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

Optimal. Leaf size=146 \[ \frac {e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac {5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac {5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6} \]

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Rubi [A]  time = 0.25, antiderivative size = 146, 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 {e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac {5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac {5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac {e^5 (a+b x)^{11}}{11 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)^2,x]

[Out]

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

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Mathematica [B]  time = 0.05, size = 413, normalized size = 2.83 \begin {gather*} a^5 d^5 x+\frac {5}{2} a^4 d^4 x^2 (a e+b d)+\frac {5}{9} b^3 e^3 x^9 \left (2 a^2 e^2+5 a b d e+2 b^2 d^2\right )+\frac {5}{3} a^3 d^3 x^3 \left (2 a^2 e^2+5 a b d e+2 b^2 d^2\right )+\frac {5}{4} b^2 e^2 x^8 \left (a^3 e^3+5 a^2 b d e^2+5 a b^2 d^2 e+b^3 d^3\right )+\frac {5}{2} a^2 d^2 x^4 \left (a^3 e^3+5 a^2 b d e^2+5 a b^2 d^2 e+b^3 d^3\right )+\frac {5}{7} b e x^7 \left (a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right )+a d x^5 \left (a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right )+\frac {1}{6} x^6 \left (a^5 e^5+25 a^4 b d e^4+100 a^3 b^2 d^2 e^3+100 a^2 b^3 d^3 e^2+25 a b^4 d^4 e+b^5 d^5\right )+\frac {1}{2} b^4 e^4 x^{10} (a e+b d)+\frac {1}{11} b^5 e^5 x^{11} \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)^2,x]

[Out]

a^5*d^5*x + (5*a^4*d^4*(b*d + a*e)*x^2)/2 + (5*a^3*d^3*(2*b^2*d^2 + 5*a*b*d*e + 2*a^2*e^2)*x^3)/3 + (5*a^2*d^2
*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3)*x^4)/2 + a*d*(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^5 + ((b^5*d^5 + 25*a*b^4*d^4*e + 100*a^2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 +
25*a^4*b*d*e^4 + a^5*e^5)*x^6)/6 + (5*b*e*(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^7)/7 + (5*b^2*e^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3)*x^8)/4 + (5*b^3*e^3*(2*b^2*d^2
+ 5*a*b*d*e + 2*a^2*e^2)*x^9)/9 + (b^4*e^4*(b*d + a*e)*x^10)/2 + (b^5*e^5*x^11)/11

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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 )^2 \, 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)^2,x]

[Out]

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

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fricas [B]  time = 0.37, size = 488, normalized size = 3.34 \begin {gather*} \frac {1}{11} x^{11} e^{5} b^{5} + \frac {1}{2} x^{10} e^{4} d b^{5} + \frac {1}{2} x^{10} e^{5} b^{4} a + \frac {10}{9} x^{9} e^{3} d^{2} b^{5} + \frac {25}{9} x^{9} e^{4} d b^{4} a + \frac {10}{9} x^{9} e^{5} b^{3} a^{2} + \frac {5}{4} x^{8} e^{2} d^{3} b^{5} + \frac {25}{4} x^{8} e^{3} d^{2} b^{4} a + \frac {25}{4} x^{8} e^{4} d b^{3} a^{2} + \frac {5}{4} x^{8} e^{5} b^{2} a^{3} + \frac {5}{7} x^{7} e d^{4} b^{5} + \frac {50}{7} x^{7} e^{2} d^{3} b^{4} a + \frac {100}{7} x^{7} e^{3} d^{2} b^{3} a^{2} + \frac {50}{7} x^{7} e^{4} d b^{2} a^{3} + \frac {5}{7} x^{7} e^{5} b a^{4} + \frac {1}{6} x^{6} d^{5} b^{5} + \frac {25}{6} x^{6} e d^{4} b^{4} a + \frac {50}{3} x^{6} e^{2} d^{3} b^{3} a^{2} + \frac {50}{3} x^{6} e^{3} d^{2} b^{2} a^{3} + \frac {25}{6} x^{6} e^{4} d b a^{4} + \frac {1}{6} x^{6} e^{5} a^{5} + x^{5} d^{5} b^{4} a + 10 x^{5} e d^{4} b^{3} a^{2} + 20 x^{5} e^{2} d^{3} b^{2} a^{3} + 10 x^{5} e^{3} d^{2} b a^{4} + x^{5} e^{4} d a^{5} + \frac {5}{2} x^{4} d^{5} b^{3} a^{2} + \frac {25}{2} x^{4} e d^{4} b^{2} a^{3} + \frac {25}{2} x^{4} e^{2} d^{3} b a^{4} + \frac {5}{2} x^{4} e^{3} d^{2} a^{5} + \frac {10}{3} x^{3} d^{5} b^{2} a^{3} + \frac {25}{3} x^{3} e d^{4} b a^{4} + \frac {10}{3} x^{3} e^{2} d^{3} a^{5} + \frac {5}{2} x^{2} d^{5} b a^{4} + \frac {5}{2} x^{2} e d^{4} a^{5} + x d^{5} a^{5} \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)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.18, size = 470, normalized size = 3.22 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} e^{5} + \frac {1}{2} \, b^{5} d x^{10} e^{4} + \frac {10}{9} \, b^{5} d^{2} x^{9} e^{3} + \frac {5}{4} \, b^{5} d^{3} x^{8} e^{2} + \frac {5}{7} \, b^{5} d^{4} x^{7} e + \frac {1}{6} \, b^{5} d^{5} x^{6} + \frac {1}{2} \, a b^{4} x^{10} e^{5} + \frac {25}{9} \, a b^{4} d x^{9} e^{4} + \frac {25}{4} \, a b^{4} d^{2} x^{8} e^{3} + \frac {50}{7} \, a b^{4} d^{3} x^{7} e^{2} + \frac {25}{6} \, a b^{4} d^{4} x^{6} e + a b^{4} d^{5} x^{5} + \frac {10}{9} \, a^{2} b^{3} x^{9} e^{5} + \frac {25}{4} \, a^{2} b^{3} d x^{8} e^{4} + \frac {100}{7} \, a^{2} b^{3} d^{2} x^{7} e^{3} + \frac {50}{3} \, a^{2} b^{3} d^{3} x^{6} e^{2} + 10 \, a^{2} b^{3} d^{4} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{5} x^{4} + \frac {5}{4} \, a^{3} b^{2} x^{8} e^{5} + \frac {50}{7} \, a^{3} b^{2} d x^{7} e^{4} + \frac {50}{3} \, a^{3} b^{2} d^{2} x^{6} e^{3} + 20 \, a^{3} b^{2} d^{3} x^{5} e^{2} + \frac {25}{2} \, a^{3} b^{2} d^{4} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{5} x^{3} + \frac {5}{7} \, a^{4} b x^{7} e^{5} + \frac {25}{6} \, a^{4} b d x^{6} e^{4} + 10 \, a^{4} b d^{2} x^{5} e^{3} + \frac {25}{2} \, a^{4} b d^{3} x^{4} e^{2} + \frac {25}{3} \, a^{4} b d^{4} x^{3} e + \frac {5}{2} \, a^{4} b d^{5} x^{2} + \frac {1}{6} \, a^{5} x^{6} e^{5} + a^{5} d x^{5} e^{4} + \frac {5}{2} \, a^{5} d^{2} x^{4} e^{3} + \frac {10}{3} \, a^{5} d^{3} x^{3} e^{2} + \frac {5}{2} \, a^{5} d^{4} x^{2} e + a^{5} 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)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.04, size = 688, normalized size = 4.71 \begin {gather*} \frac {b^{5} e^{5} x^{11}}{11}+a^{5} d^{5} x +\frac {\left (4 a \,b^{4} e^{5}+\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{4}\right ) x^{10}}{10}+\frac {\left (6 a^{2} b^{3} e^{5}+4 \left (a \,e^{5}+5 b d \,e^{4}\right ) a \,b^{3}+\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (4 a^{3} b^{2} e^{5}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2} b^{2}+4 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a \,b^{3}+\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (a^{4} b \,e^{5}+4 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{3} b +6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2} b^{2}+4 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a \,b^{3}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{4}+4 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{3} b +6 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{2} b^{2}+4 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a \,b^{3}+\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (a \,b^{4} d^{5}+\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{4}+4 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{3} b +6 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{2} b^{2}+4 \left (5 a \,d^{4} e +b \,d^{5}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 a^{2} b^{3} d^{5}+\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{4}+4 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{3} b +6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 a^{3} b^{2} d^{5}+\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{4}+4 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{4}\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)^2,x)

[Out]

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

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maxima [B]  time = 0.53, size = 427, normalized size = 2.92 \begin {gather*} \frac {1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac {1}{2} \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac {5}{4} \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac {5}{7} \, {\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} + {\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b d^{5} + a^{5} 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)^2,x, algorithm="maxima")

[Out]

1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2*b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*
b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a*b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a
*b^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6*(b^5*d^5 + 25*a*b^4*d^4*e + 100*a^
2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b
^2*d^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2*d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^
2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b*d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*d^4*e)*x^2

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mupad [B]  time = 2.08, size = 405, normalized size = 2.77 \begin {gather*} x^6\,\left (\frac {a^5\,e^5}{6}+\frac {25\,a^4\,b\,d\,e^4}{6}+\frac {50\,a^3\,b^2\,d^2\,e^3}{3}+\frac {50\,a^2\,b^3\,d^3\,e^2}{3}+\frac {25\,a\,b^4\,d^4\,e}{6}+\frac {b^5\,d^5}{6}\right )+x^5\,\left (a^5\,d\,e^4+10\,a^4\,b\,d^2\,e^3+20\,a^3\,b^2\,d^3\,e^2+10\,a^2\,b^3\,d^4\,e+a\,b^4\,d^5\right )+x^7\,\left (\frac {5\,a^4\,b\,e^5}{7}+\frac {50\,a^3\,b^2\,d\,e^4}{7}+\frac {100\,a^2\,b^3\,d^2\,e^3}{7}+\frac {50\,a\,b^4\,d^3\,e^2}{7}+\frac {5\,b^5\,d^4\,e}{7}\right )+a^5\,d^5\,x+\frac {b^5\,e^5\,x^{11}}{11}+\frac {5\,a^2\,d^2\,x^4\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{2}+\frac {5\,b^2\,e^2\,x^8\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{4}+\frac {5\,a^4\,d^4\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {b^4\,e^4\,x^{10}\,\left (a\,e+b\,d\right )}{2}+\frac {5\,a^3\,d^3\,x^3\,\left (2\,a^2\,e^2+5\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3}+\frac {5\,b^3\,e^3\,x^9\,\left (2\,a^2\,e^2+5\,a\,b\,d\,e+2\,b^2\,d^2\right )}{9} \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)^2,x)

[Out]

x^6*((a^5*e^5)/6 + (b^5*d^5)/6 + (50*a^2*b^3*d^3*e^2)/3 + (50*a^3*b^2*d^2*e^3)/3 + (25*a*b^4*d^4*e)/6 + (25*a^
4*b*d*e^4)/6) + x^5*(a*b^4*d^5 + a^5*d*e^4 + 10*a^2*b^3*d^4*e + 10*a^4*b*d^2*e^3 + 20*a^3*b^2*d^3*e^2) + x^7*(
(5*a^4*b*e^5)/7 + (5*b^5*d^4*e)/7 + (50*a*b^4*d^3*e^2)/7 + (50*a^3*b^2*d*e^4)/7 + (100*a^2*b^3*d^2*e^3)/7) + a
^5*d^5*x + (b^5*e^5*x^11)/11 + (5*a^2*d^2*x^4*(a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2))/2 + (5*b^2*
e^2*x^8*(a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2))/4 + (5*a^4*d^4*x^2*(a*e + b*d))/2 + (b^4*e^4*x^10
*(a*e + b*d))/2 + (5*a^3*d^3*x^3*(2*a^2*e^2 + 2*b^2*d^2 + 5*a*b*d*e))/3 + (5*b^3*e^3*x^9*(2*a^2*e^2 + 2*b^2*d^
2 + 5*a*b*d*e))/9

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sympy [B]  time = 0.15, size = 500, normalized size = 3.42 \begin {gather*} a^{5} d^{5} x + \frac {b^{5} e^{5} x^{11}}{11} + x^{10} \left (\frac {a b^{4} e^{5}}{2} + \frac {b^{5} d e^{4}}{2}\right ) + x^{9} \left (\frac {10 a^{2} b^{3} e^{5}}{9} + \frac {25 a b^{4} d e^{4}}{9} + \frac {10 b^{5} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac {5 a^{3} b^{2} e^{5}}{4} + \frac {25 a^{2} b^{3} d e^{4}}{4} + \frac {25 a b^{4} d^{2} e^{3}}{4} + \frac {5 b^{5} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac {5 a^{4} b e^{5}}{7} + \frac {50 a^{3} b^{2} d e^{4}}{7} + \frac {100 a^{2} b^{3} d^{2} e^{3}}{7} + \frac {50 a b^{4} d^{3} e^{2}}{7} + \frac {5 b^{5} d^{4} e}{7}\right ) + x^{6} \left (\frac {a^{5} e^{5}}{6} + \frac {25 a^{4} b d e^{4}}{6} + \frac {50 a^{3} b^{2} d^{2} e^{3}}{3} + \frac {50 a^{2} b^{3} d^{3} e^{2}}{3} + \frac {25 a b^{4} d^{4} e}{6} + \frac {b^{5} d^{5}}{6}\right ) + x^{5} \left (a^{5} d e^{4} + 10 a^{4} b d^{2} e^{3} + 20 a^{3} b^{2} d^{3} e^{2} + 10 a^{2} b^{3} d^{4} e + a b^{4} d^{5}\right ) + x^{4} \left (\frac {5 a^{5} d^{2} e^{3}}{2} + \frac {25 a^{4} b d^{3} e^{2}}{2} + \frac {25 a^{3} b^{2} d^{4} e}{2} + \frac {5 a^{2} b^{3} d^{5}}{2}\right ) + x^{3} \left (\frac {10 a^{5} d^{3} e^{2}}{3} + \frac {25 a^{4} b d^{4} e}{3} + \frac {10 a^{3} b^{2} d^{5}}{3}\right ) + x^{2} \left (\frac {5 a^{5} d^{4} e}{2} + \frac {5 a^{4} 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)**2,x)

[Out]

a**5*d**5*x + b**5*e**5*x**11/11 + x**10*(a*b**4*e**5/2 + b**5*d*e**4/2) + x**9*(10*a**2*b**3*e**5/9 + 25*a*b*
*4*d*e**4/9 + 10*b**5*d**2*e**3/9) + x**8*(5*a**3*b**2*e**5/4 + 25*a**2*b**3*d*e**4/4 + 25*a*b**4*d**2*e**3/4
+ 5*b**5*d**3*e**2/4) + x**7*(5*a**4*b*e**5/7 + 50*a**3*b**2*d*e**4/7 + 100*a**2*b**3*d**2*e**3/7 + 50*a*b**4*
d**3*e**2/7 + 5*b**5*d**4*e/7) + x**6*(a**5*e**5/6 + 25*a**4*b*d*e**4/6 + 50*a**3*b**2*d**2*e**3/3 + 50*a**2*b
**3*d**3*e**2/3 + 25*a*b**4*d**4*e/6 + b**5*d**5/6) + x**5*(a**5*d*e**4 + 10*a**4*b*d**2*e**3 + 20*a**3*b**2*d
**3*e**2 + 10*a**2*b**3*d**4*e + a*b**4*d**5) + x**4*(5*a**5*d**2*e**3/2 + 25*a**4*b*d**3*e**2/2 + 25*a**3*b**
2*d**4*e/2 + 5*a**2*b**3*d**5/2) + x**3*(10*a**5*d**3*e**2/3 + 25*a**4*b*d**4*e/3 + 10*a**3*b**2*d**5/3) + x**
2*(5*a**5*d**4*e/2 + 5*a**4*b*d**5/2)

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