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

Optimal. Leaf size=65 \[ \frac {2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac {(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3} \]

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Rubi [A]  time = 0.15, antiderivative size = 65, 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 {2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac {(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac {e^2 (a+b x)^{10}}{10 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B]  time = 0.07, size = 229, normalized size = 3.52 \begin {gather*} \frac {1}{360} x \left (120 a^7 \left (3 d^2+3 d e x+e^2 x^2\right )+210 a^6 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+252 a^5 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+210 a^4 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+120 a^3 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+45 a^2 b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+10 a b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )+b^7 x^7 \left (45 d^2+80 d e x+36 e^2 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(120*a^7*(3*d^2 + 3*d*e*x + e^2*x^2) + 210*a^6*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 252*a^5*b^2*x^2*(10*d^2
+ 15*d*e*x + 6*e^2*x^2) + 210*a^4*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 120*a^3*b^4*x^4*(21*d^2 + 35*d*e*
x + 15*e^2*x^2) + 45*a^2*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 10*a*b^6*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x
^2) + b^7*x^7*(45*d^2 + 80*d*e*x + 36*e^2*x^2)))/360

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

[Out]

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

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fricas [B]  time = 0.40, size = 294, normalized size = 4.52 \begin {gather*} \frac {1}{10} x^{10} e^{2} b^{7} + \frac {2}{9} x^{9} e d b^{7} + \frac {7}{9} x^{9} e^{2} b^{6} a + \frac {1}{8} x^{8} d^{2} b^{7} + \frac {7}{4} x^{8} e d b^{6} a + \frac {21}{8} x^{8} e^{2} b^{5} a^{2} + x^{7} d^{2} b^{6} a + 6 x^{7} e d b^{5} a^{2} + 5 x^{7} e^{2} b^{4} a^{3} + \frac {7}{2} x^{6} d^{2} b^{5} a^{2} + \frac {35}{3} x^{6} e d b^{4} a^{3} + \frac {35}{6} x^{6} e^{2} b^{3} a^{4} + 7 x^{5} d^{2} b^{4} a^{3} + 14 x^{5} e d b^{3} a^{4} + \frac {21}{5} x^{5} e^{2} b^{2} a^{5} + \frac {35}{4} x^{4} d^{2} b^{3} a^{4} + \frac {21}{2} x^{4} e d b^{2} a^{5} + \frac {7}{4} x^{4} e^{2} b a^{6} + 7 x^{3} d^{2} b^{2} a^{5} + \frac {14}{3} x^{3} e d b a^{6} + \frac {1}{3} x^{3} e^{2} a^{7} + \frac {7}{2} x^{2} d^{2} b a^{6} + x^{2} e d a^{7} + x d^{2} a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^2*b^7 + 2/9*x^9*e*d*b^7 + 7/9*x^9*e^2*b^6*a + 1/8*x^8*d^2*b^7 + 7/4*x^8*e*d*b^6*a + 21/8*x^8*e^2*b
^5*a^2 + x^7*d^2*b^6*a + 6*x^7*e*d*b^5*a^2 + 5*x^7*e^2*b^4*a^3 + 7/2*x^6*d^2*b^5*a^2 + 35/3*x^6*e*d*b^4*a^3 +
35/6*x^6*e^2*b^3*a^4 + 7*x^5*d^2*b^4*a^3 + 14*x^5*e*d*b^3*a^4 + 21/5*x^5*e^2*b^2*a^5 + 35/4*x^4*d^2*b^3*a^4 +
21/2*x^4*e*d*b^2*a^5 + 7/4*x^4*e^2*b*a^6 + 7*x^3*d^2*b^2*a^5 + 14/3*x^3*e*d*b*a^6 + 1/3*x^3*e^2*a^7 + 7/2*x^2*
d^2*b*a^6 + x^2*e*d*a^7 + x*d^2*a^7

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giac [B]  time = 0.19, size = 294, normalized size = 4.52 \begin {gather*} \frac {1}{10} \, b^{7} x^{10} e^{2} + \frac {2}{9} \, b^{7} d x^{9} e + \frac {1}{8} \, b^{7} d^{2} x^{8} + \frac {7}{9} \, a b^{6} x^{9} e^{2} + \frac {7}{4} \, a b^{6} d x^{8} e + a b^{6} d^{2} x^{7} + \frac {21}{8} \, a^{2} b^{5} x^{8} e^{2} + 6 \, a^{2} b^{5} d x^{7} e + \frac {7}{2} \, a^{2} b^{5} d^{2} x^{6} + 5 \, a^{3} b^{4} x^{7} e^{2} + \frac {35}{3} \, a^{3} b^{4} d x^{6} e + 7 \, a^{3} b^{4} d^{2} x^{5} + \frac {35}{6} \, a^{4} b^{3} x^{6} e^{2} + 14 \, a^{4} b^{3} d x^{5} e + \frac {35}{4} \, a^{4} b^{3} d^{2} x^{4} + \frac {21}{5} \, a^{5} b^{2} x^{5} e^{2} + \frac {21}{2} \, a^{5} b^{2} d x^{4} e + 7 \, a^{5} b^{2} d^{2} x^{3} + \frac {7}{4} \, a^{6} b x^{4} e^{2} + \frac {14}{3} \, a^{6} b d x^{3} e + \frac {7}{2} \, a^{6} b d^{2} x^{2} + \frac {1}{3} \, a^{7} x^{3} e^{2} + a^{7} d x^{2} e + a^{7} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^7*x^10*e^2 + 2/9*b^7*d*x^9*e + 1/8*b^7*d^2*x^8 + 7/9*a*b^6*x^9*e^2 + 7/4*a*b^6*d*x^8*e + a*b^6*d^2*x^7
+ 21/8*a^2*b^5*x^8*e^2 + 6*a^2*b^5*d*x^7*e + 7/2*a^2*b^5*d^2*x^6 + 5*a^3*b^4*x^7*e^2 + 35/3*a^3*b^4*d*x^6*e +
7*a^3*b^4*d^2*x^5 + 35/6*a^4*b^3*x^6*e^2 + 14*a^4*b^3*d*x^5*e + 35/4*a^4*b^3*d^2*x^4 + 21/5*a^5*b^2*x^5*e^2 +
21/2*a^5*b^2*d*x^4*e + 7*a^5*b^2*d^2*x^3 + 7/4*a^6*b*x^4*e^2 + 14/3*a^6*b*d*x^3*e + 7/2*a^6*b*d^2*x^2 + 1/3*a^
7*x^3*e^2 + a^7*d*x^2*e + a^7*d^2*x

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maple [B]  time = 0.05, size = 433, normalized size = 6.66 \begin {gather*} \frac {b^{7} e^{2} x^{10}}{10}+a^{7} d^{2} x +\frac {\left (6 a \,b^{6} e^{2}+\left (a \,e^{2}+2 b d e \right ) b^{6}\right ) x^{9}}{9}+\frac {\left (15 a^{2} b^{5} e^{2}+6 \left (a \,e^{2}+2 b d e \right ) a \,b^{5}+\left (2 a d e +b \,d^{2}\right ) b^{6}\right ) x^{8}}{8}+\frac {\left (20 a^{3} b^{4} e^{2}+a \,b^{6} d^{2}+15 \left (a \,e^{2}+2 b d e \right ) a^{2} b^{4}+6 \left (2 a d e +b \,d^{2}\right ) a \,b^{5}\right ) x^{7}}{7}+\frac {\left (15 a^{4} b^{3} e^{2}+6 a^{2} b^{5} d^{2}+20 \left (a \,e^{2}+2 b d e \right ) a^{3} b^{3}+15 \left (2 a d e +b \,d^{2}\right ) a^{2} b^{4}\right ) x^{6}}{6}+\frac {\left (6 a^{5} b^{2} e^{2}+15 a^{3} b^{4} d^{2}+15 \left (a \,e^{2}+2 b d e \right ) a^{4} b^{2}+20 \left (2 a d e +b \,d^{2}\right ) a^{3} b^{3}\right ) x^{5}}{5}+\frac {\left (a^{6} b \,e^{2}+20 a^{4} b^{3} d^{2}+6 \left (a \,e^{2}+2 b d e \right ) a^{5} b +15 \left (2 a d e +b \,d^{2}\right ) a^{4} b^{2}\right ) x^{4}}{4}+\frac {\left (15 a^{5} b^{2} d^{2}+\left (a \,e^{2}+2 b d e \right ) a^{6}+6 \left (2 a d e +b \,d^{2}\right ) a^{5} b \right ) x^{3}}{3}+\frac {\left (6 a^{6} b \,d^{2}+\left (2 a d e +b \,d^{2}\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)^2*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

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maxima [B]  time = 0.56, size = 273, normalized size = 4.20 \begin {gather*} \frac {1}{10} \, b^{7} e^{2} x^{10} + a^{7} d^{2} x + \frac {1}{9} \, {\left (2 \, b^{7} d e + 7 \, a b^{6} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{2} + 14 \, a b^{6} d e + 21 \, a^{2} b^{5} e^{2}\right )} x^{8} + {\left (a b^{6} d^{2} + 6 \, a^{2} b^{5} d e + 5 \, a^{3} b^{4} e^{2}\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d^{2} + 10 \, a^{3} b^{4} d e + 5 \, a^{4} b^{3} e^{2}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{2} + 10 \, a^{4} b^{3} d e + 3 \, a^{5} b^{2} e^{2}\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d^{2} + 6 \, a^{5} b^{2} d e + a^{6} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{2} + 14 \, a^{6} b d e + a^{7} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{2} + 2 \, a^{7} d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^7*e^2*x^10 + a^7*d^2*x + 1/9*(2*b^7*d*e + 7*a*b^6*e^2)*x^9 + 1/8*(b^7*d^2 + 14*a*b^6*d*e + 21*a^2*b^5*e
^2)*x^8 + (a*b^6*d^2 + 6*a^2*b^5*d*e + 5*a^3*b^4*e^2)*x^7 + 7/6*(3*a^2*b^5*d^2 + 10*a^3*b^4*d*e + 5*a^4*b^3*e^
2)*x^6 + 7/5*(5*a^3*b^4*d^2 + 10*a^4*b^3*d*e + 3*a^5*b^2*e^2)*x^5 + 7/4*(5*a^4*b^3*d^2 + 6*a^5*b^2*d*e + a^6*b
*e^2)*x^4 + 1/3*(21*a^5*b^2*d^2 + 14*a^6*b*d*e + a^7*e^2)*x^3 + 1/2*(7*a^6*b*d^2 + 2*a^7*d*e)*x^2

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mupad [B]  time = 2.09, size = 249, normalized size = 3.83 \begin {gather*} x^3\,\left (\frac {a^7\,e^2}{3}+\frac {14\,a^6\,b\,d\,e}{3}+7\,a^5\,b^2\,d^2\right )+x^8\,\left (\frac {21\,a^2\,b^5\,e^2}{8}+\frac {7\,a\,b^6\,d\,e}{4}+\frac {b^7\,d^2}{8}\right )+a^7\,d^2\,x+\frac {b^7\,e^2\,x^{10}}{10}+\frac {a^6\,d\,x^2\,\left (2\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e\,x^9\,\left (7\,a\,e+2\,b\,d\right )}{9}+\frac {7\,a^4\,b\,x^4\,\left (a^2\,e^2+6\,a\,b\,d\,e+5\,b^2\,d^2\right )}{4}+a\,b^4\,x^7\,\left (5\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )+\frac {7\,a^3\,b^2\,x^5\,\left (3\,a^2\,e^2+10\,a\,b\,d\,e+5\,b^2\,d^2\right )}{5}+\frac {7\,a^2\,b^3\,x^6\,\left (5\,a^2\,e^2+10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((a^7*e^2)/3 + 7*a^5*b^2*d^2 + (14*a^6*b*d*e)/3) + x^8*((b^7*d^2)/8 + (21*a^2*b^5*e^2)/8 + (7*a*b^6*d*e)/4
) + a^7*d^2*x + (b^7*e^2*x^10)/10 + (a^6*d*x^2*(2*a*e + 7*b*d))/2 + (b^6*e*x^9*(7*a*e + 2*b*d))/9 + (7*a^4*b*x
^4*(a^2*e^2 + 5*b^2*d^2 + 6*a*b*d*e))/4 + a*b^4*x^7*(5*a^2*e^2 + b^2*d^2 + 6*a*b*d*e) + (7*a^3*b^2*x^5*(3*a^2*
e^2 + 5*b^2*d^2 + 10*a*b*d*e))/5 + (7*a^2*b^3*x^6*(5*a^2*e^2 + 3*b^2*d^2 + 10*a*b*d*e))/6

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sympy [B]  time = 0.12, size = 303, normalized size = 4.66 \begin {gather*} a^{7} d^{2} x + \frac {b^{7} e^{2} x^{10}}{10} + x^{9} \left (\frac {7 a b^{6} e^{2}}{9} + \frac {2 b^{7} d e}{9}\right ) + x^{8} \left (\frac {21 a^{2} b^{5} e^{2}}{8} + \frac {7 a b^{6} d e}{4} + \frac {b^{7} d^{2}}{8}\right ) + x^{7} \left (5 a^{3} b^{4} e^{2} + 6 a^{2} b^{5} d e + a b^{6} d^{2}\right ) + x^{6} \left (\frac {35 a^{4} b^{3} e^{2}}{6} + \frac {35 a^{3} b^{4} d e}{3} + \frac {7 a^{2} b^{5} d^{2}}{2}\right ) + x^{5} \left (\frac {21 a^{5} b^{2} e^{2}}{5} + 14 a^{4} b^{3} d e + 7 a^{3} b^{4} d^{2}\right ) + x^{4} \left (\frac {7 a^{6} b e^{2}}{4} + \frac {21 a^{5} b^{2} d e}{2} + \frac {35 a^{4} b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{7} e^{2}}{3} + \frac {14 a^{6} b d e}{3} + 7 a^{5} b^{2} d^{2}\right ) + x^{2} \left (a^{7} d e + \frac {7 a^{6} b d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**7*d**2*x + b**7*e**2*x**10/10 + x**9*(7*a*b**6*e**2/9 + 2*b**7*d*e/9) + x**8*(21*a**2*b**5*e**2/8 + 7*a*b**
6*d*e/4 + b**7*d**2/8) + x**7*(5*a**3*b**4*e**2 + 6*a**2*b**5*d*e + a*b**6*d**2) + x**6*(35*a**4*b**3*e**2/6 +
35*a**3*b**4*d*e/3 + 7*a**2*b**5*d**2/2) + x**5*(21*a**5*b**2*e**2/5 + 14*a**4*b**3*d*e + 7*a**3*b**4*d**2) +
x**4*(7*a**6*b*e**2/4 + 21*a**5*b**2*d*e/2 + 35*a**4*b**3*d**2/4) + x**3*(a**7*e**2/3 + 14*a**6*b*d*e/3 + 7*a
**5*b**2*d**2) + x**2*(a**7*d*e + 7*a**6*b*d**2/2)

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