3.17.92 \(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=143 \[ \frac {5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac {10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac {e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac {5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac {(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6} \]

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Rubi [A]  time = 0.36, 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, 43} \begin {gather*} \frac {5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac {10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac {e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac {5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac {(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^5*(a + b*x)^8)/(8*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^9)/(9*b^6) + (e^2*(b*d - a*e)^3*(a + b*x)^1
0)/b^6 + (10*e^3*(b*d - a*e)^2*(a + b*x)^11)/(11*b^6) + (5*e^4*(b*d - a*e)*(a + b*x)^12)/(12*b^6) + (e^5*(a +
b*x)^13)/(13*b^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 43

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)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^5 \, dx\\ &=\int \left (\frac {(b d-a e)^5 (a+b x)^7}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^8}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^9}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^{10}}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^{11}}{b^5}+\frac {e^5 (a+b x)^{12}}{b^5}\right ) \, dx\\ &=\frac {(b d-a e)^5 (a+b x)^8}{8 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e)^3 (a+b x)^{10}}{b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^{11}}{11 b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{12}}{12 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6}\\ \end {align*}

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Mathematica [B]  time = 0.14, size = 493, normalized size = 3.45 \begin {gather*} \frac {x \left (1716 a^7 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+1716 a^6 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+1287 a^5 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+715 a^4 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+286 a^3 b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+78 a^2 b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )+13 a b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )+b^7 x^7 \left (1287 d^5+5720 d^4 e x+10296 d^3 e^2 x^2+9360 d^2 e^3 x^3+4290 d e^4 x^4+792 e^5 x^5\right )\right )}{10296} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x*(1716*a^7*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 1716*a^6*b*x*(21
*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 1287*a^5*b^2*x^2*(56*d^5 +
210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 715*a^4*b^3*x^3*(126*d^5 + 504
*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 286*a^3*b^4*x^4*(252*d^5 + 1050*d
^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 78*a^2*b^5*x^5*(462*d^5 + 1980*d
^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5) + 13*a*b^6*x^6*(792*d^5 + 3465*d^
4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + b^7*x^7*(1287*d^5 + 5720*d^4*e*x
+ 10296*d^3*e^2*x^2 + 9360*d^2*e^3*x^3 + 4290*d*e^4*x^4 + 792*e^5*x^5)))/10296

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.39, size = 670, normalized size = 4.69 \begin {gather*} \frac {1}{13} x^{13} e^{5} b^{7} + \frac {5}{12} x^{12} e^{4} d b^{7} + \frac {7}{12} x^{12} e^{5} b^{6} a + \frac {10}{11} x^{11} e^{3} d^{2} b^{7} + \frac {35}{11} x^{11} e^{4} d b^{6} a + \frac {21}{11} x^{11} e^{5} b^{5} a^{2} + x^{10} e^{2} d^{3} b^{7} + 7 x^{10} e^{3} d^{2} b^{6} a + \frac {21}{2} x^{10} e^{4} d b^{5} a^{2} + \frac {7}{2} x^{10} e^{5} b^{4} a^{3} + \frac {5}{9} x^{9} e d^{4} b^{7} + \frac {70}{9} x^{9} e^{2} d^{3} b^{6} a + \frac {70}{3} x^{9} e^{3} d^{2} b^{5} a^{2} + \frac {175}{9} x^{9} e^{4} d b^{4} a^{3} + \frac {35}{9} x^{9} e^{5} b^{3} a^{4} + \frac {1}{8} x^{8} d^{5} b^{7} + \frac {35}{8} x^{8} e d^{4} b^{6} a + \frac {105}{4} x^{8} e^{2} d^{3} b^{5} a^{2} + \frac {175}{4} x^{8} e^{3} d^{2} b^{4} a^{3} + \frac {175}{8} x^{8} e^{4} d b^{3} a^{4} + \frac {21}{8} x^{8} e^{5} b^{2} a^{5} + x^{7} d^{5} b^{6} a + 15 x^{7} e d^{4} b^{5} a^{2} + 50 x^{7} e^{2} d^{3} b^{4} a^{3} + 50 x^{7} e^{3} d^{2} b^{3} a^{4} + 15 x^{7} e^{4} d b^{2} a^{5} + x^{7} e^{5} b a^{6} + \frac {7}{2} x^{6} d^{5} b^{5} a^{2} + \frac {175}{6} x^{6} e d^{4} b^{4} a^{3} + \frac {175}{3} x^{6} e^{2} d^{3} b^{3} a^{4} + 35 x^{6} e^{3} d^{2} b^{2} a^{5} + \frac {35}{6} x^{6} e^{4} d b a^{6} + \frac {1}{6} x^{6} e^{5} a^{7} + 7 x^{5} d^{5} b^{4} a^{3} + 35 x^{5} e d^{4} b^{3} a^{4} + 42 x^{5} e^{2} d^{3} b^{2} a^{5} + 14 x^{5} e^{3} d^{2} b a^{6} + x^{5} e^{4} d a^{7} + \frac {35}{4} x^{4} d^{5} b^{3} a^{4} + \frac {105}{4} x^{4} e d^{4} b^{2} a^{5} + \frac {35}{2} x^{4} e^{2} d^{3} b a^{6} + \frac {5}{2} x^{4} e^{3} d^{2} a^{7} + 7 x^{3} d^{5} b^{2} a^{5} + \frac {35}{3} x^{3} e d^{4} b a^{6} + \frac {10}{3} x^{3} e^{2} d^{3} a^{7} + \frac {7}{2} x^{2} d^{5} b a^{6} + \frac {5}{2} x^{2} e d^{4} a^{7} + x d^{5} a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/13*x^13*e^5*b^7 + 5/12*x^12*e^4*d*b^7 + 7/12*x^12*e^5*b^6*a + 10/11*x^11*e^3*d^2*b^7 + 35/11*x^11*e^4*d*b^6*
a + 21/11*x^11*e^5*b^5*a^2 + x^10*e^2*d^3*b^7 + 7*x^10*e^3*d^2*b^6*a + 21/2*x^10*e^4*d*b^5*a^2 + 7/2*x^10*e^5*
b^4*a^3 + 5/9*x^9*e*d^4*b^7 + 70/9*x^9*e^2*d^3*b^6*a + 70/3*x^9*e^3*d^2*b^5*a^2 + 175/9*x^9*e^4*d*b^4*a^3 + 35
/9*x^9*e^5*b^3*a^4 + 1/8*x^8*d^5*b^7 + 35/8*x^8*e*d^4*b^6*a + 105/4*x^8*e^2*d^3*b^5*a^2 + 175/4*x^8*e^3*d^2*b^
4*a^3 + 175/8*x^8*e^4*d*b^3*a^4 + 21/8*x^8*e^5*b^2*a^5 + x^7*d^5*b^6*a + 15*x^7*e*d^4*b^5*a^2 + 50*x^7*e^2*d^3
*b^4*a^3 + 50*x^7*e^3*d^2*b^3*a^4 + 15*x^7*e^4*d*b^2*a^5 + x^7*e^5*b*a^6 + 7/2*x^6*d^5*b^5*a^2 + 175/6*x^6*e*d
^4*b^4*a^3 + 175/3*x^6*e^2*d^3*b^3*a^4 + 35*x^6*e^3*d^2*b^2*a^5 + 35/6*x^6*e^4*d*b*a^6 + 1/6*x^6*e^5*a^7 + 7*x
^5*d^5*b^4*a^3 + 35*x^5*e*d^4*b^3*a^4 + 42*x^5*e^2*d^3*b^2*a^5 + 14*x^5*e^3*d^2*b*a^6 + x^5*e^4*d*a^7 + 35/4*x
^4*d^5*b^3*a^4 + 105/4*x^4*e*d^4*b^2*a^5 + 35/2*x^4*e^2*d^3*b*a^6 + 5/2*x^4*e^3*d^2*a^7 + 7*x^3*d^5*b^2*a^5 +
35/3*x^3*e*d^4*b*a^6 + 10/3*x^3*e^2*d^3*a^7 + 7/2*x^2*d^5*b*a^6 + 5/2*x^2*e*d^4*a^7 + x*d^5*a^7

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giac [B]  time = 0.16, size = 646, normalized size = 4.52 \begin {gather*} \frac {1}{13} \, b^{7} x^{13} e^{5} + \frac {5}{12} \, b^{7} d x^{12} e^{4} + \frac {10}{11} \, b^{7} d^{2} x^{11} e^{3} + b^{7} d^{3} x^{10} e^{2} + \frac {5}{9} \, b^{7} d^{4} x^{9} e + \frac {1}{8} \, b^{7} d^{5} x^{8} + \frac {7}{12} \, a b^{6} x^{12} e^{5} + \frac {35}{11} \, a b^{6} d x^{11} e^{4} + 7 \, a b^{6} d^{2} x^{10} e^{3} + \frac {70}{9} \, a b^{6} d^{3} x^{9} e^{2} + \frac {35}{8} \, a b^{6} d^{4} x^{8} e + a b^{6} d^{5} x^{7} + \frac {21}{11} \, a^{2} b^{5} x^{11} e^{5} + \frac {21}{2} \, a^{2} b^{5} d x^{10} e^{4} + \frac {70}{3} \, a^{2} b^{5} d^{2} x^{9} e^{3} + \frac {105}{4} \, a^{2} b^{5} d^{3} x^{8} e^{2} + 15 \, a^{2} b^{5} d^{4} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{5} x^{6} + \frac {7}{2} \, a^{3} b^{4} x^{10} e^{5} + \frac {175}{9} \, a^{3} b^{4} d x^{9} e^{4} + \frac {175}{4} \, a^{3} b^{4} d^{2} x^{8} e^{3} + 50 \, a^{3} b^{4} d^{3} x^{7} e^{2} + \frac {175}{6} \, a^{3} b^{4} d^{4} x^{6} e + 7 \, a^{3} b^{4} d^{5} x^{5} + \frac {35}{9} \, a^{4} b^{3} x^{9} e^{5} + \frac {175}{8} \, a^{4} b^{3} d x^{8} e^{4} + 50 \, a^{4} b^{3} d^{2} x^{7} e^{3} + \frac {175}{3} \, a^{4} b^{3} d^{3} x^{6} e^{2} + 35 \, a^{4} b^{3} d^{4} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{5} x^{4} + \frac {21}{8} \, a^{5} b^{2} x^{8} e^{5} + 15 \, a^{5} b^{2} d x^{7} e^{4} + 35 \, a^{5} b^{2} d^{2} x^{6} e^{3} + 42 \, a^{5} b^{2} d^{3} x^{5} e^{2} + \frac {105}{4} \, a^{5} b^{2} d^{4} x^{4} e + 7 \, a^{5} b^{2} d^{5} x^{3} + a^{6} b x^{7} e^{5} + \frac {35}{6} \, a^{6} b d x^{6} e^{4} + 14 \, a^{6} b d^{2} x^{5} e^{3} + \frac {35}{2} \, a^{6} b d^{3} x^{4} e^{2} + \frac {35}{3} \, a^{6} b d^{4} x^{3} e + \frac {7}{2} \, a^{6} b d^{5} x^{2} + \frac {1}{6} \, a^{7} x^{6} e^{5} + a^{7} d x^{5} e^{4} + \frac {5}{2} \, a^{7} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{7} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{7} d^{4} x^{2} e + a^{7} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/13*b^7*x^13*e^5 + 5/12*b^7*d*x^12*e^4 + 10/11*b^7*d^2*x^11*e^3 + b^7*d^3*x^10*e^2 + 5/9*b^7*d^4*x^9*e + 1/8*
b^7*d^5*x^8 + 7/12*a*b^6*x^12*e^5 + 35/11*a*b^6*d*x^11*e^4 + 7*a*b^6*d^2*x^10*e^3 + 70/9*a*b^6*d^3*x^9*e^2 + 3
5/8*a*b^6*d^4*x^8*e + a*b^6*d^5*x^7 + 21/11*a^2*b^5*x^11*e^5 + 21/2*a^2*b^5*d*x^10*e^4 + 70/3*a^2*b^5*d^2*x^9*
e^3 + 105/4*a^2*b^5*d^3*x^8*e^2 + 15*a^2*b^5*d^4*x^7*e + 7/2*a^2*b^5*d^5*x^6 + 7/2*a^3*b^4*x^10*e^5 + 175/9*a^
3*b^4*d*x^9*e^4 + 175/4*a^3*b^4*d^2*x^8*e^3 + 50*a^3*b^4*d^3*x^7*e^2 + 175/6*a^3*b^4*d^4*x^6*e + 7*a^3*b^4*d^5
*x^5 + 35/9*a^4*b^3*x^9*e^5 + 175/8*a^4*b^3*d*x^8*e^4 + 50*a^4*b^3*d^2*x^7*e^3 + 175/3*a^4*b^3*d^3*x^6*e^2 + 3
5*a^4*b^3*d^4*x^5*e + 35/4*a^4*b^3*d^5*x^4 + 21/8*a^5*b^2*x^8*e^5 + 15*a^5*b^2*d*x^7*e^4 + 35*a^5*b^2*d^2*x^6*
e^3 + 42*a^5*b^2*d^3*x^5*e^2 + 105/4*a^5*b^2*d^4*x^4*e + 7*a^5*b^2*d^5*x^3 + a^6*b*x^7*e^5 + 35/6*a^6*b*d*x^6*
e^4 + 14*a^6*b*d^2*x^5*e^3 + 35/2*a^6*b*d^3*x^4*e^2 + 35/3*a^6*b*d^4*x^3*e + 7/2*a^6*b*d^5*x^2 + 1/6*a^7*x^6*e
^5 + a^7*d*x^5*e^4 + 5/2*a^7*d^2*x^4*e^3 + 10/3*a^7*d^3*x^3*e^2 + 5/2*a^7*d^4*x^2*e + a^7*d^5*x

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maple [B]  time = 0.05, size = 982, normalized size = 6.87 \begin {gather*} \frac {b^{7} e^{5} x^{13}}{13}+a^{7} d^{5} x +\frac {\left (6 a \,b^{6} e^{5}+\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{6}\right ) x^{12}}{12}+\frac {\left (15 a^{2} b^{5} e^{5}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a \,b^{5}+\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{6}\right ) x^{11}}{11}+\frac {\left (20 a^{3} b^{4} e^{5}+15 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2} b^{4}+6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a \,b^{5}+\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) b^{6}\right ) x^{10}}{10}+\frac {\left (15 a^{4} b^{3} e^{5}+20 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{3} b^{3}+15 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2} b^{4}+6 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a \,b^{5}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) b^{6}\right ) x^{9}}{9}+\frac {\left (6 a^{5} b^{2} e^{5}+15 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{4} b^{2}+20 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{3} b^{3}+15 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{2} b^{4}+6 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a \,b^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{6}\right ) x^{8}}{8}+\frac {\left (a^{6} b \,e^{5}+a \,b^{6} d^{5}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{5} b +15 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{4} b^{2}+20 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{3} b^{3}+15 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{2} b^{4}+6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{5} d^{5}+\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{6}+6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{5} b +15 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{4} b^{2}+20 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{3} b^{3}+15 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (15 a^{3} b^{4} d^{5}+\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{6}+6 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{5} b +15 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{4} b^{2}+20 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (20 a^{4} b^{3} d^{5}+\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{6}+6 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{5} b +15 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (15 a^{5} b^{2} d^{5}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{6}+6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b \,d^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{6}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/13*b^7*e^5*x^13+1/12*((a*e^5+5*b*d*e^4)*b^6+6*b^6*e^5*a)*x^12+1/11*((5*a*d*e^4+10*b*d^2*e^3)*b^6+6*(a*e^5+5*
b*d*e^4)*a*b^5+15*b^5*e^5*a^2)*x^11+1/10*((10*a*d^2*e^3+10*b*d^3*e^2)*b^6+6*(5*a*d*e^4+10*b*d^2*e^3)*a*b^5+15*
(a*e^5+5*b*d*e^4)*a^2*b^4+20*b^4*e^5*a^3)*x^10+1/9*((10*a*d^3*e^2+5*b*d^4*e)*b^6+6*(10*a*d^2*e^3+10*b*d^3*e^2)
*a*b^5+15*(5*a*d*e^4+10*b*d^2*e^3)*a^2*b^4+20*(a*e^5+5*b*d*e^4)*a^3*b^3+15*b^3*e^5*a^4)*x^9+1/8*((5*a*d^4*e+b*
d^5)*b^6+6*(10*a*d^3*e^2+5*b*d^4*e)*a*b^5+15*(10*a*d^2*e^3+10*b*d^3*e^2)*a^2*b^4+20*(5*a*d*e^4+10*b*d^2*e^3)*a
^3*b^3+15*(a*e^5+5*b*d*e^4)*a^4*b^2+6*b^2*e^5*a^5)*x^8+1/7*(a*d^5*b^6+6*(5*a*d^4*e+b*d^5)*a*b^5+15*(10*a*d^3*e
^2+5*b*d^4*e)*a^2*b^4+20*(10*a*d^2*e^3+10*b*d^3*e^2)*a^3*b^3+15*(5*a*d*e^4+10*b*d^2*e^3)*a^4*b^2+6*(a*e^5+5*b*
d*e^4)*a^5*b+b*e^5*a^6)*x^7+1/6*(6*a^2*d^5*b^5+15*(5*a*d^4*e+b*d^5)*a^2*b^4+20*(10*a*d^3*e^2+5*b*d^4*e)*a^3*b^
3+15*(10*a*d^2*e^3+10*b*d^3*e^2)*a^4*b^2+6*(5*a*d*e^4+10*b*d^2*e^3)*a^5*b+(a*e^5+5*b*d*e^4)*a^6)*x^6+1/5*(15*a
^3*d^5*b^4+20*(5*a*d^4*e+b*d^5)*a^3*b^3+15*(10*a*d^3*e^2+5*b*d^4*e)*a^4*b^2+6*(10*a*d^2*e^3+10*b*d^3*e^2)*a^5*
b+(5*a*d*e^4+10*b*d^2*e^3)*a^6)*x^5+1/4*(20*a^4*d^5*b^3+15*(5*a*d^4*e+b*d^5)*a^4*b^2+6*(10*a*d^3*e^2+5*b*d^4*e
)*a^5*b+(10*a*d^2*e^3+10*b*d^3*e^2)*a^6)*x^4+1/3*(15*a^5*d^5*b^2+6*(5*a*d^4*e+b*d^5)*a^5*b+(10*a*d^3*e^2+5*b*d
^4*e)*a^6)*x^3+1/2*(6*a^6*d^5*b+(5*a*d^4*e+b*d^5)*a^6)*x^2+a^7*d^5*x

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maxima [B]  time = 0.57, size = 594, normalized size = 4.15 \begin {gather*} \frac {1}{13} \, b^{7} e^{5} x^{13} + a^{7} d^{5} x + \frac {1}{12} \, {\left (5 \, b^{7} d e^{4} + 7 \, a b^{6} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, b^{7} d^{2} e^{3} + 35 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{7} d^{3} e^{2} + 14 \, a b^{6} d^{2} e^{3} + 21 \, a^{2} b^{5} d e^{4} + 7 \, a^{3} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (b^{7} d^{4} e + 14 \, a b^{6} d^{3} e^{2} + 42 \, a^{2} b^{5} d^{2} e^{3} + 35 \, a^{3} b^{4} d e^{4} + 7 \, a^{4} b^{3} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{5} + 35 \, a b^{6} d^{4} e + 210 \, a^{2} b^{5} d^{3} e^{2} + 350 \, a^{3} b^{4} d^{2} e^{3} + 175 \, a^{4} b^{3} d e^{4} + 21 \, a^{5} b^{2} e^{5}\right )} x^{8} + {\left (a b^{6} d^{5} + 15 \, a^{2} b^{5} d^{4} e + 50 \, a^{3} b^{4} d^{3} e^{2} + 50 \, a^{4} b^{3} d^{2} e^{3} + 15 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (21 \, a^{2} b^{5} d^{5} + 175 \, a^{3} b^{4} d^{4} e + 350 \, a^{4} b^{3} d^{3} e^{2} + 210 \, a^{5} b^{2} d^{2} e^{3} + 35 \, a^{6} b d e^{4} + a^{7} e^{5}\right )} x^{6} + {\left (7 \, a^{3} b^{4} d^{5} + 35 \, a^{4} b^{3} d^{4} e + 42 \, a^{5} b^{2} d^{3} e^{2} + 14 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (7 \, a^{4} b^{3} d^{5} + 21 \, a^{5} b^{2} d^{4} e + 14 \, a^{6} b d^{3} e^{2} + 2 \, a^{7} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{5} + 35 \, a^{6} b d^{4} e + 10 \, a^{7} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{5} + 5 \, a^{7} d^{4} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/13*b^7*e^5*x^13 + a^7*d^5*x + 1/12*(5*b^7*d*e^4 + 7*a*b^6*e^5)*x^12 + 1/11*(10*b^7*d^2*e^3 + 35*a*b^6*d*e^4
+ 21*a^2*b^5*e^5)*x^11 + 1/2*(2*b^7*d^3*e^2 + 14*a*b^6*d^2*e^3 + 21*a^2*b^5*d*e^4 + 7*a^3*b^4*e^5)*x^10 + 5/9*
(b^7*d^4*e + 14*a*b^6*d^3*e^2 + 42*a^2*b^5*d^2*e^3 + 35*a^3*b^4*d*e^4 + 7*a^4*b^3*e^5)*x^9 + 1/8*(b^7*d^5 + 35
*a*b^6*d^4*e + 210*a^2*b^5*d^3*e^2 + 350*a^3*b^4*d^2*e^3 + 175*a^4*b^3*d*e^4 + 21*a^5*b^2*e^5)*x^8 + (a*b^6*d^
5 + 15*a^2*b^5*d^4*e + 50*a^3*b^4*d^3*e^2 + 50*a^4*b^3*d^2*e^3 + 15*a^5*b^2*d*e^4 + a^6*b*e^5)*x^7 + 1/6*(21*a
^2*b^5*d^5 + 175*a^3*b^4*d^4*e + 350*a^4*b^3*d^3*e^2 + 210*a^5*b^2*d^2*e^3 + 35*a^6*b*d*e^4 + a^7*e^5)*x^6 + (
7*a^3*b^4*d^5 + 35*a^4*b^3*d^4*e + 42*a^5*b^2*d^3*e^2 + 14*a^6*b*d^2*e^3 + a^7*d*e^4)*x^5 + 5/4*(7*a^4*b^3*d^5
+ 21*a^5*b^2*d^4*e + 14*a^6*b*d^3*e^2 + 2*a^7*d^2*e^3)*x^4 + 1/3*(21*a^5*b^2*d^5 + 35*a^6*b*d^4*e + 10*a^7*d^
3*e^2)*x^3 + 1/2*(7*a^6*b*d^5 + 5*a^7*d^4*e)*x^2

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mupad [B]  time = 2.17, size = 570, normalized size = 3.99 \begin {gather*} x^5\,\left (a^7\,d\,e^4+14\,a^6\,b\,d^2\,e^3+42\,a^5\,b^2\,d^3\,e^2+35\,a^4\,b^3\,d^4\,e+7\,a^3\,b^4\,d^5\right )+x^9\,\left (\frac {35\,a^4\,b^3\,e^5}{9}+\frac {175\,a^3\,b^4\,d\,e^4}{9}+\frac {70\,a^2\,b^5\,d^2\,e^3}{3}+\frac {70\,a\,b^6\,d^3\,e^2}{9}+\frac {5\,b^7\,d^4\,e}{9}\right )+x^7\,\left (a^6\,b\,e^5+15\,a^5\,b^2\,d\,e^4+50\,a^4\,b^3\,d^2\,e^3+50\,a^3\,b^4\,d^3\,e^2+15\,a^2\,b^5\,d^4\,e+a\,b^6\,d^5\right )+x^6\,\left (\frac {a^7\,e^5}{6}+\frac {35\,a^6\,b\,d\,e^4}{6}+35\,a^5\,b^2\,d^2\,e^3+\frac {175\,a^4\,b^3\,d^3\,e^2}{3}+\frac {175\,a^3\,b^4\,d^4\,e}{6}+\frac {7\,a^2\,b^5\,d^5}{2}\right )+x^8\,\left (\frac {21\,a^5\,b^2\,e^5}{8}+\frac {175\,a^4\,b^3\,d\,e^4}{8}+\frac {175\,a^3\,b^4\,d^2\,e^3}{4}+\frac {105\,a^2\,b^5\,d^3\,e^2}{4}+\frac {35\,a\,b^6\,d^4\,e}{8}+\frac {b^7\,d^5}{8}\right )+a^7\,d^5\,x+\frac {b^7\,e^5\,x^{13}}{13}+\frac {5\,a^4\,d^2\,x^4\,\left (2\,a^3\,e^3+14\,a^2\,b\,d\,e^2+21\,a\,b^2\,d^2\,e+7\,b^3\,d^3\right )}{4}+\frac {b^4\,e^2\,x^{10}\,\left (7\,a^3\,e^3+21\,a^2\,b\,d\,e^2+14\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{2}+\frac {a^6\,d^4\,x^2\,\left (5\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^4\,x^{12}\,\left (7\,a\,e+5\,b\,d\right )}{12}+\frac {a^5\,d^3\,x^3\,\left (10\,a^2\,e^2+35\,a\,b\,d\,e+21\,b^2\,d^2\right )}{3}+\frac {b^5\,e^3\,x^{11}\,\left (21\,a^2\,e^2+35\,a\,b\,d\,e+10\,b^2\,d^2\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^5*(a^7*d*e^4 + 7*a^3*b^4*d^5 + 35*a^4*b^3*d^4*e + 14*a^6*b*d^2*e^3 + 42*a^5*b^2*d^3*e^2) + x^9*((5*b^7*d^4*e
)/9 + (35*a^4*b^3*e^5)/9 + (70*a*b^6*d^3*e^2)/9 + (175*a^3*b^4*d*e^4)/9 + (70*a^2*b^5*d^2*e^3)/3) + x^7*(a*b^6
*d^5 + a^6*b*e^5 + 15*a^2*b^5*d^4*e + 15*a^5*b^2*d*e^4 + 50*a^3*b^4*d^3*e^2 + 50*a^4*b^3*d^2*e^3) + x^6*((a^7*
e^5)/6 + (7*a^2*b^5*d^5)/2 + (175*a^3*b^4*d^4*e)/6 + (175*a^4*b^3*d^3*e^2)/3 + 35*a^5*b^2*d^2*e^3 + (35*a^6*b*
d*e^4)/6) + x^8*((b^7*d^5)/8 + (21*a^5*b^2*e^5)/8 + (175*a^4*b^3*d*e^4)/8 + (105*a^2*b^5*d^3*e^2)/4 + (175*a^3
*b^4*d^2*e^3)/4 + (35*a*b^6*d^4*e)/8) + a^7*d^5*x + (b^7*e^5*x^13)/13 + (5*a^4*d^2*x^4*(2*a^3*e^3 + 7*b^3*d^3
+ 21*a*b^2*d^2*e + 14*a^2*b*d*e^2))/4 + (b^4*e^2*x^10*(7*a^3*e^3 + 2*b^3*d^3 + 14*a*b^2*d^2*e + 21*a^2*b*d*e^2
))/2 + (a^6*d^4*x^2*(5*a*e + 7*b*d))/2 + (b^6*e^4*x^12*(7*a*e + 5*b*d))/12 + (a^5*d^3*x^3*(10*a^2*e^2 + 21*b^2
*d^2 + 35*a*b*d*e))/3 + (b^5*e^3*x^11*(21*a^2*e^2 + 10*b^2*d^2 + 35*a*b*d*e))/11

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sympy [B]  time = 0.18, size = 673, normalized size = 4.71 \begin {gather*} a^{7} d^{5} x + \frac {b^{7} e^{5} x^{13}}{13} + x^{12} \left (\frac {7 a b^{6} e^{5}}{12} + \frac {5 b^{7} d e^{4}}{12}\right ) + x^{11} \left (\frac {21 a^{2} b^{5} e^{5}}{11} + \frac {35 a b^{6} d e^{4}}{11} + \frac {10 b^{7} d^{2} e^{3}}{11}\right ) + x^{10} \left (\frac {7 a^{3} b^{4} e^{5}}{2} + \frac {21 a^{2} b^{5} d e^{4}}{2} + 7 a b^{6} d^{2} e^{3} + b^{7} d^{3} e^{2}\right ) + x^{9} \left (\frac {35 a^{4} b^{3} e^{5}}{9} + \frac {175 a^{3} b^{4} d e^{4}}{9} + \frac {70 a^{2} b^{5} d^{2} e^{3}}{3} + \frac {70 a b^{6} d^{3} e^{2}}{9} + \frac {5 b^{7} d^{4} e}{9}\right ) + x^{8} \left (\frac {21 a^{5} b^{2} e^{5}}{8} + \frac {175 a^{4} b^{3} d e^{4}}{8} + \frac {175 a^{3} b^{4} d^{2} e^{3}}{4} + \frac {105 a^{2} b^{5} d^{3} e^{2}}{4} + \frac {35 a b^{6} d^{4} e}{8} + \frac {b^{7} d^{5}}{8}\right ) + x^{7} \left (a^{6} b e^{5} + 15 a^{5} b^{2} d e^{4} + 50 a^{4} b^{3} d^{2} e^{3} + 50 a^{3} b^{4} d^{3} e^{2} + 15 a^{2} b^{5} d^{4} e + a b^{6} d^{5}\right ) + x^{6} \left (\frac {a^{7} e^{5}}{6} + \frac {35 a^{6} b d e^{4}}{6} + 35 a^{5} b^{2} d^{2} e^{3} + \frac {175 a^{4} b^{3} d^{3} e^{2}}{3} + \frac {175 a^{3} b^{4} d^{4} e}{6} + \frac {7 a^{2} b^{5} d^{5}}{2}\right ) + x^{5} \left (a^{7} d e^{4} + 14 a^{6} b d^{2} e^{3} + 42 a^{5} b^{2} d^{3} e^{2} + 35 a^{4} b^{3} d^{4} e + 7 a^{3} b^{4} d^{5}\right ) + x^{4} \left (\frac {5 a^{7} d^{2} e^{3}}{2} + \frac {35 a^{6} b d^{3} e^{2}}{2} + \frac {105 a^{5} b^{2} d^{4} e}{4} + \frac {35 a^{4} b^{3} d^{5}}{4}\right ) + x^{3} \left (\frac {10 a^{7} d^{3} e^{2}}{3} + \frac {35 a^{6} b d^{4} e}{3} + 7 a^{5} b^{2} d^{5}\right ) + x^{2} \left (\frac {5 a^{7} d^{4} e}{2} + \frac {7 a^{6} b d^{5}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**7*d**5*x + b**7*e**5*x**13/13 + x**12*(7*a*b**6*e**5/12 + 5*b**7*d*e**4/12) + x**11*(21*a**2*b**5*e**5/11 +
35*a*b**6*d*e**4/11 + 10*b**7*d**2*e**3/11) + x**10*(7*a**3*b**4*e**5/2 + 21*a**2*b**5*d*e**4/2 + 7*a*b**6*d*
*2*e**3 + b**7*d**3*e**2) + x**9*(35*a**4*b**3*e**5/9 + 175*a**3*b**4*d*e**4/9 + 70*a**2*b**5*d**2*e**3/3 + 70
*a*b**6*d**3*e**2/9 + 5*b**7*d**4*e/9) + x**8*(21*a**5*b**2*e**5/8 + 175*a**4*b**3*d*e**4/8 + 175*a**3*b**4*d*
*2*e**3/4 + 105*a**2*b**5*d**3*e**2/4 + 35*a*b**6*d**4*e/8 + b**7*d**5/8) + x**7*(a**6*b*e**5 + 15*a**5*b**2*d
*e**4 + 50*a**4*b**3*d**2*e**3 + 50*a**3*b**4*d**3*e**2 + 15*a**2*b**5*d**4*e + a*b**6*d**5) + x**6*(a**7*e**5
/6 + 35*a**6*b*d*e**4/6 + 35*a**5*b**2*d**2*e**3 + 175*a**4*b**3*d**3*e**2/3 + 175*a**3*b**4*d**4*e/6 + 7*a**2
*b**5*d**5/2) + x**5*(a**7*d*e**4 + 14*a**6*b*d**2*e**3 + 42*a**5*b**2*d**3*e**2 + 35*a**4*b**3*d**4*e + 7*a**
3*b**4*d**5) + x**4*(5*a**7*d**2*e**3/2 + 35*a**6*b*d**3*e**2/2 + 105*a**5*b**2*d**4*e/4 + 35*a**4*b**3*d**5/4
) + x**3*(10*a**7*d**3*e**2/3 + 35*a**6*b*d**4*e/3 + 7*a**5*b**2*d**5) + x**2*(5*a**7*d**4*e/2 + 7*a**6*b*d**5
/2)

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