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

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

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Rubi [A]  time = 0.31, antiderivative size = 143, 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, 43} \begin {gather*} \frac {5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac {e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac {10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac {5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac {(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac {e^5 (a+b x)^{12}}{12 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Mathematica [B]  time = 0.07, size = 501, normalized size = 3.50 \begin {gather*} a^6 d^5 x+\frac {1}{2} a^5 d^4 x^2 (5 a e+6 b d)+\frac {1}{2} b^4 e^3 x^{10} \left (3 a^2 e^2+6 a b d e+2 b^2 d^2\right )+\frac {5}{3} a^4 d^3 x^3 \left (2 a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac {5}{9} b^3 e^2 x^9 \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 )+\frac {5}{4} a^3 d^2 x^4 \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 )+\frac {5}{8} b^2 e x^8 \left (3 a^4 e^4+20 a^3 b d e^3+30 a^2 b^2 d^2 e^2+12 a b^3 d^3 e+b^4 d^4\right )+a^2 d x^5 \left (a^4 e^4+12 a^3 b d e^3+30 a^2 b^2 d^2 e^2+20 a b^3 d^3 e+3 b^4 d^4\right )+\frac {1}{7} b x^7 \left (6 a^5 e^5+75 a^4 b d e^4+200 a^3 b^2 d^2 e^3+150 a^2 b^3 d^3 e^2+30 a b^4 d^4 e+b^5 d^5\right )+\frac {1}{6} a x^6 \left (a^5 e^5+30 a^4 b d e^4+150 a^3 b^2 d^2 e^3+200 a^2 b^3 d^3 e^2+75 a b^4 d^4 e+6 b^5 d^5\right )+\frac {1}{11} b^5 e^4 x^{11} (6 a e+5 b d)+\frac {1}{12} b^6 e^5 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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fricas [B]  time = 0.36, size = 579, normalized size = 4.05 \begin {gather*} \frac {1}{12} x^{12} e^{5} b^{6} + \frac {5}{11} x^{11} e^{4} d b^{6} + \frac {6}{11} x^{11} e^{5} b^{5} a + x^{10} e^{3} d^{2} b^{6} + 3 x^{10} e^{4} d b^{5} a + \frac {3}{2} x^{10} e^{5} b^{4} a^{2} + \frac {10}{9} x^{9} e^{2} d^{3} b^{6} + \frac {20}{3} x^{9} e^{3} d^{2} b^{5} a + \frac {25}{3} x^{9} e^{4} d b^{4} a^{2} + \frac {20}{9} x^{9} e^{5} b^{3} a^{3} + \frac {5}{8} x^{8} e d^{4} b^{6} + \frac {15}{2} x^{8} e^{2} d^{3} b^{5} a + \frac {75}{4} x^{8} e^{3} d^{2} b^{4} a^{2} + \frac {25}{2} x^{8} e^{4} d b^{3} a^{3} + \frac {15}{8} x^{8} e^{5} b^{2} a^{4} + \frac {1}{7} x^{7} d^{5} b^{6} + \frac {30}{7} x^{7} e d^{4} b^{5} a + \frac {150}{7} x^{7} e^{2} d^{3} b^{4} a^{2} + \frac {200}{7} x^{7} e^{3} d^{2} b^{3} a^{3} + \frac {75}{7} x^{7} e^{4} d b^{2} a^{4} + \frac {6}{7} x^{7} e^{5} b a^{5} + x^{6} d^{5} b^{5} a + \frac {25}{2} x^{6} e d^{4} b^{4} a^{2} + \frac {100}{3} x^{6} e^{2} d^{3} b^{3} a^{3} + 25 x^{6} e^{3} d^{2} b^{2} a^{4} + 5 x^{6} e^{4} d b a^{5} + \frac {1}{6} x^{6} e^{5} a^{6} + 3 x^{5} d^{5} b^{4} a^{2} + 20 x^{5} e d^{4} b^{3} a^{3} + 30 x^{5} e^{2} d^{3} b^{2} a^{4} + 12 x^{5} e^{3} d^{2} b a^{5} + x^{5} e^{4} d a^{6} + 5 x^{4} d^{5} b^{3} a^{3} + \frac {75}{4} x^{4} e d^{4} b^{2} a^{4} + 15 x^{4} e^{2} d^{3} b a^{5} + \frac {5}{2} x^{4} e^{3} d^{2} a^{6} + 5 x^{3} d^{5} b^{2} a^{4} + 10 x^{3} e d^{4} b a^{5} + \frac {10}{3} x^{3} e^{2} d^{3} a^{6} + 3 x^{2} d^{5} b a^{5} + \frac {5}{2} x^{2} e d^{4} a^{6} + x d^{5} a^{6} \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)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^5*b^6 + 5/11*x^11*e^4*d*b^6 + 6/11*x^11*e^5*b^5*a + x^10*e^3*d^2*b^6 + 3*x^10*e^4*d*b^5*a + 3/2*x^
10*e^5*b^4*a^2 + 10/9*x^9*e^2*d^3*b^6 + 20/3*x^9*e^3*d^2*b^5*a + 25/3*x^9*e^4*d*b^4*a^2 + 20/9*x^9*e^5*b^3*a^3
+ 5/8*x^8*e*d^4*b^6 + 15/2*x^8*e^2*d^3*b^5*a + 75/4*x^8*e^3*d^2*b^4*a^2 + 25/2*x^8*e^4*d*b^3*a^3 + 15/8*x^8*e
^5*b^2*a^4 + 1/7*x^7*d^5*b^6 + 30/7*x^7*e*d^4*b^5*a + 150/7*x^7*e^2*d^3*b^4*a^2 + 200/7*x^7*e^3*d^2*b^3*a^3 +
75/7*x^7*e^4*d*b^2*a^4 + 6/7*x^7*e^5*b*a^5 + x^6*d^5*b^5*a + 25/2*x^6*e*d^4*b^4*a^2 + 100/3*x^6*e^2*d^3*b^3*a^
3 + 25*x^6*e^3*d^2*b^2*a^4 + 5*x^6*e^4*d*b*a^5 + 1/6*x^6*e^5*a^6 + 3*x^5*d^5*b^4*a^2 + 20*x^5*e*d^4*b^3*a^3 +
30*x^5*e^2*d^3*b^2*a^4 + 12*x^5*e^3*d^2*b*a^5 + x^5*e^4*d*a^6 + 5*x^4*d^5*b^3*a^3 + 75/4*x^4*e*d^4*b^2*a^4 + 1
5*x^4*e^2*d^3*b*a^5 + 5/2*x^4*e^3*d^2*a^6 + 5*x^3*d^5*b^2*a^4 + 10*x^3*e*d^4*b*a^5 + 10/3*x^3*e^2*d^3*a^6 + 3*
x^2*d^5*b*a^5 + 5/2*x^2*e*d^4*a^6 + x*d^5*a^6

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giac [B]  time = 0.20, size = 558, normalized size = 3.90 \begin {gather*} \frac {1}{12} \, b^{6} x^{12} e^{5} + \frac {5}{11} \, b^{6} d x^{11} e^{4} + b^{6} d^{2} x^{10} e^{3} + \frac {10}{9} \, b^{6} d^{3} x^{9} e^{2} + \frac {5}{8} \, b^{6} d^{4} x^{8} e + \frac {1}{7} \, b^{6} d^{5} x^{7} + \frac {6}{11} \, a b^{5} x^{11} e^{5} + 3 \, a b^{5} d x^{10} e^{4} + \frac {20}{3} \, a b^{5} d^{2} x^{9} e^{3} + \frac {15}{2} \, a b^{5} d^{3} x^{8} e^{2} + \frac {30}{7} \, a b^{5} d^{4} x^{7} e + a b^{5} d^{5} x^{6} + \frac {3}{2} \, a^{2} b^{4} x^{10} e^{5} + \frac {25}{3} \, a^{2} b^{4} d x^{9} e^{4} + \frac {75}{4} \, a^{2} b^{4} d^{2} x^{8} e^{3} + \frac {150}{7} \, a^{2} b^{4} d^{3} x^{7} e^{2} + \frac {25}{2} \, a^{2} b^{4} d^{4} x^{6} e + 3 \, a^{2} b^{4} d^{5} x^{5} + \frac {20}{9} \, a^{3} b^{3} x^{9} e^{5} + \frac {25}{2} \, a^{3} b^{3} d x^{8} e^{4} + \frac {200}{7} \, a^{3} b^{3} d^{2} x^{7} e^{3} + \frac {100}{3} \, a^{3} b^{3} d^{3} x^{6} e^{2} + 20 \, a^{3} b^{3} d^{4} x^{5} e + 5 \, a^{3} b^{3} d^{5} x^{4} + \frac {15}{8} \, a^{4} b^{2} x^{8} e^{5} + \frac {75}{7} \, a^{4} b^{2} d x^{7} e^{4} + 25 \, a^{4} b^{2} d^{2} x^{6} e^{3} + 30 \, a^{4} b^{2} d^{3} x^{5} e^{2} + \frac {75}{4} \, a^{4} b^{2} d^{4} x^{4} e + 5 \, a^{4} b^{2} d^{5} x^{3} + \frac {6}{7} \, a^{5} b x^{7} e^{5} + 5 \, a^{5} b d x^{6} e^{4} + 12 \, a^{5} b d^{2} x^{5} e^{3} + 15 \, a^{5} b d^{3} x^{4} e^{2} + 10 \, a^{5} b d^{4} x^{3} e + 3 \, a^{5} b d^{5} x^{2} + \frac {1}{6} \, a^{6} x^{6} e^{5} + a^{6} d x^{5} e^{4} + \frac {5}{2} \, a^{6} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{6} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{6} d^{4} x^{2} e + a^{6} 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)^3,x, algorithm="giac")

[Out]

1/12*b^6*x^12*e^5 + 5/11*b^6*d*x^11*e^4 + b^6*d^2*x^10*e^3 + 10/9*b^6*d^3*x^9*e^2 + 5/8*b^6*d^4*x^8*e + 1/7*b^
6*d^5*x^7 + 6/11*a*b^5*x^11*e^5 + 3*a*b^5*d*x^10*e^4 + 20/3*a*b^5*d^2*x^9*e^3 + 15/2*a*b^5*d^3*x^8*e^2 + 30/7*
a*b^5*d^4*x^7*e + a*b^5*d^5*x^6 + 3/2*a^2*b^4*x^10*e^5 + 25/3*a^2*b^4*d*x^9*e^4 + 75/4*a^2*b^4*d^2*x^8*e^3 + 1
50/7*a^2*b^4*d^3*x^7*e^2 + 25/2*a^2*b^4*d^4*x^6*e + 3*a^2*b^4*d^5*x^5 + 20/9*a^3*b^3*x^9*e^5 + 25/2*a^3*b^3*d*
x^8*e^4 + 200/7*a^3*b^3*d^2*x^7*e^3 + 100/3*a^3*b^3*d^3*x^6*e^2 + 20*a^3*b^3*d^4*x^5*e + 5*a^3*b^3*d^5*x^4 + 1
5/8*a^4*b^2*x^8*e^5 + 75/7*a^4*b^2*d*x^7*e^4 + 25*a^4*b^2*d^2*x^6*e^3 + 30*a^4*b^2*d^3*x^5*e^2 + 75/4*a^4*b^2*
d^4*x^4*e + 5*a^4*b^2*d^5*x^3 + 6/7*a^5*b*x^7*e^5 + 5*a^5*b*d*x^6*e^4 + 12*a^5*b*d^2*x^5*e^3 + 15*a^5*b*d^3*x^
4*e^2 + 10*a^5*b*d^4*x^3*e + 3*a^5*b*d^5*x^2 + 1/6*a^6*x^6*e^5 + a^6*d*x^5*e^4 + 5/2*a^6*d^2*x^4*e^3 + 10/3*a^
6*d^3*x^3*e^2 + 5/2*a^6*d^4*x^2*e + a^6*d^5*x

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maple [B]  time = 0.05, size = 521, normalized size = 3.64 \begin {gather*} \frac {b^{6} e^{5} x^{12}}{12}+a^{6} d^{5} x +\frac {\left (6 e^{5} a \,b^{5}+5 d \,e^{4} b^{6}\right ) x^{11}}{11}+\frac {\left (15 e^{5} a^{2} b^{4}+30 d \,e^{4} a \,b^{5}+10 d^{2} e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (20 e^{5} a^{3} b^{3}+75 d \,e^{4} a^{2} b^{4}+60 d^{2} e^{3} a \,b^{5}+10 d^{3} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (15 e^{5} a^{4} b^{2}+100 d \,e^{4} a^{3} b^{3}+150 d^{2} e^{3} a^{2} b^{4}+60 d^{3} e^{2} a \,b^{5}+5 d^{4} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (6 e^{5} a^{5} b +75 d \,e^{4} a^{4} b^{2}+200 d^{2} e^{3} a^{3} b^{3}+150 d^{3} e^{2} a^{2} b^{4}+30 d^{4} e a \,b^{5}+d^{5} b^{6}\right ) x^{7}}{7}+\frac {\left (e^{5} a^{6}+30 d \,e^{4} a^{5} b +150 d^{2} e^{3} a^{4} b^{2}+200 d^{3} e^{2} a^{3} b^{3}+75 d^{4} e \,a^{2} b^{4}+6 d^{5} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{6}+60 d^{2} e^{3} a^{5} b +150 d^{3} e^{2} a^{4} b^{2}+100 d^{4} e \,a^{3} b^{3}+15 d^{5} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{6}+60 d^{3} e^{2} a^{5} b +75 d^{4} e \,a^{4} b^{2}+20 d^{5} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{6}+30 d^{4} e \,a^{5} b +15 d^{5} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (5 d^{4} e \,a^{6}+6 d^{5} a^{5} b \right ) x^{2}}{2} \end {gather*}

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)^3,x)

[Out]

1/12*b^6*e^5*x^12+1/11*(6*a*b^5*e^5+5*b^6*d*e^4)*x^11+1/10*(15*a^2*b^4*e^5+30*a*b^5*d*e^4+10*b^6*d^2*e^3)*x^10
+1/9*(20*a^3*b^3*e^5+75*a^2*b^4*d*e^4+60*a*b^5*d^2*e^3+10*b^6*d^3*e^2)*x^9+1/8*(15*a^4*b^2*e^5+100*a^3*b^3*d*e
^4+150*a^2*b^4*d^2*e^3+60*a*b^5*d^3*e^2+5*b^6*d^4*e)*x^8+1/7*(6*a^5*b*e^5+75*a^4*b^2*d*e^4+200*a^3*b^3*d^2*e^3
+150*a^2*b^4*d^3*e^2+30*a*b^5*d^4*e+b^6*d^5)*x^7+1/6*(a^6*e^5+30*a^5*b*d*e^4+150*a^4*b^2*d^2*e^3+200*a^3*b^3*d
^3*e^2+75*a^2*b^4*d^4*e+6*a*b^5*d^5)*x^6+1/5*(5*a^6*d*e^4+60*a^5*b*d^2*e^3+150*a^4*b^2*d^3*e^2+100*a^3*b^3*d^4
*e+15*a^2*b^4*d^5)*x^5+1/4*(10*a^6*d^2*e^3+60*a^5*b*d^3*e^2+75*a^4*b^2*d^4*e+20*a^3*b^3*d^5)*x^4+1/3*(10*a^6*d
^3*e^2+30*a^5*b*d^4*e+15*a^4*b^2*d^5)*x^3+1/2*(5*a^6*d^4*e+6*a^5*b*d^5)*x^2+d^5*a^6*x

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maxima [B]  time = 1.48, size = 517, normalized size = 3.62 \begin {gather*} \frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} 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)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 492, normalized size = 3.44 \begin {gather*} x^5\,\left (a^6\,d\,e^4+12\,a^5\,b\,d^2\,e^3+30\,a^4\,b^2\,d^3\,e^2+20\,a^3\,b^3\,d^4\,e+3\,a^2\,b^4\,d^5\right )+x^8\,\left (\frac {15\,a^4\,b^2\,e^5}{8}+\frac {25\,a^3\,b^3\,d\,e^4}{2}+\frac {75\,a^2\,b^4\,d^2\,e^3}{4}+\frac {15\,a\,b^5\,d^3\,e^2}{2}+\frac {5\,b^6\,d^4\,e}{8}\right )+x^6\,\left (\frac {a^6\,e^5}{6}+5\,a^5\,b\,d\,e^4+25\,a^4\,b^2\,d^2\,e^3+\frac {100\,a^3\,b^3\,d^3\,e^2}{3}+\frac {25\,a^2\,b^4\,d^4\,e}{2}+a\,b^5\,d^5\right )+x^7\,\left (\frac {6\,a^5\,b\,e^5}{7}+\frac {75\,a^4\,b^2\,d\,e^4}{7}+\frac {200\,a^3\,b^3\,d^2\,e^3}{7}+\frac {150\,a^2\,b^4\,d^3\,e^2}{7}+\frac {30\,a\,b^5\,d^4\,e}{7}+\frac {b^6\,d^5}{7}\right )+a^6\,d^5\,x+\frac {b^6\,e^5\,x^{12}}{12}+\frac {5\,a^3\,d^2\,x^4\,\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 )}{4}+\frac {5\,b^3\,e^2\,x^9\,\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 )}{9}+\frac {a^5\,d^4\,x^2\,\left (5\,a\,e+6\,b\,d\right )}{2}+\frac {b^5\,e^4\,x^{11}\,\left (6\,a\,e+5\,b\,d\right )}{11}+\frac {5\,a^4\,d^3\,x^3\,\left (2\,a^2\,e^2+6\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b^4\,e^3\,x^{10}\,\left (3\,a^2\,e^2+6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{2} \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)^3,x)

[Out]

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

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sympy [B]  time = 0.16, size = 580, normalized size = 4.06 \begin {gather*} a^{6} d^{5} x + \frac {b^{6} e^{5} x^{12}}{12} + x^{11} \left (\frac {6 a b^{5} e^{5}}{11} + \frac {5 b^{6} d e^{4}}{11}\right ) + x^{10} \left (\frac {3 a^{2} b^{4} e^{5}}{2} + 3 a b^{5} d e^{4} + b^{6} d^{2} e^{3}\right ) + x^{9} \left (\frac {20 a^{3} b^{3} e^{5}}{9} + \frac {25 a^{2} b^{4} d e^{4}}{3} + \frac {20 a b^{5} d^{2} e^{3}}{3} + \frac {10 b^{6} d^{3} e^{2}}{9}\right ) + x^{8} \left (\frac {15 a^{4} b^{2} e^{5}}{8} + \frac {25 a^{3} b^{3} d e^{4}}{2} + \frac {75 a^{2} b^{4} d^{2} e^{3}}{4} + \frac {15 a b^{5} d^{3} e^{2}}{2} + \frac {5 b^{6} d^{4} e}{8}\right ) + x^{7} \left (\frac {6 a^{5} b e^{5}}{7} + \frac {75 a^{4} b^{2} d e^{4}}{7} + \frac {200 a^{3} b^{3} d^{2} e^{3}}{7} + \frac {150 a^{2} b^{4} d^{3} e^{2}}{7} + \frac {30 a b^{5} d^{4} e}{7} + \frac {b^{6} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{6} e^{5}}{6} + 5 a^{5} b d e^{4} + 25 a^{4} b^{2} d^{2} e^{3} + \frac {100 a^{3} b^{3} d^{3} e^{2}}{3} + \frac {25 a^{2} b^{4} d^{4} e}{2} + a b^{5} d^{5}\right ) + x^{5} \left (a^{6} d e^{4} + 12 a^{5} b d^{2} e^{3} + 30 a^{4} b^{2} d^{3} e^{2} + 20 a^{3} b^{3} d^{4} e + 3 a^{2} b^{4} d^{5}\right ) + x^{4} \left (\frac {5 a^{6} d^{2} e^{3}}{2} + 15 a^{5} b d^{3} e^{2} + \frac {75 a^{4} b^{2} d^{4} e}{4} + 5 a^{3} b^{3} d^{5}\right ) + x^{3} \left (\frac {10 a^{6} d^{3} e^{2}}{3} + 10 a^{5} b d^{4} e + 5 a^{4} b^{2} d^{5}\right ) + x^{2} \left (\frac {5 a^{6} d^{4} e}{2} + 3 a^{5} 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)**3,x)

[Out]

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

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