3.17.82 \(\int (a+b x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=119 \[ \frac {4 e^3 (a+b x)^9 (b d-a e)}{9 b^5}+\frac {3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac {4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5} \]

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Rubi [A]  time = 0.22, antiderivative size = 119, 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 {4 e^3 (a+b x)^9 (b d-a e)}{9 b^5}+\frac {3 e^2 (a+b x)^8 (b d-a e)^2}{4 b^5}+\frac {4 e (a+b x)^7 (b d-a e)^3}{7 b^5}+\frac {(a+b x)^6 (b d-a e)^4}{6 b^5}+\frac {e^4 (a+b x)^{10}}{10 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B]  time = 0.08, size = 301, normalized size = 2.53 \begin {gather*} \frac {x \left (252 a^5 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+210 a^4 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+120 a^3 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+10 a b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(252*a^5*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 210*a^4*b*x*(15*d^4 + 40*d^3*e*x +
45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 120*a^3*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e
^3*x^3 + 15*e^4*x^4) + 45*a^2*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) +
10*a*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + b^5*x^5*(210*d^4 + 720*d
^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)))/1260

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^4 \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)^4*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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fricas [B]  time = 0.49, size = 396, normalized size = 3.33 \begin {gather*} \frac {1}{10} x^{10} e^{4} b^{5} + \frac {4}{9} x^{9} e^{3} d b^{5} + \frac {5}{9} x^{9} e^{4} b^{4} a + \frac {3}{4} x^{8} e^{2} d^{2} b^{5} + \frac {5}{2} x^{8} e^{3} d b^{4} a + \frac {5}{4} x^{8} e^{4} b^{3} a^{2} + \frac {4}{7} x^{7} e d^{3} b^{5} + \frac {30}{7} x^{7} e^{2} d^{2} b^{4} a + \frac {40}{7} x^{7} e^{3} d b^{3} a^{2} + \frac {10}{7} x^{7} e^{4} b^{2} a^{3} + \frac {1}{6} x^{6} d^{4} b^{5} + \frac {10}{3} x^{6} e d^{3} b^{4} a + 10 x^{6} e^{2} d^{2} b^{3} a^{2} + \frac {20}{3} x^{6} e^{3} d b^{2} a^{3} + \frac {5}{6} x^{6} e^{4} b a^{4} + x^{5} d^{4} b^{4} a + 8 x^{5} e d^{3} b^{3} a^{2} + 12 x^{5} e^{2} d^{2} b^{2} a^{3} + 4 x^{5} e^{3} d b a^{4} + \frac {1}{5} x^{5} e^{4} a^{5} + \frac {5}{2} x^{4} d^{4} b^{3} a^{2} + 10 x^{4} e d^{3} b^{2} a^{3} + \frac {15}{2} x^{4} e^{2} d^{2} b a^{4} + x^{4} e^{3} d a^{5} + \frac {10}{3} x^{3} d^{4} b^{2} a^{3} + \frac {20}{3} x^{3} e d^{3} b a^{4} + 2 x^{3} e^{2} d^{2} a^{5} + \frac {5}{2} x^{2} d^{4} b a^{4} + 2 x^{2} e d^{3} a^{5} + x d^{4} a^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^4*b^5 + 4/9*x^9*e^3*d*b^5 + 5/9*x^9*e^4*b^4*a + 3/4*x^8*e^2*d^2*b^5 + 5/2*x^8*e^3*d*b^4*a + 5/4*x^
8*e^4*b^3*a^2 + 4/7*x^7*e*d^3*b^5 + 30/7*x^7*e^2*d^2*b^4*a + 40/7*x^7*e^3*d*b^3*a^2 + 10/7*x^7*e^4*b^2*a^3 + 1
/6*x^6*d^4*b^5 + 10/3*x^6*e*d^3*b^4*a + 10*x^6*e^2*d^2*b^3*a^2 + 20/3*x^6*e^3*d*b^2*a^3 + 5/6*x^6*e^4*b*a^4 +
x^5*d^4*b^4*a + 8*x^5*e*d^3*b^3*a^2 + 12*x^5*e^2*d^2*b^2*a^3 + 4*x^5*e^3*d*b*a^4 + 1/5*x^5*e^4*a^5 + 5/2*x^4*d
^4*b^3*a^2 + 10*x^4*e*d^3*b^2*a^3 + 15/2*x^4*e^2*d^2*b*a^4 + x^4*e^3*d*a^5 + 10/3*x^3*d^4*b^2*a^3 + 20/3*x^3*e
*d^3*b*a^4 + 2*x^3*e^2*d^2*a^5 + 5/2*x^2*d^4*b*a^4 + 2*x^2*e*d^3*a^5 + x*d^4*a^5

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giac [B]  time = 0.16, size = 384, normalized size = 3.23 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} e^{4} + \frac {4}{9} \, b^{5} d x^{9} e^{3} + \frac {3}{4} \, b^{5} d^{2} x^{8} e^{2} + \frac {4}{7} \, b^{5} d^{3} x^{7} e + \frac {1}{6} \, b^{5} d^{4} x^{6} + \frac {5}{9} \, a b^{4} x^{9} e^{4} + \frac {5}{2} \, a b^{4} d x^{8} e^{3} + \frac {30}{7} \, a b^{4} d^{2} x^{7} e^{2} + \frac {10}{3} \, a b^{4} d^{3} x^{6} e + a b^{4} d^{4} x^{5} + \frac {5}{4} \, a^{2} b^{3} x^{8} e^{4} + \frac {40}{7} \, a^{2} b^{3} d x^{7} e^{3} + 10 \, a^{2} b^{3} d^{2} x^{6} e^{2} + 8 \, a^{2} b^{3} d^{3} x^{5} e + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} + \frac {10}{7} \, a^{3} b^{2} x^{7} e^{4} + \frac {20}{3} \, a^{3} b^{2} d x^{6} e^{3} + 12 \, a^{3} b^{2} d^{2} x^{5} e^{2} + 10 \, a^{3} b^{2} d^{3} x^{4} e + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} + \frac {5}{6} \, a^{4} b x^{6} e^{4} + 4 \, a^{4} b d x^{5} e^{3} + \frac {15}{2} \, a^{4} b d^{2} x^{4} e^{2} + \frac {20}{3} \, a^{4} b d^{3} x^{3} e + \frac {5}{2} \, a^{4} b d^{4} x^{2} + \frac {1}{5} \, a^{5} x^{5} e^{4} + a^{5} d x^{4} e^{3} + 2 \, a^{5} d^{2} x^{3} e^{2} + 2 \, a^{5} d^{3} x^{2} e + a^{5} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*x^10*e^4 + 4/9*b^5*d*x^9*e^3 + 3/4*b^5*d^2*x^8*e^2 + 4/7*b^5*d^3*x^7*e + 1/6*b^5*d^4*x^6 + 5/9*a*b^4*
x^9*e^4 + 5/2*a*b^4*d*x^8*e^3 + 30/7*a*b^4*d^2*x^7*e^2 + 10/3*a*b^4*d^3*x^6*e + a*b^4*d^4*x^5 + 5/4*a^2*b^3*x^
8*e^4 + 40/7*a^2*b^3*d*x^7*e^3 + 10*a^2*b^3*d^2*x^6*e^2 + 8*a^2*b^3*d^3*x^5*e + 5/2*a^2*b^3*d^4*x^4 + 10/7*a^3
*b^2*x^7*e^4 + 20/3*a^3*b^2*d*x^6*e^3 + 12*a^3*b^2*d^2*x^5*e^2 + 10*a^3*b^2*d^3*x^4*e + 10/3*a^3*b^2*d^4*x^3 +
5/6*a^4*b*x^6*e^4 + 4*a^4*b*d*x^5*e^3 + 15/2*a^4*b*d^2*x^4*e^2 + 20/3*a^4*b*d^3*x^3*e + 5/2*a^4*b*d^4*x^2 + 1
/5*a^5*x^5*e^4 + a^5*d*x^4*e^3 + 2*a^5*d^2*x^3*e^2 + 2*a^5*d^3*x^2*e + a^5*d^4*x

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maple [B]  time = 0.04, size = 559, normalized size = 4.70 \begin {gather*} \frac {b^{5} e^{4} x^{10}}{10}+a^{5} d^{4} x +\frac {\left (4 a \,b^{4} e^{4}+\left (a \,e^{4}+4 b d \,e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (6 a^{2} b^{3} e^{4}+4 \left (a \,e^{4}+4 b d \,e^{3}\right ) a \,b^{3}+\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (4 a^{3} b^{2} e^{4}+6 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{2} b^{2}+4 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a \,b^{3}+\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (a^{4} b \,e^{4}+4 \left (a \,e^{4}+4 b d \,e^{3}\right ) a^{3} b +6 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{2} b^{2}+4 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a \,b^{3}+\left (4 a \,d^{3} e +b \,d^{4}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (a \,b^{4} d^{4}+\left (a \,e^{4}+4 b d \,e^{3}\right ) a^{4}+4 \left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{3} b +6 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{2} b^{2}+4 \left (4 a \,d^{3} e +b \,d^{4}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 a^{2} b^{3} d^{4}+\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}\right ) a^{4}+4 \left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{3} b +6 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 a^{3} b^{2} d^{4}+\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e \right ) a^{4}+4 \left (4 a \,d^{3} e +b \,d^{4}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{4}+\left (4 a \,d^{3} e +b \,d^{4}\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)^4*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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maxima [B]  time = 0.64, size = 360, normalized size = 3.03 \begin {gather*} \frac {1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac {1}{9} \, {\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{4} + 4 \, a^{5} d^{3} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^5*e^4*x^10 + a^5*d^4*x + 1/9*(4*b^5*d*e^3 + 5*a*b^4*e^4)*x^9 + 1/4*(3*b^5*d^2*e^2 + 10*a*b^4*d*e^3 + 5*
a^2*b^3*e^4)*x^8 + 2/7*(2*b^5*d^3*e + 15*a*b^4*d^2*e^2 + 20*a^2*b^3*d*e^3 + 5*a^3*b^2*e^4)*x^7 + 1/6*(b^5*d^4
+ 20*a*b^4*d^3*e + 60*a^2*b^3*d^2*e^2 + 40*a^3*b^2*d*e^3 + 5*a^4*b*e^4)*x^6 + 1/5*(5*a*b^4*d^4 + 40*a^2*b^3*d^
3*e + 60*a^3*b^2*d^2*e^2 + 20*a^4*b*d*e^3 + a^5*e^4)*x^5 + 1/2*(5*a^2*b^3*d^4 + 20*a^3*b^2*d^3*e + 15*a^4*b*d^
2*e^2 + 2*a^5*d*e^3)*x^4 + 2/3*(5*a^3*b^2*d^4 + 10*a^4*b*d^3*e + 3*a^5*d^2*e^2)*x^3 + 1/2*(5*a^4*b*d^4 + 4*a^5
*d^3*e)*x^2

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mupad [B]  time = 2.07, size = 340, normalized size = 2.86 \begin {gather*} x^4\,\left (a^5\,d\,e^3+\frac {15\,a^4\,b\,d^2\,e^2}{2}+10\,a^3\,b^2\,d^3\,e+\frac {5\,a^2\,b^3\,d^4}{2}\right )+x^7\,\left (\frac {10\,a^3\,b^2\,e^4}{7}+\frac {40\,a^2\,b^3\,d\,e^3}{7}+\frac {30\,a\,b^4\,d^2\,e^2}{7}+\frac {4\,b^5\,d^3\,e}{7}\right )+x^5\,\left (\frac {a^5\,e^4}{5}+4\,a^4\,b\,d\,e^3+12\,a^3\,b^2\,d^2\,e^2+8\,a^2\,b^3\,d^3\,e+a\,b^4\,d^4\right )+x^6\,\left (\frac {5\,a^4\,b\,e^4}{6}+\frac {20\,a^3\,b^2\,d\,e^3}{3}+10\,a^2\,b^3\,d^2\,e^2+\frac {10\,a\,b^4\,d^3\,e}{3}+\frac {b^5\,d^4}{6}\right )+a^5\,d^4\,x+\frac {b^5\,e^4\,x^{10}}{10}+\frac {a^4\,d^3\,x^2\,\left (4\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e^3\,x^9\,\left (5\,a\,e+4\,b\,d\right )}{9}+\frac {2\,a^3\,d^2\,x^3\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{3}+\frac {b^3\,e^2\,x^8\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*(a^5*d*e^3 + (5*a^2*b^3*d^4)/2 + 10*a^3*b^2*d^3*e + (15*a^4*b*d^2*e^2)/2) + x^7*((4*b^5*d^3*e)/7 + (10*a^3
*b^2*e^4)/7 + (30*a*b^4*d^2*e^2)/7 + (40*a^2*b^3*d*e^3)/7) + x^5*((a^5*e^4)/5 + a*b^4*d^4 + 8*a^2*b^3*d^3*e +
12*a^3*b^2*d^2*e^2 + 4*a^4*b*d*e^3) + x^6*((b^5*d^4)/6 + (5*a^4*b*e^4)/6 + (20*a^3*b^2*d*e^3)/3 + 10*a^2*b^3*d
^2*e^2 + (10*a*b^4*d^3*e)/3) + a^5*d^4*x + (b^5*e^4*x^10)/10 + (a^4*d^3*x^2*(4*a*e + 5*b*d))/2 + (b^4*e^3*x^9*
(5*a*e + 4*b*d))/9 + (2*a^3*d^2*x^3*(3*a^2*e^2 + 5*b^2*d^2 + 10*a*b*d*e))/3 + (b^3*e^2*x^8*(5*a^2*e^2 + 3*b^2*
d^2 + 10*a*b*d*e))/4

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sympy [B]  time = 0.14, size = 401, normalized size = 3.37 \begin {gather*} a^{5} d^{4} x + \frac {b^{5} e^{4} x^{10}}{10} + x^{9} \left (\frac {5 a b^{4} e^{4}}{9} + \frac {4 b^{5} d e^{3}}{9}\right ) + x^{8} \left (\frac {5 a^{2} b^{3} e^{4}}{4} + \frac {5 a b^{4} d e^{3}}{2} + \frac {3 b^{5} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac {10 a^{3} b^{2} e^{4}}{7} + \frac {40 a^{2} b^{3} d e^{3}}{7} + \frac {30 a b^{4} d^{2} e^{2}}{7} + \frac {4 b^{5} d^{3} e}{7}\right ) + x^{6} \left (\frac {5 a^{4} b e^{4}}{6} + \frac {20 a^{3} b^{2} d e^{3}}{3} + 10 a^{2} b^{3} d^{2} e^{2} + \frac {10 a b^{4} d^{3} e}{3} + \frac {b^{5} d^{4}}{6}\right ) + x^{5} \left (\frac {a^{5} e^{4}}{5} + 4 a^{4} b d e^{3} + 12 a^{3} b^{2} d^{2} e^{2} + 8 a^{2} b^{3} d^{3} e + a b^{4} d^{4}\right ) + x^{4} \left (a^{5} d e^{3} + \frac {15 a^{4} b d^{2} e^{2}}{2} + 10 a^{3} b^{2} d^{3} e + \frac {5 a^{2} b^{3} d^{4}}{2}\right ) + x^{3} \left (2 a^{5} d^{2} e^{2} + \frac {20 a^{4} b d^{3} e}{3} + \frac {10 a^{3} b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{5} d^{3} e + \frac {5 a^{4} b d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d**4*x + b**5*e**4*x**10/10 + x**9*(5*a*b**4*e**4/9 + 4*b**5*d*e**3/9) + x**8*(5*a**2*b**3*e**4/4 + 5*a*b
**4*d*e**3/2 + 3*b**5*d**2*e**2/4) + x**7*(10*a**3*b**2*e**4/7 + 40*a**2*b**3*d*e**3/7 + 30*a*b**4*d**2*e**2/7
+ 4*b**5*d**3*e/7) + x**6*(5*a**4*b*e**4/6 + 20*a**3*b**2*d*e**3/3 + 10*a**2*b**3*d**2*e**2 + 10*a*b**4*d**3*
e/3 + b**5*d**4/6) + x**5*(a**5*e**4/5 + 4*a**4*b*d*e**3 + 12*a**3*b**2*d**2*e**2 + 8*a**2*b**3*d**3*e + a*b**
4*d**4) + x**4*(a**5*d*e**3 + 15*a**4*b*d**2*e**2/2 + 10*a**3*b**2*d**3*e + 5*a**2*b**3*d**4/2) + x**3*(2*a**5
*d**2*e**2 + 20*a**4*b*d**3*e/3 + 10*a**3*b**2*d**4/3) + x**2*(2*a**5*d**3*e + 5*a**4*b*d**4/2)

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