3.20.5 \(\int (a+b x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1905]

Optimal. Leaf size=143 \[ -\frac {(b d-a e)^5 (d+e x)^7}{7 e^6}+\frac {5 b (b d-a e)^4 (d+e x)^8}{8 e^6}-\frac {10 b^2 (b d-a e)^3 (d+e x)^9}{9 e^6}+\frac {b^3 (b d-a e)^2 (d+e x)^{10}}{e^6}-\frac {5 b^4 (b d-a e) (d+e x)^{11}}{11 e^6}+\frac {b^5 (d+e x)^{12}}{12 e^6} \]

[Out]

-1/7*(-a*e+b*d)^5*(e*x+d)^7/e^6+5/8*b*(-a*e+b*d)^4*(e*x+d)^8/e^6-10/9*b^2*(-a*e+b*d)^3*(e*x+d)^9/e^6+b^3*(-a*e
+b*d)^2*(e*x+d)^10/e^6-5/11*b^4*(-a*e+b*d)*(e*x+d)^11/e^6+1/12*b^5*(e*x+d)^12/e^6

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Rubi [A]
time = 0.22, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45} \begin {gather*} -\frac {5 b^4 (d+e x)^{11} (b d-a e)}{11 e^6}+\frac {b^3 (d+e x)^{10} (b d-a e)^2}{e^6}-\frac {10 b^2 (d+e x)^9 (b d-a e)^3}{9 e^6}+\frac {5 b (d+e x)^8 (b d-a e)^4}{8 e^6}-\frac {(d+e x)^7 (b d-a e)^5}{7 e^6}+\frac {b^5 (d+e x)^{12}}{12 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*((b*d - a*e)^5*(d + e*x)^7)/e^6 + (5*b*(b*d - a*e)^4*(d + e*x)^8)/(8*e^6) - (10*b^2*(b*d - a*e)^3*(d + e*
x)^9)/(9*e^6) + (b^3*(b*d - a*e)^2*(d + e*x)^10)/e^6 - (5*b^4*(b*d - a*e)*(d + e*x)^11)/(11*e^6) + (b^5*(d + e
*x)^12)/(12*e^6)

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 (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^6 \, dx\\ &=\int \left (\frac {(-b d+a e)^5 (d+e x)^6}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^7}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^8}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^9}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{10}}{e^5}+\frac {b^5 (d+e x)^{11}}{e^5}\right ) \, dx\\ &=-\frac {(b d-a e)^5 (d+e x)^7}{7 e^6}+\frac {5 b (b d-a e)^4 (d+e x)^8}{8 e^6}-\frac {10 b^2 (b d-a e)^3 (d+e x)^9}{9 e^6}+\frac {b^3 (b d-a e)^2 (d+e x)^{10}}{e^6}-\frac {5 b^4 (b d-a e) (d+e x)^{11}}{11 e^6}+\frac {b^5 (d+e x)^{12}}{12 e^6}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(143)=286\).
time = 0.05, size = 501, normalized size = 3.50 \begin {gather*} a^5 d^6 x+\frac {1}{2} a^4 d^5 (5 b d+6 a e) x^2+\frac {5}{3} a^3 d^4 \left (2 b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^3+\frac {5}{4} a^2 d^3 \left (2 b^3 d^3+12 a b^2 d^2 e+15 a^2 b d e^2+4 a^3 e^3\right ) x^4+a d^2 \left (b^4 d^4+12 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+20 a^3 b d e^3+3 a^4 e^4\right ) x^5+\frac {1}{6} d \left (b^5 d^5+30 a b^4 d^4 e+150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5\right ) x^6+\frac {1}{7} e \left (6 b^5 d^5+75 a b^4 d^4 e+200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5\right ) x^7+\frac {5}{8} b e^2 \left (3 b^4 d^4+20 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4\right ) x^8+\frac {5}{9} b^2 e^3 \left (4 b^3 d^3+15 a b^2 d^2 e+12 a^2 b d e^2+2 a^3 e^3\right ) x^9+\frac {1}{2} b^3 e^4 \left (3 b^2 d^2+6 a b d e+2 a^2 e^2\right ) x^{10}+\frac {1}{11} b^4 e^5 (6 b d+5 a e) x^{11}+\frac {1}{12} b^5 e^6 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^5*d^6*x + (a^4*d^5*(5*b*d + 6*a*e)*x^2)/2 + (5*a^3*d^4*(2*b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^3)/3 + (5*a^2*d
^3*(2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 4*a^3*e^3)*x^4)/4 + a*d^2*(b^4*d^4 + 12*a*b^3*d^3*e + 30*a^2
*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 3*a^4*e^4)*x^5 + (d*(b^5*d^5 + 30*a*b^4*d^4*e + 150*a^2*b^3*d^3*e^2 + 200*a^3*
b^2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e^5)*x^6)/6 + (e*(6*b^5*d^5 + 75*a*b^4*d^4*e + 200*a^2*b^3*d^3*e^2 + 150*
a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5)*x^7)/7 + (5*b*e^2*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2
+ 12*a^3*b*d*e^3 + a^4*e^4)*x^8)/8 + (5*b^2*e^3*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2 + 2*a^3*e^3)*x^9
)/9 + (b^3*e^4*(3*b^2*d^2 + 6*a*b*d*e + 2*a^2*e^2)*x^10)/2 + (b^4*e^5*(6*b*d + 5*a*e)*x^11)/11 + (b^5*e^6*x^12
)/12

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(816\) vs. \(2(133)=266\).
time = 0.99, size = 817, normalized size = 5.71 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^5*e^6*x^12+1/11*((a*e^6+6*b*d*e^5)*b^4+4*a*b^4*e^6)*x^11+1/10*((6*a*d*e^5+15*b*d^2*e^4)*b^4+4*(a*e^6+6*
b*d*e^5)*a*b^3+6*a^2*b^3*e^6)*x^10+1/9*((15*a*d^2*e^4+20*b*d^3*e^3)*b^4+4*(6*a*d*e^5+15*b*d^2*e^4)*a*b^3+6*(a*
e^6+6*b*d*e^5)*a^2*b^2+4*a^3*b^2*e^6)*x^9+1/8*((20*a*d^3*e^3+15*b*d^4*e^2)*b^4+4*(15*a*d^2*e^4+20*b*d^3*e^3)*a
*b^3+6*(6*a*d*e^5+15*b*d^2*e^4)*a^2*b^2+4*(a*e^6+6*b*d*e^5)*a^3*b+a^4*b*e^6)*x^8+1/7*((15*a*d^4*e^2+6*b*d^5*e)
*b^4+4*(20*a*d^3*e^3+15*b*d^4*e^2)*a*b^3+6*(15*a*d^2*e^4+20*b*d^3*e^3)*a^2*b^2+4*(6*a*d*e^5+15*b*d^2*e^4)*a^3*
b+(a*e^6+6*b*d*e^5)*a^4)*x^7+1/6*((6*a*d^5*e+b*d^6)*b^4+4*(15*a*d^4*e^2+6*b*d^5*e)*a*b^3+6*(20*a*d^3*e^3+15*b*
d^4*e^2)*a^2*b^2+4*(15*a*d^2*e^4+20*b*d^3*e^3)*a^3*b+(6*a*d*e^5+15*b*d^2*e^4)*a^4)*x^6+1/5*(a*b^4*d^6+4*(6*a*d
^5*e+b*d^6)*a*b^3+6*(15*a*d^4*e^2+6*b*d^5*e)*a^2*b^2+4*(20*a*d^3*e^3+15*b*d^4*e^2)*a^3*b+(15*a*d^2*e^4+20*b*d^
3*e^3)*a^4)*x^5+1/4*(4*a^2*b^3*d^6+6*(6*a*d^5*e+b*d^6)*a^2*b^2+4*(15*a*d^4*e^2+6*b*d^5*e)*a^3*b+(20*a*d^3*e^3+
15*b*d^4*e^2)*a^4)*x^4+1/3*(6*a^3*b^2*d^6+4*(6*a*d^5*e+b*d^6)*a^3*b+(15*a*d^4*e^2+6*b*d^5*e)*a^4)*x^3+1/2*(4*a
^4*b*d^6+(6*a*d^5*e+b*d^6)*a^4)*x^2+a^5*d^6*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (138) = 276\).
time = 0.27, size = 493, normalized size = 3.45 \begin {gather*} \frac {1}{12} \, b^{5} x^{12} e^{6} + a^{5} d^{6} x + \frac {1}{11} \, {\left (6 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{11} + \frac {1}{2} \, {\left (3 \, b^{5} d^{2} e^{4} + 6 \, a b^{4} d e^{5} + 2 \, a^{2} b^{3} e^{6}\right )} x^{10} + \frac {5}{9} \, {\left (4 \, b^{5} d^{3} e^{3} + 15 \, a b^{4} d^{2} e^{4} + 12 \, a^{2} b^{3} d e^{5} + 2 \, a^{3} b^{2} e^{6}\right )} x^{9} + \frac {5}{8} \, {\left (3 \, b^{5} d^{4} e^{2} + 20 \, a b^{4} d^{3} e^{3} + 30 \, a^{2} b^{3} d^{2} e^{4} + 12 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, b^{5} d^{5} e + 75 \, a b^{4} d^{4} e^{2} + 200 \, a^{2} b^{3} d^{3} e^{3} + 150 \, a^{3} b^{2} d^{2} e^{4} + 30 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{6} + 30 \, a b^{4} d^{5} e + 150 \, a^{2} b^{3} d^{4} e^{2} + 200 \, a^{3} b^{2} d^{3} e^{3} + 75 \, a^{4} b d^{2} e^{4} + 6 \, a^{5} d e^{5}\right )} x^{6} + {\left (a b^{4} d^{6} + 12 \, a^{2} b^{3} d^{5} e + 30 \, a^{3} b^{2} d^{4} e^{2} + 20 \, a^{4} b d^{3} e^{3} + 3 \, a^{5} d^{2} e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{6} + 12 \, a^{3} b^{2} d^{5} e + 15 \, a^{4} b d^{4} e^{2} + 4 \, a^{5} d^{3} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{6} + 6 \, a^{4} b d^{5} e + 3 \, a^{5} d^{4} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{6} + 6 \, a^{5} d^{5} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^5*x^12*e^6 + a^5*d^6*x + 1/11*(6*b^5*d*e^5 + 5*a*b^4*e^6)*x^11 + 1/2*(3*b^5*d^2*e^4 + 6*a*b^4*d*e^5 + 2
*a^2*b^3*e^6)*x^10 + 5/9*(4*b^5*d^3*e^3 + 15*a*b^4*d^2*e^4 + 12*a^2*b^3*d*e^5 + 2*a^3*b^2*e^6)*x^9 + 5/8*(3*b^
5*d^4*e^2 + 20*a*b^4*d^3*e^3 + 30*a^2*b^3*d^2*e^4 + 12*a^3*b^2*d*e^5 + a^4*b*e^6)*x^8 + 1/7*(6*b^5*d^5*e + 75*
a*b^4*d^4*e^2 + 200*a^2*b^3*d^3*e^3 + 150*a^3*b^2*d^2*e^4 + 30*a^4*b*d*e^5 + a^5*e^6)*x^7 + 1/6*(b^5*d^6 + 30*
a*b^4*d^5*e + 150*a^2*b^3*d^4*e^2 + 200*a^3*b^2*d^3*e^3 + 75*a^4*b*d^2*e^4 + 6*a^5*d*e^5)*x^6 + (a*b^4*d^6 + 1
2*a^2*b^3*d^5*e + 30*a^3*b^2*d^4*e^2 + 20*a^4*b*d^3*e^3 + 3*a^5*d^2*e^4)*x^5 + 5/4*(2*a^2*b^3*d^6 + 12*a^3*b^2
*d^5*e + 15*a^4*b*d^4*e^2 + 4*a^5*d^3*e^3)*x^4 + 5/3*(2*a^3*b^2*d^6 + 6*a^4*b*d^5*e + 3*a^5*d^4*e^2)*x^3 + 1/2
*(5*a^4*b*d^6 + 6*a^5*d^5*e)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (138) = 276\).
time = 2.10, size = 515, normalized size = 3.60 \begin {gather*} \frac {1}{6} \, b^{5} d^{6} x^{6} + a b^{4} d^{6} x^{5} + \frac {5}{2} \, a^{2} b^{3} d^{6} x^{4} + \frac {10}{3} \, a^{3} b^{2} d^{6} x^{3} + \frac {5}{2} \, a^{4} b d^{6} x^{2} + a^{5} d^{6} x + \frac {1}{5544} \, {\left (462 \, b^{5} x^{12} + 2520 \, a b^{4} x^{11} + 5544 \, a^{2} b^{3} x^{10} + 6160 \, a^{3} b^{2} x^{9} + 3465 \, a^{4} b x^{8} + 792 \, a^{5} x^{7}\right )} e^{6} + \frac {1}{462} \, {\left (252 \, b^{5} d x^{11} + 1386 \, a b^{4} d x^{10} + 3080 \, a^{2} b^{3} d x^{9} + 3465 \, a^{3} b^{2} d x^{8} + 1980 \, a^{4} b d x^{7} + 462 \, a^{5} d x^{6}\right )} e^{5} + \frac {1}{84} \, {\left (126 \, b^{5} d^{2} x^{10} + 700 \, a b^{4} d^{2} x^{9} + 1575 \, a^{2} b^{3} d^{2} x^{8} + 1800 \, a^{3} b^{2} d^{2} x^{7} + 1050 \, a^{4} b d^{2} x^{6} + 252 \, a^{5} d^{2} x^{5}\right )} e^{4} + \frac {5}{126} \, {\left (56 \, b^{5} d^{3} x^{9} + 315 \, a b^{4} d^{3} x^{8} + 720 \, a^{2} b^{3} d^{3} x^{7} + 840 \, a^{3} b^{2} d^{3} x^{6} + 504 \, a^{4} b d^{3} x^{5} + 126 \, a^{5} d^{3} x^{4}\right )} e^{3} + \frac {5}{56} \, {\left (21 \, b^{5} d^{4} x^{8} + 120 \, a b^{4} d^{4} x^{7} + 280 \, a^{2} b^{3} d^{4} x^{6} + 336 \, a^{3} b^{2} d^{4} x^{5} + 210 \, a^{4} b d^{4} x^{4} + 56 \, a^{5} d^{4} x^{3}\right )} e^{2} + \frac {1}{7} \, {\left (6 \, b^{5} d^{5} x^{7} + 35 \, a b^{4} d^{5} x^{6} + 84 \, a^{2} b^{3} d^{5} x^{5} + 105 \, a^{3} b^{2} d^{5} x^{4} + 70 \, a^{4} b d^{5} x^{3} + 21 \, a^{5} d^{5} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^5*d^6*x^6 + a*b^4*d^6*x^5 + 5/2*a^2*b^3*d^6*x^4 + 10/3*a^3*b^2*d^6*x^3 + 5/2*a^4*b*d^6*x^2 + a^5*d^6*x +
1/5544*(462*b^5*x^12 + 2520*a*b^4*x^11 + 5544*a^2*b^3*x^10 + 6160*a^3*b^2*x^9 + 3465*a^4*b*x^8 + 792*a^5*x^7)
*e^6 + 1/462*(252*b^5*d*x^11 + 1386*a*b^4*d*x^10 + 3080*a^2*b^3*d*x^9 + 3465*a^3*b^2*d*x^8 + 1980*a^4*b*d*x^7
+ 462*a^5*d*x^6)*e^5 + 1/84*(126*b^5*d^2*x^10 + 700*a*b^4*d^2*x^9 + 1575*a^2*b^3*d^2*x^8 + 1800*a^3*b^2*d^2*x^
7 + 1050*a^4*b*d^2*x^6 + 252*a^5*d^2*x^5)*e^4 + 5/126*(56*b^5*d^3*x^9 + 315*a*b^4*d^3*x^8 + 720*a^2*b^3*d^3*x^
7 + 840*a^3*b^2*d^3*x^6 + 504*a^4*b*d^3*x^5 + 126*a^5*d^3*x^4)*e^3 + 5/56*(21*b^5*d^4*x^8 + 120*a*b^4*d^4*x^7
+ 280*a^2*b^3*d^4*x^6 + 336*a^3*b^2*d^4*x^5 + 210*a^4*b*d^4*x^4 + 56*a^5*d^4*x^3)*e^2 + 1/7*(6*b^5*d^5*x^7 + 3
5*a*b^4*d^5*x^6 + 84*a^2*b^3*d^5*x^5 + 105*a^3*b^2*d^5*x^4 + 70*a^4*b*d^5*x^3 + 21*a^5*d^5*x^2)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (129) = 258\).
time = 0.06, size = 580, normalized size = 4.06 \begin {gather*} a^{5} d^{6} x + \frac {b^{5} e^{6} x^{12}}{12} + x^{11} \cdot \left (\frac {5 a b^{4} e^{6}}{11} + \frac {6 b^{5} d e^{5}}{11}\right ) + x^{10} \left (a^{2} b^{3} e^{6} + 3 a b^{4} d e^{5} + \frac {3 b^{5} d^{2} e^{4}}{2}\right ) + x^{9} \cdot \left (\frac {10 a^{3} b^{2} e^{6}}{9} + \frac {20 a^{2} b^{3} d e^{5}}{3} + \frac {25 a b^{4} d^{2} e^{4}}{3} + \frac {20 b^{5} d^{3} e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {5 a^{4} b e^{6}}{8} + \frac {15 a^{3} b^{2} d e^{5}}{2} + \frac {75 a^{2} b^{3} d^{2} e^{4}}{4} + \frac {25 a b^{4} d^{3} e^{3}}{2} + \frac {15 b^{5} d^{4} e^{2}}{8}\right ) + x^{7} \left (\frac {a^{5} e^{6}}{7} + \frac {30 a^{4} b d e^{5}}{7} + \frac {150 a^{3} b^{2} d^{2} e^{4}}{7} + \frac {200 a^{2} b^{3} d^{3} e^{3}}{7} + \frac {75 a b^{4} d^{4} e^{2}}{7} + \frac {6 b^{5} d^{5} e}{7}\right ) + x^{6} \left (a^{5} d e^{5} + \frac {25 a^{4} b d^{2} e^{4}}{2} + \frac {100 a^{3} b^{2} d^{3} e^{3}}{3} + 25 a^{2} b^{3} d^{4} e^{2} + 5 a b^{4} d^{5} e + \frac {b^{5} d^{6}}{6}\right ) + x^{5} \cdot \left (3 a^{5} d^{2} e^{4} + 20 a^{4} b d^{3} e^{3} + 30 a^{3} b^{2} d^{4} e^{2} + 12 a^{2} b^{3} d^{5} e + a b^{4} d^{6}\right ) + x^{4} \cdot \left (5 a^{5} d^{3} e^{3} + \frac {75 a^{4} b d^{4} e^{2}}{4} + 15 a^{3} b^{2} d^{5} e + \frac {5 a^{2} b^{3} d^{6}}{2}\right ) + x^{3} \cdot \left (5 a^{5} d^{4} e^{2} + 10 a^{4} b d^{5} e + \frac {10 a^{3} b^{2} d^{6}}{3}\right ) + x^{2} \cdot \left (3 a^{5} d^{5} e + \frac {5 a^{4} b d^{6}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d**6*x + b**5*e**6*x**12/12 + x**11*(5*a*b**4*e**6/11 + 6*b**5*d*e**5/11) + x**10*(a**2*b**3*e**6 + 3*a*b
**4*d*e**5 + 3*b**5*d**2*e**4/2) + x**9*(10*a**3*b**2*e**6/9 + 20*a**2*b**3*d*e**5/3 + 25*a*b**4*d**2*e**4/3 +
20*b**5*d**3*e**3/9) + x**8*(5*a**4*b*e**6/8 + 15*a**3*b**2*d*e**5/2 + 75*a**2*b**3*d**2*e**4/4 + 25*a*b**4*d
**3*e**3/2 + 15*b**5*d**4*e**2/8) + x**7*(a**5*e**6/7 + 30*a**4*b*d*e**5/7 + 150*a**3*b**2*d**2*e**4/7 + 200*a
**2*b**3*d**3*e**3/7 + 75*a*b**4*d**4*e**2/7 + 6*b**5*d**5*e/7) + x**6*(a**5*d*e**5 + 25*a**4*b*d**2*e**4/2 +
100*a**3*b**2*d**3*e**3/3 + 25*a**2*b**3*d**4*e**2 + 5*a*b**4*d**5*e + b**5*d**6/6) + x**5*(3*a**5*d**2*e**4 +
20*a**4*b*d**3*e**3 + 30*a**3*b**2*d**4*e**2 + 12*a**2*b**3*d**5*e + a*b**4*d**6) + x**4*(5*a**5*d**3*e**3 +
75*a**4*b*d**4*e**2/4 + 15*a**3*b**2*d**5*e + 5*a**2*b**3*d**6/2) + x**3*(5*a**5*d**4*e**2 + 10*a**4*b*d**5*e
+ 10*a**3*b**2*d**6/3) + x**2*(3*a**5*d**5*e + 5*a**4*b*d**6/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (138) = 276\).
time = 0.94, size = 555, normalized size = 3.88 \begin {gather*} \frac {1}{12} \, b^{5} x^{12} e^{6} + \frac {6}{11} \, b^{5} d x^{11} e^{5} + \frac {3}{2} \, b^{5} d^{2} x^{10} e^{4} + \frac {20}{9} \, b^{5} d^{3} x^{9} e^{3} + \frac {15}{8} \, b^{5} d^{4} x^{8} e^{2} + \frac {6}{7} \, b^{5} d^{5} x^{7} e + \frac {1}{6} \, b^{5} d^{6} x^{6} + \frac {5}{11} \, a b^{4} x^{11} e^{6} + 3 \, a b^{4} d x^{10} e^{5} + \frac {25}{3} \, a b^{4} d^{2} x^{9} e^{4} + \frac {25}{2} \, a b^{4} d^{3} x^{8} e^{3} + \frac {75}{7} \, a b^{4} d^{4} x^{7} e^{2} + 5 \, a b^{4} d^{5} x^{6} e + a b^{4} d^{6} x^{5} + a^{2} b^{3} x^{10} e^{6} + \frac {20}{3} \, a^{2} b^{3} d x^{9} e^{5} + \frac {75}{4} \, a^{2} b^{3} d^{2} x^{8} e^{4} + \frac {200}{7} \, a^{2} b^{3} d^{3} x^{7} e^{3} + 25 \, a^{2} b^{3} d^{4} x^{6} e^{2} + 12 \, a^{2} b^{3} d^{5} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{6} x^{4} + \frac {10}{9} \, a^{3} b^{2} x^{9} e^{6} + \frac {15}{2} \, a^{3} b^{2} d x^{8} e^{5} + \frac {150}{7} \, a^{3} b^{2} d^{2} x^{7} e^{4} + \frac {100}{3} \, a^{3} b^{2} d^{3} x^{6} e^{3} + 30 \, a^{3} b^{2} d^{4} x^{5} e^{2} + 15 \, a^{3} b^{2} d^{5} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{6} x^{3} + \frac {5}{8} \, a^{4} b x^{8} e^{6} + \frac {30}{7} \, a^{4} b d x^{7} e^{5} + \frac {25}{2} \, a^{4} b d^{2} x^{6} e^{4} + 20 \, a^{4} b d^{3} x^{5} e^{3} + \frac {75}{4} \, a^{4} b d^{4} x^{4} e^{2} + 10 \, a^{4} b d^{5} x^{3} e + \frac {5}{2} \, a^{4} b d^{6} x^{2} + \frac {1}{7} \, a^{5} x^{7} e^{6} + a^{5} d x^{6} e^{5} + 3 \, a^{5} d^{2} x^{5} e^{4} + 5 \, a^{5} d^{3} x^{4} e^{3} + 5 \, a^{5} d^{4} x^{3} e^{2} + 3 \, a^{5} d^{5} x^{2} e + a^{5} d^{6} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^5*x^12*e^6 + 6/11*b^5*d*x^11*e^5 + 3/2*b^5*d^2*x^10*e^4 + 20/9*b^5*d^3*x^9*e^3 + 15/8*b^5*d^4*x^8*e^2 +
6/7*b^5*d^5*x^7*e + 1/6*b^5*d^6*x^6 + 5/11*a*b^4*x^11*e^6 + 3*a*b^4*d*x^10*e^5 + 25/3*a*b^4*d^2*x^9*e^4 + 25/
2*a*b^4*d^3*x^8*e^3 + 75/7*a*b^4*d^4*x^7*e^2 + 5*a*b^4*d^5*x^6*e + a*b^4*d^6*x^5 + a^2*b^3*x^10*e^6 + 20/3*a^2
*b^3*d*x^9*e^5 + 75/4*a^2*b^3*d^2*x^8*e^4 + 200/7*a^2*b^3*d^3*x^7*e^3 + 25*a^2*b^3*d^4*x^6*e^2 + 12*a^2*b^3*d^
5*x^5*e + 5/2*a^2*b^3*d^6*x^4 + 10/9*a^3*b^2*x^9*e^6 + 15/2*a^3*b^2*d*x^8*e^5 + 150/7*a^3*b^2*d^2*x^7*e^4 + 10
0/3*a^3*b^2*d^3*x^6*e^3 + 30*a^3*b^2*d^4*x^5*e^2 + 15*a^3*b^2*d^5*x^4*e + 10/3*a^3*b^2*d^6*x^3 + 5/8*a^4*b*x^8
*e^6 + 30/7*a^4*b*d*x^7*e^5 + 25/2*a^4*b*d^2*x^6*e^4 + 20*a^4*b*d^3*x^5*e^3 + 75/4*a^4*b*d^4*x^4*e^2 + 10*a^4*
b*d^5*x^3*e + 5/2*a^4*b*d^6*x^2 + 1/7*a^5*x^7*e^6 + a^5*d*x^6*e^5 + 3*a^5*d^2*x^5*e^4 + 5*a^5*d^3*x^4*e^3 + 5*
a^5*d^4*x^3*e^2 + 3*a^5*d^5*x^2*e + a^5*d^6*x

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Mupad [B]
time = 2.13, size = 492, normalized size = 3.44 \begin {gather*} x^5\,\left (3\,a^5\,d^2\,e^4+20\,a^4\,b\,d^3\,e^3+30\,a^3\,b^2\,d^4\,e^2+12\,a^2\,b^3\,d^5\,e+a\,b^4\,d^6\right )+x^8\,\left (\frac {5\,a^4\,b\,e^6}{8}+\frac {15\,a^3\,b^2\,d\,e^5}{2}+\frac {75\,a^2\,b^3\,d^2\,e^4}{4}+\frac {25\,a\,b^4\,d^3\,e^3}{2}+\frac {15\,b^5\,d^4\,e^2}{8}\right )+x^6\,\left (a^5\,d\,e^5+\frac {25\,a^4\,b\,d^2\,e^4}{2}+\frac {100\,a^3\,b^2\,d^3\,e^3}{3}+25\,a^2\,b^3\,d^4\,e^2+5\,a\,b^4\,d^5\,e+\frac {b^5\,d^6}{6}\right )+x^7\,\left (\frac {a^5\,e^6}{7}+\frac {30\,a^4\,b\,d\,e^5}{7}+\frac {150\,a^3\,b^2\,d^2\,e^4}{7}+\frac {200\,a^2\,b^3\,d^3\,e^3}{7}+\frac {75\,a\,b^4\,d^4\,e^2}{7}+\frac {6\,b^5\,d^5\,e}{7}\right )+a^5\,d^6\,x+\frac {b^5\,e^6\,x^{12}}{12}+\frac {5\,a^2\,d^3\,x^4\,\left (4\,a^3\,e^3+15\,a^2\,b\,d\,e^2+12\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{4}+\frac {5\,b^2\,e^3\,x^9\,\left (2\,a^3\,e^3+12\,a^2\,b\,d\,e^2+15\,a\,b^2\,d^2\,e+4\,b^3\,d^3\right )}{9}+\frac {a^4\,d^5\,x^2\,\left (6\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e^5\,x^{11}\,\left (5\,a\,e+6\,b\,d\right )}{11}+\frac {5\,a^3\,d^4\,x^3\,\left (3\,a^2\,e^2+6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3}+\frac {b^3\,e^4\,x^{10}\,\left (2\,a^2\,e^2+6\,a\,b\,d\,e+3\,b^2\,d^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*(a*b^4*d^6 + 3*a^5*d^2*e^4 + 12*a^2*b^3*d^5*e + 20*a^4*b*d^3*e^3 + 30*a^3*b^2*d^4*e^2) + x^8*((5*a^4*b*e^6
)/8 + (15*b^5*d^4*e^2)/8 + (25*a*b^4*d^3*e^3)/2 + (15*a^3*b^2*d*e^5)/2 + (75*a^2*b^3*d^2*e^4)/4) + x^6*((b^5*d
^6)/6 + a^5*d*e^5 + (25*a^4*b*d^2*e^4)/2 + 25*a^2*b^3*d^4*e^2 + (100*a^3*b^2*d^3*e^3)/3 + 5*a*b^4*d^5*e) + x^7
*((a^5*e^6)/7 + (6*b^5*d^5*e)/7 + (75*a*b^4*d^4*e^2)/7 + (200*a^2*b^3*d^3*e^3)/7 + (150*a^3*b^2*d^2*e^4)/7 + (
30*a^4*b*d*e^5)/7) + a^5*d^6*x + (b^5*e^6*x^12)/12 + (5*a^2*d^3*x^4*(4*a^3*e^3 + 2*b^3*d^3 + 12*a*b^2*d^2*e +
15*a^2*b*d*e^2))/4 + (5*b^2*e^3*x^9*(2*a^3*e^3 + 4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2))/9 + (a^4*d^5*x^
2*(6*a*e + 5*b*d))/2 + (b^4*e^5*x^11*(5*a*e + 6*b*d))/11 + (5*a^3*d^4*x^3*(3*a^2*e^2 + 2*b^2*d^2 + 6*a*b*d*e))
/3 + (b^3*e^4*x^10*(2*a^2*e^2 + 3*b^2*d^2 + 6*a*b*d*e))/2

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