\(\int (d+e x)^2 (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1487]

Optimal result

Integrand size = 26, antiderivative size = 65 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^2 (a+b x)^7}{7 b^3}+\frac {e (b d-a e) (a+b x)^8}{4 b^3}+\frac {e^2 (a+b x)^9}{9 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {e (a+b x)^8 (b d-a e)}{4 b^3}+\frac {(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac {e^2 (a+b x)^9}{9 b^3} \]

[In]

Int[(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)^7)/(7*b^3) + (e*(b*d - a*e)*(a + b*x)^8)/(4*b^3) + (e^2*(a + b*x)^9)/(9*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 45

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(199\) vs. \(2(65)=130\).

Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.06 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{252} x \left (84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right ) \]

[In]

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

[Out]

(x*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 126*a^4*b^2*x^2*(10*d^2 +
15*d*e*x + 6*e^2*x^2) + 84*a^3*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(21*d^2 + 35*d*e*x +
15*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + b^6*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2)))/252

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs. \(2(59)=118\).

Time = 2.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.57

method	result	size
norman	\(\frac {e^{2} b^{6} x^{9}}{9}+\left (\frac {3}{4} e^{2} a \,b^{5}+\frac {1}{4} d e \,b^{6}\right ) x^{8}+\left (\frac {15}{7} e^{2} a^{2} b^{4}+\frac {12}{7} d e a \,b^{5}+\frac {1}{7} b^{6} d^{2}\right ) x^{7}+\left (\frac {10}{3} e^{2} a^{3} b^{3}+5 d e \,a^{2} b^{4}+d^{2} a \,b^{5}\right ) x^{6}+\left (3 a^{4} b^{2} e^{2}+8 a^{3} b^{3} d e +3 a^{2} b^{4} d^{2}\right ) x^{5}+\left (\frac {3}{2} e^{2} a^{5} b +\frac {15}{2} d e \,a^{4} b^{2}+5 d^{2} a^{3} b^{3}\right ) x^{4}+\left (\frac {1}{3} e^{2} a^{6}+4 d e \,a^{5} b +5 d^{2} a^{4} b^{2}\right ) x^{3}+\left (d e \,a^{6}+3 d^{2} a^{5} b \right ) x^{2}+d^{2} a^{6} x\)	\(232\)
default	\(\frac {e^{2} b^{6} x^{9}}{9}+\frac {\left (6 e^{2} a \,b^{5}+2 d e \,b^{6}\right ) x^{8}}{8}+\frac {\left (15 e^{2} a^{2} b^{4}+12 d e a \,b^{5}+b^{6} d^{2}\right ) x^{7}}{7}+\frac {\left (20 e^{2} a^{3} b^{3}+30 d e \,a^{2} b^{4}+6 d^{2} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (15 a^{4} b^{2} e^{2}+40 a^{3} b^{3} d e +15 a^{2} b^{4} d^{2}\right ) x^{5}}{5}+\frac {\left (6 e^{2} a^{5} b +30 d e \,a^{4} b^{2}+20 d^{2} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (e^{2} a^{6}+12 d e \,a^{5} b +15 d^{2} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{6}+6 d^{2} a^{5} b \right ) x^{2}}{2}+d^{2} a^{6} x\)	\(239\)
risch	\(\frac {1}{9} e^{2} b^{6} x^{9}+\frac {3}{4} x^{8} e^{2} a \,b^{5}+\frac {1}{4} x^{8} d e \,b^{6}+\frac {15}{7} x^{7} e^{2} a^{2} b^{4}+\frac {12}{7} x^{7} d e a \,b^{5}+\frac {1}{7} x^{7} b^{6} d^{2}+\frac {10}{3} x^{6} e^{2} a^{3} b^{3}+5 x^{6} d e \,a^{2} b^{4}+x^{6} d^{2} a \,b^{5}+3 a^{4} b^{2} e^{2} x^{5}+8 a^{3} b^{3} d e \,x^{5}+3 a^{2} b^{4} d^{2} x^{5}+\frac {3}{2} x^{4} e^{2} a^{5} b +\frac {15}{2} x^{4} d e \,a^{4} b^{2}+5 x^{4} d^{2} a^{3} b^{3}+\frac {1}{3} x^{3} e^{2} a^{6}+4 x^{3} d e \,a^{5} b +5 x^{3} d^{2} a^{4} b^{2}+a^{6} d e \,x^{2}+3 a^{5} b \,d^{2} x^{2}+d^{2} a^{6} x\)	\(254\)
parallelrisch	\(\frac {1}{9} e^{2} b^{6} x^{9}+\frac {3}{4} x^{8} e^{2} a \,b^{5}+\frac {1}{4} x^{8} d e \,b^{6}+\frac {15}{7} x^{7} e^{2} a^{2} b^{4}+\frac {12}{7} x^{7} d e a \,b^{5}+\frac {1}{7} x^{7} b^{6} d^{2}+\frac {10}{3} x^{6} e^{2} a^{3} b^{3}+5 x^{6} d e \,a^{2} b^{4}+x^{6} d^{2} a \,b^{5}+3 a^{4} b^{2} e^{2} x^{5}+8 a^{3} b^{3} d e \,x^{5}+3 a^{2} b^{4} d^{2} x^{5}+\frac {3}{2} x^{4} e^{2} a^{5} b +\frac {15}{2} x^{4} d e \,a^{4} b^{2}+5 x^{4} d^{2} a^{3} b^{3}+\frac {1}{3} x^{3} e^{2} a^{6}+4 x^{3} d e \,a^{5} b +5 x^{3} d^{2} a^{4} b^{2}+a^{6} d e \,x^{2}+3 a^{5} b \,d^{2} x^{2}+d^{2} a^{6} x\)	\(254\)
gosper	\(\frac {x \left (28 e^{2} b^{6} x^{8}+189 x^{7} e^{2} a \,b^{5}+63 x^{7} d e \,b^{6}+540 x^{6} e^{2} a^{2} b^{4}+432 x^{6} d e a \,b^{5}+36 x^{6} b^{6} d^{2}+840 x^{5} e^{2} a^{3} b^{3}+1260 x^{5} d e \,a^{2} b^{4}+252 x^{5} d^{2} a \,b^{5}+756 a^{4} b^{2} e^{2} x^{4}+2016 a^{3} b^{3} d e \,x^{4}+756 a^{2} b^{4} d^{2} x^{4}+378 x^{3} e^{2} a^{5} b +1890 x^{3} d e \,a^{4} b^{2}+1260 x^{3} d^{2} a^{3} b^{3}+84 x^{2} e^{2} a^{6}+1008 x^{2} d e \,a^{5} b +1260 x^{2} d^{2} a^{4} b^{2}+252 a^{6} d e x +756 a^{5} b \,d^{2} x +252 d^{2} a^{6}\right )}{252}\)	\(255\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (59) = 118\).

Time = 0.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.60 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac {1}{4} \, {\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (54) = 108\).

Time = 0.04 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.88 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{2} x + \frac {b^{6} e^{2} x^{9}}{9} + x^{8} \cdot \left (\frac {3 a b^{5} e^{2}}{4} + \frac {b^{6} d e}{4}\right ) + x^{7} \cdot \left (\frac {15 a^{2} b^{4} e^{2}}{7} + \frac {12 a b^{5} d e}{7} + \frac {b^{6} d^{2}}{7}\right ) + x^{6} \cdot \left (\frac {10 a^{3} b^{3} e^{2}}{3} + 5 a^{2} b^{4} d e + a b^{5} d^{2}\right ) + x^{5} \cdot \left (3 a^{4} b^{2} e^{2} + 8 a^{3} b^{3} d e + 3 a^{2} b^{4} d^{2}\right ) + x^{4} \cdot \left (\frac {3 a^{5} b e^{2}}{2} + \frac {15 a^{4} b^{2} d e}{2} + 5 a^{3} b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{6} e^{2}}{3} + 4 a^{5} b d e + 5 a^{4} b^{2} d^{2}\right ) + x^{2} \left (a^{6} d e + 3 a^{5} b d^{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (59) = 118\).

Time = 0.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.60 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac {1}{4} \, {\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (59) = 118\).

Time = 0.26 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.89 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{9} \, b^{6} e^{2} x^{9} + \frac {1}{4} \, b^{6} d e x^{8} + \frac {3}{4} \, a b^{5} e^{2} x^{8} + \frac {1}{7} \, b^{6} d^{2} x^{7} + \frac {12}{7} \, a b^{5} d e x^{7} + \frac {15}{7} \, a^{2} b^{4} e^{2} x^{7} + a b^{5} d^{2} x^{6} + 5 \, a^{2} b^{4} d e x^{6} + \frac {10}{3} \, a^{3} b^{3} e^{2} x^{6} + 3 \, a^{2} b^{4} d^{2} x^{5} + 8 \, a^{3} b^{3} d e x^{5} + 3 \, a^{4} b^{2} e^{2} x^{5} + 5 \, a^{3} b^{3} d^{2} x^{4} + \frac {15}{2} \, a^{4} b^{2} d e x^{4} + \frac {3}{2} \, a^{5} b e^{2} x^{4} + 5 \, a^{4} b^{2} d^{2} x^{3} + 4 \, a^{5} b d e x^{3} + \frac {1}{3} \, a^{6} e^{2} x^{3} + 3 \, a^{5} b d^{2} x^{2} + a^{6} d e x^{2} + a^{6} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.29 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^3\,\left (\frac {a^6\,e^2}{3}+4\,a^5\,b\,d\,e+5\,a^4\,b^2\,d^2\right )+x^7\,\left (\frac {15\,a^2\,b^4\,e^2}{7}+\frac {12\,a\,b^5\,d\,e}{7}+\frac {b^6\,d^2}{7}\right )+a^6\,d^2\,x+\frac {b^6\,e^2\,x^9}{9}+a^5\,d\,x^2\,\left (a\,e+3\,b\,d\right )+\frac {b^5\,e\,x^8\,\left (3\,a\,e+b\,d\right )}{4}+\frac {a^3\,b\,x^4\,\left (3\,a^2\,e^2+15\,a\,b\,d\,e+10\,b^2\,d^2\right )}{2}+\frac {a\,b^3\,x^6\,\left (10\,a^2\,e^2+15\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+a^2\,b^2\,x^5\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+3\,b^2\,d^2\right ) \]

[In]

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

[Out]

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