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

Optimal. Leaf size=92 \[ \frac {3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac {e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac {(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4} \]

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Rubi [A]  time = 0.22, antiderivative size = 92, 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 {3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac {e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac {(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^3*(a + b*x)^8)/(8*b^4) + (e*(b*d - a*e)^2*(a + b*x)^9)/(3*b^4) + (3*e^2*(b*d - a*e)*(a + b*x)^10)
/(10*b^4) + (e^3*(a + b*x)^11)/(11*b^4)

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

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Mathematica [B]  time = 0.05, size = 360, normalized size = 3.91 \begin {gather*} a^7 d^3 x+\frac {1}{2} a^6 d^2 x^2 (3 a e+7 b d)+\frac {1}{3} b^5 e x^9 \left (7 a^2 e^2+7 a b d e+b^2 d^2\right )+a^5 d x^3 \left (a^2 e^2+7 a b d e+7 b^2 d^2\right )+a b^3 x^7 \left (5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+\frac {7}{2} a^2 b^2 x^6 \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 {7}{5} a^3 b x^5 \left (a^3 e^3+9 a^2 b d e^2+15 a b^2 d^2 e+5 b^3 d^3\right )+\frac {1}{8} b^4 x^8 \left (35 a^3 e^3+63 a^2 b d e^2+21 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{4} a^4 x^4 \left (a^3 e^3+21 a^2 b d e^2+63 a b^2 d^2 e+35 b^3 d^3\right )+\frac {1}{10} b^6 e^2 x^{10} (7 a e+3 b d)+\frac {1}{11} b^7 e^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^7*d^3*x + (a^6*d^2*(7*b*d + 3*a*e)*x^2)/2 + a^5*d*(7*b^2*d^2 + 7*a*b*d*e + a^2*e^2)*x^3 + (a^4*(35*b^3*d^3 +
63*a*b^2*d^2*e + 21*a^2*b*d*e^2 + a^3*e^3)*x^4)/4 + (7*a^3*b*(5*b^3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^
3*e^3)*x^5)/5 + (7*a^2*b^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3)*x^6)/2 + a*b^3*(b^3*d^3 + 9*a*b
^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x^7 + (b^4*(b^3*d^3 + 21*a*b^2*d^2*e + 63*a^2*b*d*e^2 + 35*a^3*e^3)*x^8
)/8 + (b^5*e*(b^2*d^2 + 7*a*b*d*e + 7*a^2*e^2)*x^9)/3 + (b^6*e^2*(3*b*d + 7*a*e)*x^10)/10 + (b^7*e^3*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)^3 \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)^3*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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fricas [B]  time = 0.44, size = 420, normalized size = 4.57 \begin {gather*} \frac {1}{11} x^{11} e^{3} b^{7} + \frac {3}{10} x^{10} e^{2} d b^{7} + \frac {7}{10} x^{10} e^{3} b^{6} a + \frac {1}{3} x^{9} e d^{2} b^{7} + \frac {7}{3} x^{9} e^{2} d b^{6} a + \frac {7}{3} x^{9} e^{3} b^{5} a^{2} + \frac {1}{8} x^{8} d^{3} b^{7} + \frac {21}{8} x^{8} e d^{2} b^{6} a + \frac {63}{8} x^{8} e^{2} d b^{5} a^{2} + \frac {35}{8} x^{8} e^{3} b^{4} a^{3} + x^{7} d^{3} b^{6} a + 9 x^{7} e d^{2} b^{5} a^{2} + 15 x^{7} e^{2} d b^{4} a^{3} + 5 x^{7} e^{3} b^{3} a^{4} + \frac {7}{2} x^{6} d^{3} b^{5} a^{2} + \frac {35}{2} x^{6} e d^{2} b^{4} a^{3} + \frac {35}{2} x^{6} e^{2} d b^{3} a^{4} + \frac {7}{2} x^{6} e^{3} b^{2} a^{5} + 7 x^{5} d^{3} b^{4} a^{3} + 21 x^{5} e d^{2} b^{3} a^{4} + \frac {63}{5} x^{5} e^{2} d b^{2} a^{5} + \frac {7}{5} x^{5} e^{3} b a^{6} + \frac {35}{4} x^{4} d^{3} b^{3} a^{4} + \frac {63}{4} x^{4} e d^{2} b^{2} a^{5} + \frac {21}{4} x^{4} e^{2} d b a^{6} + \frac {1}{4} x^{4} e^{3} a^{7} + 7 x^{3} d^{3} b^{2} a^{5} + 7 x^{3} e d^{2} b a^{6} + x^{3} e^{2} d a^{7} + \frac {7}{2} x^{2} d^{3} b a^{6} + \frac {3}{2} x^{2} e d^{2} a^{7} + x d^{3} a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e^3*b^7 + 3/10*x^10*e^2*d*b^7 + 7/10*x^10*e^3*b^6*a + 1/3*x^9*e*d^2*b^7 + 7/3*x^9*e^2*d*b^6*a + 7/3*
x^9*e^3*b^5*a^2 + 1/8*x^8*d^3*b^7 + 21/8*x^8*e*d^2*b^6*a + 63/8*x^8*e^2*d*b^5*a^2 + 35/8*x^8*e^3*b^4*a^3 + x^7
*d^3*b^6*a + 9*x^7*e*d^2*b^5*a^2 + 15*x^7*e^2*d*b^4*a^3 + 5*x^7*e^3*b^3*a^4 + 7/2*x^6*d^3*b^5*a^2 + 35/2*x^6*e
*d^2*b^4*a^3 + 35/2*x^6*e^2*d*b^3*a^4 + 7/2*x^6*e^3*b^2*a^5 + 7*x^5*d^3*b^4*a^3 + 21*x^5*e*d^2*b^3*a^4 + 63/5*
x^5*e^2*d*b^2*a^5 + 7/5*x^5*e^3*b*a^6 + 35/4*x^4*d^3*b^3*a^4 + 63/4*x^4*e*d^2*b^2*a^5 + 21/4*x^4*e^2*d*b*a^6 +
1/4*x^4*e^3*a^7 + 7*x^3*d^3*b^2*a^5 + 7*x^3*e*d^2*b*a^6 + x^3*e^2*d*a^7 + 7/2*x^2*d^3*b*a^6 + 3/2*x^2*e*d^2*a
^7 + x*d^3*a^7

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giac [B]  time = 0.17, size = 412, normalized size = 4.48 \begin {gather*} \frac {1}{11} \, b^{7} x^{11} e^{3} + \frac {3}{10} \, b^{7} d x^{10} e^{2} + \frac {1}{3} \, b^{7} d^{2} x^{9} e + \frac {1}{8} \, b^{7} d^{3} x^{8} + \frac {7}{10} \, a b^{6} x^{10} e^{3} + \frac {7}{3} \, a b^{6} d x^{9} e^{2} + \frac {21}{8} \, a b^{6} d^{2} x^{8} e + a b^{6} d^{3} x^{7} + \frac {7}{3} \, a^{2} b^{5} x^{9} e^{3} + \frac {63}{8} \, a^{2} b^{5} d x^{8} e^{2} + 9 \, a^{2} b^{5} d^{2} x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{3} x^{6} + \frac {35}{8} \, a^{3} b^{4} x^{8} e^{3} + 15 \, a^{3} b^{4} d x^{7} e^{2} + \frac {35}{2} \, a^{3} b^{4} d^{2} x^{6} e + 7 \, a^{3} b^{4} d^{3} x^{5} + 5 \, a^{4} b^{3} x^{7} e^{3} + \frac {35}{2} \, a^{4} b^{3} d x^{6} e^{2} + 21 \, a^{4} b^{3} d^{2} x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{3} x^{4} + \frac {7}{2} \, a^{5} b^{2} x^{6} e^{3} + \frac {63}{5} \, a^{5} b^{2} d x^{5} e^{2} + \frac {63}{4} \, a^{5} b^{2} d^{2} x^{4} e + 7 \, a^{5} b^{2} d^{3} x^{3} + \frac {7}{5} \, a^{6} b x^{5} e^{3} + \frac {21}{4} \, a^{6} b d x^{4} e^{2} + 7 \, a^{6} b d^{2} x^{3} e + \frac {7}{2} \, a^{6} b d^{3} x^{2} + \frac {1}{4} \, a^{7} x^{4} e^{3} + a^{7} d x^{3} e^{2} + \frac {3}{2} \, a^{7} d^{2} x^{2} e + a^{7} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^7*x^11*e^3 + 3/10*b^7*d*x^10*e^2 + 1/3*b^7*d^2*x^9*e + 1/8*b^7*d^3*x^8 + 7/10*a*b^6*x^10*e^3 + 7/3*a*b^
6*d*x^9*e^2 + 21/8*a*b^6*d^2*x^8*e + a*b^6*d^3*x^7 + 7/3*a^2*b^5*x^9*e^3 + 63/8*a^2*b^5*d*x^8*e^2 + 9*a^2*b^5*
d^2*x^7*e + 7/2*a^2*b^5*d^3*x^6 + 35/8*a^3*b^4*x^8*e^3 + 15*a^3*b^4*d*x^7*e^2 + 35/2*a^3*b^4*d^2*x^6*e + 7*a^3
*b^4*d^3*x^5 + 5*a^4*b^3*x^7*e^3 + 35/2*a^4*b^3*d*x^6*e^2 + 21*a^4*b^3*d^2*x^5*e + 35/4*a^4*b^3*d^3*x^4 + 7/2*
a^5*b^2*x^6*e^3 + 63/5*a^5*b^2*d*x^5*e^2 + 63/4*a^5*b^2*d^2*x^4*e + 7*a^5*b^2*d^3*x^3 + 7/5*a^6*b*x^5*e^3 + 21
/4*a^6*b*d*x^4*e^2 + 7*a^6*b*d^2*x^3*e + 7/2*a^6*b*d^3*x^2 + 1/4*a^7*x^4*e^3 + a^7*d*x^3*e^2 + 3/2*a^7*d^2*x^2
*e + a^7*d^3*x

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maple [B]  time = 0.04, size = 616, normalized size = 6.70 \begin {gather*} \frac {b^{7} e^{3} x^{11}}{11}+a^{7} d^{3} x +\frac {\left (6 a \,b^{6} e^{3}+\left (a \,e^{3}+3 b d \,e^{2}\right ) b^{6}\right ) x^{10}}{10}+\frac {\left (15 a^{2} b^{5} e^{3}+6 \left (a \,e^{3}+3 b d \,e^{2}\right ) a \,b^{5}+\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) b^{6}\right ) x^{9}}{9}+\frac {\left (20 a^{3} b^{4} e^{3}+15 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{2} b^{4}+6 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a \,b^{5}+\left (3 a \,d^{2} e +b \,d^{3}\right ) b^{6}\right ) x^{8}}{8}+\frac {\left (15 a^{4} b^{3} e^{3}+a \,b^{6} d^{3}+20 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{3} b^{3}+15 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{2} b^{4}+6 \left (3 a \,d^{2} e +b \,d^{3}\right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (6 a^{5} b^{2} e^{3}+6 a^{2} b^{5} d^{3}+15 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{4} b^{2}+20 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{3} b^{3}+15 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (a^{6} b \,e^{3}+15 a^{3} b^{4} d^{3}+6 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{5} b +15 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{4} b^{2}+20 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (20 a^{4} b^{3} d^{3}+\left (a \,e^{3}+3 b d \,e^{2}\right ) a^{6}+6 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{5} b +15 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (15 a^{5} b^{2} d^{3}+\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{6}+6 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b \,d^{3}+\left (3 a \,d^{2} e +b \,d^{3}\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)^3*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

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maxima [B]  time = 0.58, size = 376, normalized size = 4.09 \begin {gather*} \frac {1}{11} \, b^{7} e^{3} x^{11} + a^{7} d^{3} x + \frac {1}{10} \, {\left (3 \, b^{7} d e^{2} + 7 \, a b^{6} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (b^{7} d^{2} e + 7 \, a b^{6} d e^{2} + 7 \, a^{2} b^{5} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{3} + 21 \, a b^{6} d^{2} e + 63 \, a^{2} b^{5} d e^{2} + 35 \, a^{3} b^{4} e^{3}\right )} x^{8} + {\left (a b^{6} d^{3} + 9 \, a^{2} b^{5} d^{2} e + 15 \, a^{3} b^{4} d e^{2} + 5 \, a^{4} b^{3} e^{3}\right )} x^{7} + \frac {7}{2} \, {\left (a^{2} b^{5} d^{3} + 5 \, a^{3} b^{4} d^{2} e + 5 \, a^{4} b^{3} d e^{2} + a^{5} b^{2} e^{3}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{3} + 15 \, a^{4} b^{3} d^{2} e + 9 \, a^{5} b^{2} d e^{2} + a^{6} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, a^{4} b^{3} d^{3} + 63 \, a^{5} b^{2} d^{2} e + 21 \, a^{6} b d e^{2} + a^{7} e^{3}\right )} x^{4} + {\left (7 \, a^{5} b^{2} d^{3} + 7 \, a^{6} b d^{2} e + a^{7} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{3} + 3 \, a^{7} d^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^7*e^3*x^11 + a^7*d^3*x + 1/10*(3*b^7*d*e^2 + 7*a*b^6*e^3)*x^10 + 1/3*(b^7*d^2*e + 7*a*b^6*d*e^2 + 7*a^2
*b^5*e^3)*x^9 + 1/8*(b^7*d^3 + 21*a*b^6*d^2*e + 63*a^2*b^5*d*e^2 + 35*a^3*b^4*e^3)*x^8 + (a*b^6*d^3 + 9*a^2*b^
5*d^2*e + 15*a^3*b^4*d*e^2 + 5*a^4*b^3*e^3)*x^7 + 7/2*(a^2*b^5*d^3 + 5*a^3*b^4*d^2*e + 5*a^4*b^3*d*e^2 + a^5*b
^2*e^3)*x^6 + 7/5*(5*a^3*b^4*d^3 + 15*a^4*b^3*d^2*e + 9*a^5*b^2*d*e^2 + a^6*b*e^3)*x^5 + 1/4*(35*a^4*b^3*d^3 +
63*a^5*b^2*d^2*e + 21*a^6*b*d*e^2 + a^7*e^3)*x^4 + (7*a^5*b^2*d^3 + 7*a^6*b*d^2*e + a^7*d*e^2)*x^3 + 1/2*(7*a
^6*b*d^3 + 3*a^7*d^2*e)*x^2

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mupad [B]  time = 2.17, size = 356, normalized size = 3.87 \begin {gather*} x^7\,\left (5\,a^4\,b^3\,e^3+15\,a^3\,b^4\,d\,e^2+9\,a^2\,b^5\,d^2\,e+a\,b^6\,d^3\right )+x^5\,\left (\frac {7\,a^6\,b\,e^3}{5}+\frac {63\,a^5\,b^2\,d\,e^2}{5}+21\,a^4\,b^3\,d^2\,e+7\,a^3\,b^4\,d^3\right )+x^4\,\left (\frac {a^7\,e^3}{4}+\frac {21\,a^6\,b\,d\,e^2}{4}+\frac {63\,a^5\,b^2\,d^2\,e}{4}+\frac {35\,a^4\,b^3\,d^3}{4}\right )+x^8\,\left (\frac {35\,a^3\,b^4\,e^3}{8}+\frac {63\,a^2\,b^5\,d\,e^2}{8}+\frac {21\,a\,b^6\,d^2\,e}{8}+\frac {b^7\,d^3}{8}\right )+a^7\,d^3\,x+\frac {b^7\,e^3\,x^{11}}{11}+\frac {7\,a^2\,b^2\,x^6\,\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 {a^6\,d^2\,x^2\,\left (3\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^2\,x^{10}\,\left (7\,a\,e+3\,b\,d\right )}{10}+a^5\,d\,x^3\,\left (a^2\,e^2+7\,a\,b\,d\,e+7\,b^2\,d^2\right )+\frac {b^5\,e\,x^9\,\left (7\,a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^7*(a*b^6*d^3 + 5*a^4*b^3*e^3 + 9*a^2*b^5*d^2*e + 15*a^3*b^4*d*e^2) + x^5*((7*a^6*b*e^3)/5 + 7*a^3*b^4*d^3 +
21*a^4*b^3*d^2*e + (63*a^5*b^2*d*e^2)/5) + x^4*((a^7*e^3)/4 + (35*a^4*b^3*d^3)/4 + (63*a^5*b^2*d^2*e)/4 + (21*
a^6*b*d*e^2)/4) + x^8*((b^7*d^3)/8 + (35*a^3*b^4*e^3)/8 + (63*a^2*b^5*d*e^2)/8 + (21*a*b^6*d^2*e)/8) + a^7*d^3
*x + (b^7*e^3*x^11)/11 + (7*a^2*b^2*x^6*(a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2))/2 + (a^6*d^2*x^2*
(3*a*e + 7*b*d))/2 + (b^6*e^2*x^10*(7*a*e + 3*b*d))/10 + a^5*d*x^3*(a^2*e^2 + 7*b^2*d^2 + 7*a*b*d*e) + (b^5*e*
x^9*(7*a^2*e^2 + b^2*d^2 + 7*a*b*d*e))/3

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sympy [B]  time = 0.14, size = 427, normalized size = 4.64 \begin {gather*} a^{7} d^{3} x + \frac {b^{7} e^{3} x^{11}}{11} + x^{10} \left (\frac {7 a b^{6} e^{3}}{10} + \frac {3 b^{7} d e^{2}}{10}\right ) + x^{9} \left (\frac {7 a^{2} b^{5} e^{3}}{3} + \frac {7 a b^{6} d e^{2}}{3} + \frac {b^{7} d^{2} e}{3}\right ) + x^{8} \left (\frac {35 a^{3} b^{4} e^{3}}{8} + \frac {63 a^{2} b^{5} d e^{2}}{8} + \frac {21 a b^{6} d^{2} e}{8} + \frac {b^{7} d^{3}}{8}\right ) + x^{7} \left (5 a^{4} b^{3} e^{3} + 15 a^{3} b^{4} d e^{2} + 9 a^{2} b^{5} d^{2} e + a b^{6} d^{3}\right ) + x^{6} \left (\frac {7 a^{5} b^{2} e^{3}}{2} + \frac {35 a^{4} b^{3} d e^{2}}{2} + \frac {35 a^{3} b^{4} d^{2} e}{2} + \frac {7 a^{2} b^{5} d^{3}}{2}\right ) + x^{5} \left (\frac {7 a^{6} b e^{3}}{5} + \frac {63 a^{5} b^{2} d e^{2}}{5} + 21 a^{4} b^{3} d^{2} e + 7 a^{3} b^{4} d^{3}\right ) + x^{4} \left (\frac {a^{7} e^{3}}{4} + \frac {21 a^{6} b d e^{2}}{4} + \frac {63 a^{5} b^{2} d^{2} e}{4} + \frac {35 a^{4} b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{7} d e^{2} + 7 a^{6} b d^{2} e + 7 a^{5} b^{2} d^{3}\right ) + x^{2} \left (\frac {3 a^{7} d^{2} e}{2} + \frac {7 a^{6} b d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**7*d**3*x + b**7*e**3*x**11/11 + x**10*(7*a*b**6*e**3/10 + 3*b**7*d*e**2/10) + x**9*(7*a**2*b**5*e**3/3 + 7*
a*b**6*d*e**2/3 + b**7*d**2*e/3) + x**8*(35*a**3*b**4*e**3/8 + 63*a**2*b**5*d*e**2/8 + 21*a*b**6*d**2*e/8 + b*
*7*d**3/8) + x**7*(5*a**4*b**3*e**3 + 15*a**3*b**4*d*e**2 + 9*a**2*b**5*d**2*e + a*b**6*d**3) + x**6*(7*a**5*b
**2*e**3/2 + 35*a**4*b**3*d*e**2/2 + 35*a**3*b**4*d**2*e/2 + 7*a**2*b**5*d**3/2) + x**5*(7*a**6*b*e**3/5 + 63*
a**5*b**2*d*e**2/5 + 21*a**4*b**3*d**2*e + 7*a**3*b**4*d**3) + x**4*(a**7*e**3/4 + 21*a**6*b*d*e**2/4 + 63*a**
5*b**2*d**2*e/4 + 35*a**4*b**3*d**3/4) + x**3*(a**7*d*e**2 + 7*a**6*b*d**2*e + 7*a**5*b**2*d**3) + x**2*(3*a**
7*d**2*e/2 + 7*a**6*b*d**3/2)

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