\(\int (a+b x) (d+e x) (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1921]

Optimal result

Integrand size = 29, antiderivative size = 38 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e) (a+b x)^8}{8 b^2}+\frac {e (a+b x)^9}{9 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 45} \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(a+b x)^8 (b d-a e)}{8 b^2}+\frac {e (a+b x)^9}{9 b^2} \]

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(38)=76\).

Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.97 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^7 d x+\frac {1}{2} a^6 (7 b d+a e) x^2+\frac {7}{3} a^5 b (3 b d+a e) x^3+\frac {7}{4} a^4 b^2 (5 b d+3 a e) x^4+7 a^3 b^3 (b d+a e) x^5+\frac {7}{6} a^2 b^4 (3 b d+5 a e) x^6+a b^5 (b d+3 a e) x^7+\frac {1}{8} b^6 (b d+7 a e) x^8+\frac {1}{9} b^7 e x^9 \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(162\) vs. \(2(34)=68\).

Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.29

method	result	size
norman	\(\frac {b^{7} e \,x^{9}}{9}+\left (\frac {7}{8} e a \,b^{6}+\frac {1}{8} d \,b^{7}\right ) x^{8}+\left (3 e \,a^{2} b^{5}+d a \,b^{6}\right ) x^{7}+\left (\frac {35}{6} e \,a^{3} b^{4}+\frac {7}{2} d \,a^{2} b^{5}\right ) x^{6}+\left (7 e \,a^{4} b^{3}+7 d \,a^{3} b^{4}\right ) x^{5}+\left (\frac {21}{4} e \,a^{5} b^{2}+\frac {35}{4} d \,a^{4} b^{3}\right ) x^{4}+\left (\frac {7}{3} e \,a^{6} b +7 d \,a^{5} b^{2}\right ) x^{3}+\left (\frac {1}{2} e \,a^{7}+\frac {7}{2} d \,a^{6} b \right ) x^{2}+d \,a^{7} x\)	\(163\)
gosper	\(\frac {x \left (8 b^{7} e \,x^{8}+63 x^{7} e a \,b^{6}+9 x^{7} d \,b^{7}+216 a^{2} b^{5} e \,x^{6}+72 a \,b^{6} d \,x^{6}+420 x^{5} e \,a^{3} b^{4}+252 x^{5} d \,a^{2} b^{5}+504 a^{4} b^{3} e \,x^{4}+504 a^{3} b^{4} d \,x^{4}+378 x^{3} e \,a^{5} b^{2}+630 x^{3} d \,a^{4} b^{3}+168 x^{2} e \,a^{6} b +504 x^{2} d \,a^{5} b^{2}+36 x e \,a^{7}+252 x d \,a^{6} b +72 d \,a^{7}\right )}{72}\)	\(170\)
risch	\(\frac {1}{9} b^{7} e \,x^{9}+\frac {7}{8} x^{8} e a \,b^{6}+\frac {1}{8} x^{8} d \,b^{7}+3 a^{2} b^{5} e \,x^{7}+a \,b^{6} d \,x^{7}+\frac {35}{6} x^{6} e \,a^{3} b^{4}+\frac {7}{2} x^{6} d \,a^{2} b^{5}+7 a^{4} b^{3} e \,x^{5}+7 a^{3} b^{4} d \,x^{5}+\frac {21}{4} x^{4} e \,a^{5} b^{2}+\frac {35}{4} x^{4} d \,a^{4} b^{3}+\frac {7}{3} x^{3} e \,a^{6} b +7 x^{3} d \,a^{5} b^{2}+\frac {1}{2} x^{2} e \,a^{7}+\frac {7}{2} x^{2} d \,a^{6} b +d \,a^{7} x\)	\(170\)
parallelrisch	\(\frac {1}{9} b^{7} e \,x^{9}+\frac {7}{8} x^{8} e a \,b^{6}+\frac {1}{8} x^{8} d \,b^{7}+3 a^{2} b^{5} e \,x^{7}+a \,b^{6} d \,x^{7}+\frac {35}{6} x^{6} e \,a^{3} b^{4}+\frac {7}{2} x^{6} d \,a^{2} b^{5}+7 a^{4} b^{3} e \,x^{5}+7 a^{3} b^{4} d \,x^{5}+\frac {21}{4} x^{4} e \,a^{5} b^{2}+\frac {35}{4} x^{4} d \,a^{4} b^{3}+\frac {7}{3} x^{3} e \,a^{6} b +7 x^{3} d \,a^{5} b^{2}+\frac {1}{2} x^{2} e \,a^{7}+\frac {7}{2} x^{2} d \,a^{6} b +d \,a^{7} x\)	\(170\)
default	\(\frac {b^{7} e \,x^{9}}{9}+\frac {\left (\left (a e +b d \right ) b^{6}+6 e a \,b^{6}\right ) x^{8}}{8}+\frac {\left (d a \,b^{6}+6 \left (a e +b d \right ) b^{5} a +15 e \,a^{2} b^{5}\right ) x^{7}}{7}+\frac {\left (6 d \,a^{2} b^{5}+15 \left (a e +b d \right ) b^{4} a^{2}+20 e \,a^{3} b^{4}\right ) x^{6}}{6}+\frac {\left (15 d \,a^{3} b^{4}+20 \left (a e +b d \right ) a^{3} b^{3}+15 e \,a^{4} b^{3}\right ) x^{5}}{5}+\frac {\left (20 d \,a^{4} b^{3}+15 \left (a e +b d \right ) a^{4} b^{2}+6 e \,a^{5} b^{2}\right ) x^{4}}{4}+\frac {\left (15 d \,a^{5} b^{2}+6 \left (a e +b d \right ) a^{5} b +e \,a^{6} b \right ) x^{3}}{3}+\frac {\left (6 d \,a^{6} b +\left (a e +b d \right ) a^{6}\right ) x^{2}}{2}+d \,a^{7} x\)	\(250\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (34) = 68\).

Time = 0.35 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.29 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{9} \, b^{7} e x^{9} + a^{7} d x + \frac {1}{8} \, {\left (b^{7} d + 7 \, a b^{6} e\right )} x^{8} + {\left (a b^{6} d + 3 \, a^{2} b^{5} e\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d + 5 \, a^{3} b^{4} e\right )} x^{6} + 7 \, {\left (a^{3} b^{4} d + a^{4} b^{3} e\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d + 3 \, a^{5} b^{2} e\right )} x^{4} + \frac {7}{3} \, {\left (3 \, a^{5} b^{2} d + a^{6} b e\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d + a^{7} e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (32) = 64\).

Time = 0.04 (sec) , antiderivative size = 178, normalized size of antiderivative = 4.68 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{7} d x + \frac {b^{7} e x^{9}}{9} + x^{8} \cdot \left (\frac {7 a b^{6} e}{8} + \frac {b^{7} d}{8}\right ) + x^{7} \cdot \left (3 a^{2} b^{5} e + a b^{6} d\right ) + x^{6} \cdot \left (\frac {35 a^{3} b^{4} e}{6} + \frac {7 a^{2} b^{5} d}{2}\right ) + x^{5} \cdot \left (7 a^{4} b^{3} e + 7 a^{3} b^{4} d\right ) + x^{4} \cdot \left (\frac {21 a^{5} b^{2} e}{4} + \frac {35 a^{4} b^{3} d}{4}\right ) + x^{3} \cdot \left (\frac {7 a^{6} b e}{3} + 7 a^{5} b^{2} d\right ) + x^{2} \left (\frac {a^{7} e}{2} + \frac {7 a^{6} b d}{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (34) = 68\).

Time = 0.18 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.29 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{9} \, b^{7} e x^{9} + a^{7} d x + \frac {1}{8} \, {\left (b^{7} d + 7 \, a b^{6} e\right )} x^{8} + {\left (a b^{6} d + 3 \, a^{2} b^{5} e\right )} x^{7} + \frac {7}{6} \, {\left (3 \, a^{2} b^{5} d + 5 \, a^{3} b^{4} e\right )} x^{6} + 7 \, {\left (a^{3} b^{4} d + a^{4} b^{3} e\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{3} d + 3 \, a^{5} b^{2} e\right )} x^{4} + \frac {7}{3} \, {\left (3 \, a^{5} b^{2} d + a^{6} b e\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d + a^{7} e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (34) = 68\).

Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.45 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{9} \, b^{7} e x^{9} + \frac {1}{8} \, b^{7} d x^{8} + \frac {7}{8} \, a b^{6} e x^{8} + a b^{6} d x^{7} + 3 \, a^{2} b^{5} e x^{7} + \frac {7}{2} \, a^{2} b^{5} d x^{6} + \frac {35}{6} \, a^{3} b^{4} e x^{6} + 7 \, a^{3} b^{4} d x^{5} + 7 \, a^{4} b^{3} e x^{5} + \frac {35}{4} \, a^{4} b^{3} d x^{4} + \frac {21}{4} \, a^{5} b^{2} e x^{4} + 7 \, a^{5} b^{2} d x^{3} + \frac {7}{3} \, a^{6} b e x^{3} + \frac {7}{2} \, a^{6} b d x^{2} + \frac {1}{2} \, a^{7} e x^{2} + a^{7} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.76 \[ \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^2\,\left (\frac {e\,a^7}{2}+\frac {7\,b\,d\,a^6}{2}\right )+x^8\,\left (\frac {d\,b^7}{8}+\frac {7\,a\,e\,b^6}{8}\right )+\frac {b^7\,e\,x^9}{9}+a^7\,d\,x+\frac {7\,a^5\,b\,x^3\,\left (a\,e+3\,b\,d\right )}{3}+a\,b^5\,x^7\,\left (3\,a\,e+b\,d\right )+7\,a^3\,b^3\,x^5\,\left (a\,e+b\,d\right )+\frac {7\,a^4\,b^2\,x^4\,\left (3\,a\,e+5\,b\,d\right )}{4}+\frac {7\,a^2\,b^4\,x^6\,\left (5\,a\,e+3\,b\,d\right )}{6} \]

[In]

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

[Out]

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