\(\int (d+e x) (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1488]

Optimal result

Integrand size = 24, antiderivative size = 38 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e) (a+b x)^7}{7 b^2}+\frac {e (a+b x)^8}{8 b^2} \]

[Out]

1/7*(-a*e+b*d)*(b*x+a)^7/b^2+1/8*e*(b*x+a)^8/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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(a+b x)^7 (b d-a e)}{7 b^2}+\frac {e (a+b x)^8}{8 b^2} \]

[In]

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

[Out]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(38)=76\).

Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.21 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{56} x \left (28 a^6 (2 d+e x)+56 a^5 b x (3 d+2 e x)+70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right ) \]

[In]

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

[Out]

(x*(28*a^6*(2*d + e*x) + 56*a^5*b*x*(3*d + 2*e*x) + 70*a^4*b^2*x^2*(4*d + 3*e*x) + 56*a^3*b^3*x^3*(5*d + 4*e*x
) + 28*a^2*b^4*x^4*(6*d + 5*e*x) + 8*a*b^5*x^5*(7*d + 6*e*x) + b^6*x^6*(8*d + 7*e*x)))/56

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(139\) vs. \(2(34)=68\).

Time = 2.52 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.68

method	result	size
norman	\(\frac {b^{6} e \,x^{8}}{8}+\left (\frac {6}{7} e a \,b^{5}+\frac {1}{7} d \,b^{6}\right ) x^{7}+\left (\frac {5}{2} e \,a^{2} b^{4}+d a \,b^{5}\right ) x^{6}+\left (4 e \,a^{3} b^{3}+3 d \,a^{2} b^{4}\right ) x^{5}+\left (\frac {15}{4} a^{4} e \,b^{2}+5 a^{3} d \,b^{3}\right ) x^{4}+\left (2 a^{5} b e +5 a^{4} b^{2} d \right ) x^{3}+\left (\frac {1}{2} e \,a^{6}+3 d \,a^{5} b \right ) x^{2}+d \,a^{6} x\)	\(140\)
default	\(\frac {b^{6} e \,x^{8}}{8}+\frac {\left (6 e a \,b^{5}+d \,b^{6}\right ) x^{7}}{7}+\frac {\left (15 e \,a^{2} b^{4}+6 d a \,b^{5}\right ) x^{6}}{6}+\frac {\left (20 e \,a^{3} b^{3}+15 d \,a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (15 a^{4} e \,b^{2}+20 a^{3} d \,b^{3}\right ) x^{4}}{4}+\frac {\left (6 a^{5} b e +15 a^{4} b^{2} d \right ) x^{3}}{3}+\frac {\left (e \,a^{6}+6 d \,a^{5} b \right ) x^{2}}{2}+d \,a^{6} x\)	\(145\)
gosper	\(\frac {x \left (7 b^{6} e \,x^{7}+48 x^{6} e a \,b^{5}+8 x^{6} d \,b^{6}+140 x^{5} e \,a^{2} b^{4}+56 x^{5} d a \,b^{5}+224 a^{3} b^{3} e \,x^{4}+168 a^{2} b^{4} d \,x^{4}+210 x^{3} a^{4} e \,b^{2}+280 x^{3} a^{3} d \,b^{3}+112 a^{5} b e \,x^{2}+280 a^{4} b^{2} d \,x^{2}+28 x e \,a^{6}+168 x d \,a^{5} b +56 d \,a^{6}\right )}{56}\)	\(146\)
risch	\(\frac {1}{8} b^{6} e \,x^{8}+\frac {6}{7} x^{7} e a \,b^{5}+\frac {1}{7} x^{7} d \,b^{6}+\frac {5}{2} x^{6} e \,a^{2} b^{4}+x^{6} d a \,b^{5}+4 a^{3} b^{3} e \,x^{5}+3 a^{2} b^{4} d \,x^{5}+\frac {15}{4} x^{4} a^{4} e \,b^{2}+5 x^{4} a^{3} d \,b^{3}+2 a^{5} b e \,x^{3}+5 a^{4} b^{2} d \,x^{3}+\frac {1}{2} x^{2} e \,a^{6}+3 x^{2} d \,a^{5} b +d \,a^{6} x\)	\(146\)
parallelrisch	\(\frac {1}{8} b^{6} e \,x^{8}+\frac {6}{7} x^{7} e a \,b^{5}+\frac {1}{7} x^{7} d \,b^{6}+\frac {5}{2} x^{6} e \,a^{2} b^{4}+x^{6} d a \,b^{5}+4 a^{3} b^{3} e \,x^{5}+3 a^{2} b^{4} d \,x^{5}+\frac {15}{4} x^{4} a^{4} e \,b^{2}+5 x^{4} a^{3} d \,b^{3}+2 a^{5} b e \,x^{3}+5 a^{4} b^{2} d \,x^{3}+\frac {1}{2} x^{2} e \,a^{6}+3 x^{2} d \,a^{5} b +d \,a^{6} x\)	\(146\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (34) = 68\).

Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.74 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac {1}{7} \, {\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} + {\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} + {\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d + a^{6} e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (32) = 64\).

Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.89 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d x + \frac {b^{6} e x^{8}}{8} + x^{7} \cdot \left (\frac {6 a b^{5} e}{7} + \frac {b^{6} d}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{2} b^{4} e}{2} + a b^{5} d\right ) + x^{5} \cdot \left (4 a^{3} b^{3} e + 3 a^{2} b^{4} d\right ) + x^{4} \cdot \left (\frac {15 a^{4} b^{2} e}{4} + 5 a^{3} b^{3} d\right ) + x^{3} \cdot \left (2 a^{5} b e + 5 a^{4} b^{2} d\right ) + x^{2} \left (\frac {a^{6} e}{2} + 3 a^{5} b d\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (34) = 68\).

Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.74 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac {1}{7} \, {\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} + {\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} + {\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d + a^{6} e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (34) = 68\).

Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.82 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{8} \, b^{6} e x^{8} + \frac {1}{7} \, b^{6} d x^{7} + \frac {6}{7} \, a b^{5} e x^{7} + a b^{5} d x^{6} + \frac {5}{2} \, a^{2} b^{4} e x^{6} + 3 \, a^{2} b^{4} d x^{5} + 4 \, a^{3} b^{3} e x^{5} + 5 \, a^{3} b^{3} d x^{4} + \frac {15}{4} \, a^{4} b^{2} e x^{4} + 5 \, a^{4} b^{2} d x^{3} + 2 \, a^{5} b e x^{3} + 3 \, a^{5} b d x^{2} + \frac {1}{2} \, a^{6} e x^{2} + a^{6} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.32 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^2\,\left (\frac {e\,a^6}{2}+3\,b\,d\,a^5\right )+x^7\,\left (\frac {d\,b^6}{7}+\frac {6\,a\,e\,b^5}{7}\right )+\frac {b^6\,e\,x^8}{8}+a^6\,d\,x+a^4\,b\,x^3\,\left (2\,a\,e+5\,b\,d\right )+\frac {a\,b^4\,x^6\,\left (5\,a\,e+2\,b\,d\right )}{2}+\frac {5\,a^3\,b^2\,x^4\,\left (3\,a\,e+4\,b\,d\right )}{4}+a^2\,b^3\,x^5\,\left (4\,a\,e+3\,b\,d\right ) \]

[In]

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

[Out]

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