left (d \,e^{3} a^{6}+9 d^{2} e^{2} a^{5} b +15 d^{3} e \,a^{4} b^{2}+5 d^{4} a^{3} b^{3}\right ) x^{4}+\left (2 d^{2} e^{2} a^{6}+8 d^{3} e \,a^{5} b +5 d^{4} a^{4} b^{2}\right ) x^{3}+\left (2 d^{3} e \,a^{6}+3 d^{4} a^{5} b \right ) x^{2}+d^{4} a^{6} x\)	\(417\)
default	\(\frac {e^{4} b^{6} x^{11}}{11}+\frac {\left (6 e^{4} a \,b^{5}+4 d \,e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (15 e^{4} a^{2} b^{4}+24 d \,e^{3} a \,b^{5}+6 d^{2} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (20 e^{4} a^{3} b^{3}+60 d \,e^{3} a^{2} b^{4}+36 d^{2} e^{2} a \,b^{5}+4 d^{3} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (15 e^{4} a^{4} b^{2}+80 d \,e^{3} a^{3} b^{3}+90 d^{2} e^{2} a^{2} b^{4}+24 d^{3} e a \,b^{5}+d^{4} b^{6}\right ) x^{7}}{7}+\frac {\left (6 e^{4} a^{5} b +60 d \,e^{3} a^{4} b^{2}+120 d^{2} e^{2} a^{3} b^{3}+60 d^{3} e \,a^{2} b^{4}+6 d^{4} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (e^{4} a^{6}+24 d \,e^{3} a^{5} b +90 d^{2} e^{2} a^{4} b^{2}+80 d^{3} e \,a^{3} b^{3}+15 b^{4} d^{4} a^{2}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{6}+36 d^{2} e^{2} a^{5} b +60 d^{3} e \,a^{4} b^{2}+20 d^{4} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{6}+24 d^{3} e \,a^{5} b +15 d^{4} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{6}+6 d^{4} a^{5} b \right ) x^{2}}{2}+d^{4} a^{6} x\)	\(427\)
risch	\(\frac {1}{11} e^{4} b^{6} x^{11}+d^{4} a^{6} x +\frac {1}{7} x^{7} d^{4} b^{6}+\frac {1}{5} x^{5} e^{4} a^{6}+\frac {24}{5} x^{5} d \,e^{3} a^{5} b +18 x^{5} d^{2} e^{2} a^{4} b^{2}+16 x^{5} d^{3} e \,a^{3} b^{3}+10 a^{4} b^{2} d \,e^{3} x^{6}+20 a^{3} b^{3} d^{2} e^{2} x^{6}+10 a^{2} b^{4} d^{3} e \,x^{6}+9 a^{5} b \,d^{2} e^{2} x^{4}+15 a^{4} b^{2} d^{3} e \,x^{4}+8 a^{5} b \,d^{3} e \,x^{3}+\frac {3}{5} x^{10} e^{4} a \,b^{5}+\frac {2}{5} x^{10} d \,e^{3} b^{6}+\frac {5}{3} x^{9} e^{4} a^{2} b^{4}+\frac {2}{3} x^{9} d^{2} e^{2} b^{6}+\frac {5}{2} x^{8} e^{4} a^{3} b^{3}+\frac {1}{2} x^{8} d^{3} e \,b^{6}+\frac {15}{7} x^{7} e^{4} a^{4} b^{2}+3 x^{5} b^{4} d^{4} a^{2}+a^{5} b \,e^{4} x^{6}+a \,b^{5} d^{4} x^{6}+a^{6} d \,e^{3} x^{4}+5 a^{3} b^{3} d^{4} x^{4}+2 a^{6} d^{2} e^{2} x^{3}+5 a^{4} b^{2} d^{4} x^{3}+2 a^{6} d^{3} e \,x^{2}+3 a^{5} b \,d^{4} x^{2}+\frac {8}{3} x^{9} d \,e^{3} a \,b^{5}+\frac {15}{2} x^{8} d \,e^{3} a^{2} b^{4}+\frac {9}{2} x^{8} d^{2} e^{2} a \,b^{5}+\frac {80}{7} x^{7} d \,e^{3} a^{3} b^{3}+\frac {90}{7} x^{7} d^{2} e^{2} a^{2} b^{4}+\frac {24}{7} x^{7} d^{3} e a \,b^{5}\)	\(471\)
gosper	\(\frac {x \left (210 e^{4} b^{6} x^{10}+1386 x^{9} e^{4} a \,b^{5}+924 x^{9} d \,e^{3} b^{6}+3850 x^{8} e^{4} a^{2} b^{4}+6160 x^{8} d \,e^{3} a \,b^{5}+1540 x^{8} d^{2} e^{2} b^{6}+5775 x^{7} e^{4} a^{3} b^{3}+17325 x^{7} d \,e^{3} a^{2} b^{4}+10395 x^{7} d^{2} e^{2} a \,b^{5}+1155 x^{7} d^{3} e \,b^{6}+4950 x^{6} e^{4} a^{4} b^{2}+26400 x^{6} d \,e^{3} a^{3} b^{3}+29700 x^{6} d^{2} e^{2} a^{2} b^{4}+7920 x^{6} d^{3} e a \,b^{5}+330 x^{6} d^{4} b^{6}+2310 a^{5} b \,e^{4} x^{5}+23100 a^{4} b^{2} d \,e^{3} x^{5}+46200 a^{3} b^{3} d^{2} e^{2} x^{5}+23100 a^{2} b^{4} d^{3} e \,x^{5}+2310 a \,b^{5} d^{4} x^{5}+462 x^{4} e^{4} a^{6}+11088 x^{4} d \,e^{3} a^{5} b +41580 x^{4} d^{2} e^{2} a^{4} b^{2}+36960 x^{4} d^{3} e \,a^{3} b^{3}+6930 x^{4} b^{4} d^{4} a^{2}+2310 a^{6} d \,e^{3} x^{3}+20790 a^{5} b \,d^{2} e^{2} x^{3}+34650 a^{4} b^{2} d^{3} e \,x^{3}+11550 a^{3} b^{3} d^{4} x^{3}+4620 a^{6} d^{2} e^{2} x^{2}+18480 a^{5} b \,d^{3} e \,x^{2}+11550 a^{4} b^{2} d^{4} x^{2}+4620 a^{6} d^{3} e x +6930 a^{5} b \,d^{4} x +2310 d^{4} a^{6}\right )}{2310}\)	\(473\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/11*e^4*b^6*x^11+1/10*(6*a*b^5*e^4+4*b^6*d*e^3)*x^10+1/9*(15*a^2*b^4*e^4+24*a*b^5*d*e^3+6*b^6*d^2*e^2)*x^9+1/
8*(20*a^3*b^3*e^4+60*a^2*b^4*d*e^3+36*a*b^5*d^2*e^2+4*b^6*d^3*e)*x^8+1/7*(15*a^4*b^2*e^4+80*a^3*b^3*d*e^3+90*a
^2*b^4*d^2*e^2+24*a*b^5*d^3*e+b^6*d^4)*x^7+1/6*(6*a^5*b*e^4+60*a^4*b^2*d*e^3+120*a^3*b^3*d^2*e^2+60*a^2*b^4*d^
3*e+6*a*b^5*d^4)*x^6+1/5*(a^6*e^4+24*a^5*b*d*e^3+90*a^4*b^2*d^2*e^2+80*a^3*b^3*d^3*e+15*a^2*b^4*d^4)*x^5+1/4*(
4*a^6*d*e^3+36*a^5*b*d^2*e^2+60*a^4*b^2*d^3*e+20*a^3*b^3*d^4)*x^4+1/3*(6*a^6*d^2*e^2+24*a^5*b*d^3*e+15*a^4*b^2
*d^4)*x^3+1/2*(4*a^6*d^3*e+6*a^5*b*d^4)*x^2+d^4*a^6*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (111) = 222\).
time = 0.30, size = 404, normalized size = 3.39 \begin {gather*} \frac {1}{11} \, b^{6} x^{11} e^{4} + a^{6} d^{4} x + \frac {1}{5} \, {\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} + {\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/11*b^6*x^11*e^4 + a^6*d^4*x + 1/5*(2*b^6*d*e^3 + 3*a*b^5*e^4)*x^10 + 1/3*(2*b^6*d^2*e^2 + 8*a*b^5*d*e^3 + 5*
a^2*b^4*e^4)*x^9 + 1/2*(b^6*d^3*e + 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 + 5*a^3*b^3*e^4)*x^8 + 1/7*(b^6*d^4 + 2
4*a*b^5*d^3*e + 90*a^2*b^4*d^2*e^2 + 80*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x^7 + (a*b^5*d^4 + 10*a^2*b^4*d^3*e +
20*a^3*b^3*d^2*e^2 + 10*a^4*b^2*d*e^3 + a^5*b*e^4)*x^6 + 1/5*(15*a^2*b^4*d^4 + 80*a^3*b^3*d^3*e + 90*a^4*b^2*d
^2*e^2 + 24*a^5*b*d*e^3 + a^6*e^4)*x^5 + (5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2 + a^6*d*e^3)*x^4
+ (5*a^4*b^2*d^4 + 8*a^5*b*d^3*e + 2*a^6*d^2*e^2)*x^3 + (3*a^5*b*d^4 + 2*a^6*d^3*e)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (111) = 222\).
time = 2.06, size = 422, normalized size = 3.55 \begin {gather*} \frac {1}{7} \, b^{6} d^{4} x^{7} + a b^{5} d^{4} x^{6} + 3 \, a^{2} b^{4} d^{4} x^{5} + 5 \, a^{3} b^{3} d^{4} x^{4} + 5 \, a^{4} b^{2} d^{4} x^{3} + 3 \, a^{5} b d^{4} x^{2} + a^{6} d^{4} x + \frac {1}{2310} \, {\left (210 \, b^{6} x^{11} + 1386 \, a b^{5} x^{10} + 3850 \, a^{2} b^{4} x^{9} + 5775 \, a^{3} b^{3} x^{8} + 4950 \, a^{4} b^{2} x^{7} + 2310 \, a^{5} b x^{6} + 462 \, a^{6} x^{5}\right )} e^{4} + \frac {1}{210} \, {\left (84 \, b^{6} d x^{10} + 560 \, a b^{5} d x^{9} + 1575 \, a^{2} b^{4} d x^{8} + 2400 \, a^{3} b^{3} d x^{7} + 2100 \, a^{4} b^{2} d x^{6} + 1008 \, a^{5} b d x^{5} + 210 \, a^{6} d x^{4}\right )} e^{3} + \frac {1}{42} \, {\left (28 \, b^{6} d^{2} x^{9} + 189 \, a b^{5} d^{2} x^{8} + 540 \, a^{2} b^{4} d^{2} x^{7} + 840 \, a^{3} b^{3} d^{2} x^{6} + 756 \, a^{4} b^{2} d^{2} x^{5} + 378 \, a^{5} b d^{2} x^{4} + 84 \, a^{6} d^{2} x^{3}\right )} e^{2} + \frac {1}{14} \, {\left (7 \, b^{6} d^{3} x^{8} + 48 \, a b^{5} d^{3} x^{7} + 140 \, a^{2} b^{4} d^{3} x^{6} + 224 \, a^{3} b^{3} d^{3} x^{5} + 210 \, a^{4} b^{2} d^{3} x^{4} + 112 \, a^{5} b d^{3} x^{3} + 28 \, a^{6} d^{3} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/7*b^6*d^4*x^7 + a*b^5*d^4*x^6 + 3*a^2*b^4*d^4*x^5 + 5*a^3*b^3*d^4*x^4 + 5*a^4*b^2*d^4*x^3 + 3*a^5*b*d^4*x^2
+ a^6*d^4*x + 1/2310*(210*b^6*x^11 + 1386*a*b^5*x^10 + 3850*a^2*b^4*x^9 + 5775*a^3*b^3*x^8 + 4950*a^4*b^2*x^7
+ 2310*a^5*b*x^6 + 462*a^6*x^5)*e^4 + 1/210*(84*b^6*d*x^10 + 560*a*b^5*d*x^9 + 1575*a^2*b^4*d*x^8 + 2400*a^3*b
^3*d*x^7 + 2100*a^4*b^2*d*x^6 + 1008*a^5*b*d*x^5 + 210*a^6*d*x^4)*e^3 + 1/42*(28*b^6*d^2*x^9 + 189*a*b^5*d^2*x
^8 + 540*a^2*b^4*d^2*x^7 + 840*a^3*b^3*d^2*x^6 + 756*a^4*b^2*d^2*x^5 + 378*a^5*b*d^2*x^4 + 84*a^6*d^2*x^3)*e^2
+ 1/14*(7*b^6*d^3*x^8 + 48*a*b^5*d^3*x^7 + 140*a^2*b^4*d^3*x^6 + 224*a^3*b^3*d^3*x^5 + 210*a^4*b^2*d^3*x^4 +
112*a^5*b*d^3*x^3 + 28*a^6*d^3*x^2)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (105) = 210\).
time = 0.04, size = 462, normalized size = 3.88 \begin {gather*} a^{6} d^{4} x + \frac {b^{6} e^{4} x^{11}}{11} + x^{10} \cdot \left (\frac {3 a b^{5} e^{4}}{5} + \frac {2 b^{6} d e^{3}}{5}\right ) + x^{9} \cdot \left (\frac {5 a^{2} b^{4} e^{4}}{3} + \frac {8 a b^{5} d e^{3}}{3} + \frac {2 b^{6} d^{2} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {5 a^{3} b^{3} e^{4}}{2} + \frac {15 a^{2} b^{4} d e^{3}}{2} + \frac {9 a b^{5} d^{2} e^{2}}{2} + \frac {b^{6} d^{3} e}{2}\right ) + x^{7} \cdot \left (\frac {15 a^{4} b^{2} e^{4}}{7} + \frac {80 a^{3} b^{3} d e^{3}}{7} + \frac {90 a^{2} b^{4} d^{2} e^{2}}{7} + \frac {24 a b^{5} d^{3} e}{7} + \frac {b^{6} d^{4}}{7}\right ) + x^{6} \left (a^{5} b e^{4} + 10 a^{4} b^{2} d e^{3} + 20 a^{3} b^{3} d^{2} e^{2} + 10 a^{2} b^{4} d^{3} e + a b^{5} d^{4}\right ) + x^{5} \left (\frac {a^{6} e^{4}}{5} + \frac {24 a^{5} b d e^{3}}{5} + 18 a^{4} b^{2} d^{2} e^{2} + 16 a^{3} b^{3} d^{3} e + 3 a^{2} b^{4} d^{4}\right ) + x^{4} \left (a^{6} d e^{3} + 9 a^{5} b d^{2} e^{2} + 15 a^{4} b^{2} d^{3} e + 5 a^{3} b^{3} d^{4}\right ) + x^{3} \cdot \left (2 a^{6} d^{2} e^{2} + 8 a^{5} b d^{3} e + 5 a^{4} b^{2} d^{4}\right ) + x^{2} \cdot \left (2 a^{6} d^{3} e + 3 a^{5} b d^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**4*x + b**6*e**4*x**11/11 + x**10*(3*a*b**5*e**4/5 + 2*b**6*d*e**3/5) + x**9*(5*a**2*b**4*e**4/3 + 8*a*
b**5*d*e**3/3 + 2*b**6*d**2*e**2/3) + x**8*(5*a**3*b**3*e**4/2 + 15*a**2*b**4*d*e**3/2 + 9*a*b**5*d**2*e**2/2
+ b**6*d**3*e/2) + x**7*(15*a**4*b**2*e**4/7 + 80*a**3*b**3*d*e**3/7 + 90*a**2*b**4*d**2*e**2/7 + 24*a*b**5*d*
*3*e/7 + b**6*d**4/7) + x**6*(a**5*b*e**4 + 10*a**4*b**2*d*e**3 + 20*a**3*b**3*d**2*e**2 + 10*a**2*b**4*d**3*e
+ a*b**5*d**4) + x**5*(a**6*e**4/5 + 24*a**5*b*d*e**3/5 + 18*a**4*b**2*d**2*e**2 + 16*a**3*b**3*d**3*e + 3*a*
*2*b**4*d**4) + x**4*(a**6*d*e**3 + 9*a**5*b*d**2*e**2 + 15*a**4*b**2*d**3*e + 5*a**3*b**3*d**4) + x**3*(2*a**
6*d**2*e**2 + 8*a**5*b*d**3*e + 5*a**4*b**2*d**4) + x**2*(2*a**6*d**3*e + 3*a**5*b*d**4)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (111) = 222\).
time = 2.07, size = 456, normalized size = 3.83 \begin {gather*} \frac {1}{11} \, b^{6} x^{11} e^{4} + \frac {2}{5} \, b^{6} d x^{10} e^{3} + \frac {2}{3} \, b^{6} d^{2} x^{9} e^{2} + \frac {1}{2} \, b^{6} d^{3} x^{8} e + \frac {1}{7} \, b^{6} d^{4} x^{7} + \frac {3}{5} \, a b^{5} x^{10} e^{4} + \frac {8}{3} \, a b^{5} d x^{9} e^{3} + \frac {9}{2} \, a b^{5} d^{2} x^{8} e^{2} + \frac {24}{7} \, a b^{5} d^{3} x^{7} e + a b^{5} d^{4} x^{6} + \frac {5}{3} \, a^{2} b^{4} x^{9} e^{4} + \frac {15}{2} \, a^{2} b^{4} d x^{8} e^{3} + \frac {90}{7} \, a^{2} b^{4} d^{2} x^{7} e^{2} + 10 \, a^{2} b^{4} d^{3} x^{6} e + 3 \, a^{2} b^{4} d^{4} x^{5} + \frac {5}{2} \, a^{3} b^{3} x^{8} e^{4} + \frac {80}{7} \, a^{3} b^{3} d x^{7} e^{3} + 20 \, a^{3} b^{3} d^{2} x^{6} e^{2} + 16 \, a^{3} b^{3} d^{3} x^{5} e + 5 \, a^{3} b^{3} d^{4} x^{4} + \frac {15}{7} \, a^{4} b^{2} x^{7} e^{4} + 10 \, a^{4} b^{2} d x^{6} e^{3} + 18 \, a^{4} b^{2} d^{2} x^{5} e^{2} + 15 \, a^{4} b^{2} d^{3} x^{4} e + 5 \, a^{4} b^{2} d^{4} x^{3} + a^{5} b x^{6} e^{4} + \frac {24}{5} \, a^{5} b d x^{5} e^{3} + 9 \, a^{5} b d^{2} x^{4} e^{2} + 8 \, a^{5} b d^{3} x^{3} e + 3 \, a^{5} b d^{4} x^{2} + \frac {1}{5} \, a^{6} x^{5} e^{4} + a^{6} d x^{4} e^{3} + 2 \, a^{6} d^{2} x^{3} e^{2} + 2 \, a^{6} d^{3} x^{2} e + a^{6} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/11*b^6*x^11*e^4 + 2/5*b^6*d*x^10*e^3 + 2/3*b^6*d^2*x^9*e^2 + 1/2*b^6*d^3*x^8*e + 1/7*b^6*d^4*x^7 + 3/5*a*b^5
*x^10*e^4 + 8/3*a*b^5*d*x^9*e^3 + 9/2*a*b^5*d^2*x^8*e^2 + 24/7*a*b^5*d^3*x^7*e + a*b^5*d^4*x^6 + 5/3*a^2*b^4*x
^9*e^4 + 15/2*a^2*b^4*d*x^8*e^3 + 90/7*a^2*b^4*d^2*x^7*e^2 + 10*a^2*b^4*d^3*x^6*e + 3*a^2*b^4*d^4*x^5 + 5/2*a^
3*b^3*x^8*e^4 + 80/7*a^3*b^3*d*x^7*e^3 + 20*a^3*b^3*d^2*x^6*e^2 + 16*a^3*b^3*d^3*x^5*e + 5*a^3*b^3*d^4*x^4 + 1
5/7*a^4*b^2*x^7*e^4 + 10*a^4*b^2*d*x^6*e^3 + 18*a^4*b^2*d^2*x^5*e^2 + 15*a^4*b^2*d^3*x^4*e + 5*a^4*b^2*d^4*x^3
+ a^5*b*x^6*e^4 + 24/5*a^5*b*d*x^5*e^3 + 9*a^5*b*d^2*x^4*e^2 + 8*a^5*b*d^3*x^3*e + 3*a^5*b*d^4*x^2 + 1/5*a^6*
x^5*e^4 + a^6*d*x^4*e^3 + 2*a^6*d^2*x^3*e^2 + 2*a^6*d^3*x^2*e + a^6*d^4*x

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Mupad [B]
time = 0.61, size = 402, normalized size = 3.38 \begin {gather*} x^5\,\left (\frac {a^6\,e^4}{5}+\frac {24\,a^5\,b\,d\,e^3}{5}+18\,a^4\,b^2\,d^2\,e^2+16\,a^3\,b^3\,d^3\,e+3\,a^2\,b^4\,d^4\right )+x^7\,\left (\frac {15\,a^4\,b^2\,e^4}{7}+\frac {80\,a^3\,b^3\,d\,e^3}{7}+\frac {90\,a^2\,b^4\,d^2\,e^2}{7}+\frac {24\,a\,b^5\,d^3\,e}{7}+\frac {b^6\,d^4}{7}\right )+x^4\,\left (a^6\,d\,e^3+9\,a^5\,b\,d^2\,e^2+15\,a^4\,b^2\,d^3\,e+5\,a^3\,b^3\,d^4\right )+x^8\,\left (\frac {5\,a^3\,b^3\,e^4}{2}+\frac {15\,a^2\,b^4\,d\,e^3}{2}+\frac {9\,a\,b^5\,d^2\,e^2}{2}+\frac {b^6\,d^3\,e}{2}\right )+x^6\,\left (a^5\,b\,e^4+10\,a^4\,b^2\,d\,e^3+20\,a^3\,b^3\,d^2\,e^2+10\,a^2\,b^4\,d^3\,e+a\,b^5\,d^4\right )+a^6\,d^4\,x+\frac {b^6\,e^4\,x^{11}}{11}+a^5\,d^3\,x^2\,\left (2\,a\,e+3\,b\,d\right )+\frac {b^5\,e^3\,x^{10}\,\left (3\,a\,e+2\,b\,d\right )}{5}+a^4\,d^2\,x^3\,\left (2\,a^2\,e^2+8\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {b^4\,e^2\,x^9\,\left (5\,a^2\,e^2+8\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^5*((a^6*e^4)/5 + 3*a^2*b^4*d^4 + 16*a^3*b^3*d^3*e + 18*a^4*b^2*d^2*e^2 + (24*a^5*b*d*e^3)/5) + x^7*((b^6*d^4
)/7 + (15*a^4*b^2*e^4)/7 + (80*a^3*b^3*d*e^3)/7 + (90*a^2*b^4*d^2*e^2)/7 + (24*a*b^5*d^3*e)/7) + x^4*(a^6*d*e^
3 + 5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2) + x^8*((b^6*d^3*e)/2 + (5*a^3*b^3*e^4)/2 + (9*a*b^5*d^
2*e^2)/2 + (15*a^2*b^4*d*e^3)/2) + x^6*(a*b^5*d^4 + a^5*b*e^4 + 10*a^2*b^4*d^3*e + 10*a^4*b^2*d*e^3 + 20*a^3*b
^3*d^2*e^2) + a^6*d^4*x + (b^6*e^4*x^11)/11 + a^5*d^3*x^2*(2*a*e + 3*b*d) + (b^5*e^3*x^10*(3*a*e + 2*b*d))/5 +
a^4*d^2*x^3*(2*a^2*e^2 + 5*b^2*d^2 + 8*a*b*d*e) + (b^4*e^2*x^9*(5*a^2*e^2 + 2*b^2*d^2 + 8*a*b*d*e))/3

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