\(\int (d+e x) (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1468]

Optimal result

Integrand size = 24, antiderivative size = 38 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e) (a+b x)^5}{5 b^2}+\frac {e (a+b x)^6}{6 b^2} \]

[Out]

1/5*(-a*e+b*d)*(b*x+a)^5/b^2+1/6*e*(b*x+a)^6/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} \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(a+b x)^5 (b d-a e)}{5 b^2}+\frac {e (a+b x)^6}{6 b^2} \]

[In]

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

[Out]

((b*d - a*e)*(a + b*x)^5)/(5*b^2) + (e*(a + b*x)^6)/(6*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)^4 (d+e x) \, dx \\ & = \int \left (\frac {(b d-a e) (a+b x)^4}{b}+\frac {e (a+b x)^5}{b}\right ) \, dx \\ & = \frac {(b d-a e) (a+b x)^5}{5 b^2}+\frac {e (a+b x)^6}{6 b^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(38)=76\).

Time = 0.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.21 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{30} x \left (15 a^4 (2 d+e x)+20 a^3 b x (3 d+2 e x)+15 a^2 b^2 x^2 (4 d+3 e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right ) \]

[In]

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

[Out]

(x*(15*a^4*(2*d + e*x) + 20*a^3*b*x*(3*d + 2*e*x) + 15*a^2*b^2*x^2*(4*d + 3*e*x) + 6*a*b^3*x^3*(5*d + 4*e*x) +
b^4*x^4*(6*d + 5*e*x)))/30

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).

Time = 2.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47

method	result	size
norman	\(\frac {e \,b^{4} x^{6}}{6}+\left (\frac {4}{5} e a \,b^{3}+\frac {1}{5} d \,b^{4}\right ) x^{5}+\left (\frac {3}{2} a^{2} e \,b^{2}+a \,b^{3} d \right ) x^{4}+\left (\frac {4}{3} a^{3} b e +2 a^{2} b^{2} d \right ) x^{3}+\left (\frac {1}{2} e \,a^{4}+2 d \,a^{3} b \right ) x^{2}+a^{4} d x\)	\(94\)
default	\(\frac {e \,b^{4} x^{6}}{6}+\frac {\left (4 e a \,b^{3}+d \,b^{4}\right ) x^{5}}{5}+\frac {\left (6 a^{2} e \,b^{2}+4 a \,b^{3} d \right ) x^{4}}{4}+\frac {\left (4 a^{3} b e +6 a^{2} b^{2} d \right ) x^{3}}{3}+\frac {\left (e \,a^{4}+4 d \,a^{3} b \right ) x^{2}}{2}+a^{4} d x\)	\(97\)
gosper	\(\frac {x \left (5 e \,b^{4} x^{5}+24 x^{4} e a \,b^{3}+6 x^{4} d \,b^{4}+45 a^{2} b^{2} e \,x^{3}+30 a \,b^{3} d \,x^{3}+40 a^{3} b e \,x^{2}+60 x^{2} a^{2} b^{2} d +15 a^{4} e x +60 x d \,a^{3} b +30 a^{4} d \right )}{30}\)	\(98\)
risch	\(\frac {1}{6} e \,b^{4} x^{6}+\frac {4}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}+\frac {3}{2} a^{2} b^{2} e \,x^{4}+a \,b^{3} d \,x^{4}+\frac {4}{3} a^{3} b e \,x^{3}+2 x^{3} a^{2} b^{2} d +\frac {1}{2} a^{4} e \,x^{2}+2 a^{3} b d \,x^{2}+a^{4} d x\)	\(98\)
parallelrisch	\(\frac {1}{6} e \,b^{4} x^{6}+\frac {4}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}+\frac {3}{2} a^{2} b^{2} e \,x^{4}+a \,b^{3} d \,x^{4}+\frac {4}{3} a^{3} b e \,x^{3}+2 x^{3} a^{2} b^{2} d +\frac {1}{2} a^{4} e \,x^{2}+2 a^{3} b d \,x^{2}+a^{4} d x\)	\(98\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).

Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac {1}{5} \, {\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (32) = 64\).

Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.63 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d x + \frac {b^{4} e x^{6}}{6} + x^{5} \cdot \left (\frac {4 a b^{3} e}{5} + \frac {b^{4} d}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b^{2} e}{2} + a b^{3} d\right ) + x^{3} \cdot \left (\frac {4 a^{3} b e}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac {a^{4} e}{2} + 2 a^{3} b d\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).

Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac {1}{5} \, {\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (34) = 68\).

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.55 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{6} \, b^{4} e x^{6} + \frac {1}{5} \, b^{4} d x^{5} + \frac {4}{5} \, a b^{3} e x^{5} + a b^{3} d x^{4} + \frac {3}{2} \, a^{2} b^{2} e x^{4} + 2 \, a^{2} b^{2} d x^{3} + \frac {4}{3} \, a^{3} b e x^{3} + 2 \, a^{3} b d x^{2} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.62 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^2\,\left (\frac {e\,a^4}{2}+2\,b\,d\,a^3\right )+x^5\,\left (\frac {d\,b^4}{5}+\frac {4\,a\,e\,b^3}{5}\right )+\frac {b^4\,e\,x^6}{6}+a^4\,d\,x+\frac {2\,a^2\,b\,x^3\,\left (2\,a\,e+3\,b\,d\right )}{3}+\frac {a\,b^2\,x^4\,\left (3\,a\,e+2\,b\,d\right )}{2} \]

[In]

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

[Out]

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