\(\int (d+e x)^2 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1467]

Optimal result

Integrand size = 26, antiderivative size = 65 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^2 (a+b x)^5}{5 b^3}+\frac {e (b d-a e) (a+b x)^6}{3 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (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} \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {e (a+b x)^6 (b d-a e)}{3 b^3}+\frac {(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \]

[In]

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

[Out]

((b*d - a*e)^2*(a + b*x)^5)/(5*b^3) + (e*(b*d - a*e)*(a + b*x)^6)/(3*b^3) + (e^2*(a + b*x)^7)/(7*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)^4 (d+e x)^2 \, dx \\ & = \int \left (\frac {(b d-a e)^2 (a+b x)^4}{b^2}+\frac {2 e (b d-a e) (a+b x)^5}{b^2}+\frac {e^2 (a+b x)^6}{b^2}\right ) \, dx \\ & = \frac {(b d-a e)^2 (a+b x)^5}{5 b^3}+\frac {e (b d-a e) (a+b x)^6}{3 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(65)=130\).

Time = 0.02 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.28 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 d^2 x+a^3 d (2 b d+a e) x^2+\frac {1}{3} a^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right ) x^3+a b \left (b^2 d^2+3 a b d e+a^2 e^2\right ) x^4+\frac {1}{5} b^2 \left (b^2 d^2+8 a b d e+6 a^2 e^2\right ) x^5+\frac {1}{3} b^3 e (b d+2 a e) x^6+\frac {1}{7} b^4 e^2 x^7 \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(59)=118\).

Time = 2.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.42

method	result	size
norman	\(\frac {e^{2} b^{4} x^{7}}{7}+\left (\frac {2}{3} e^{2} a \,b^{3}+\frac {1}{3} b^{4} d e \right ) x^{6}+\left (\frac {6}{5} a^{2} b^{2} e^{2}+\frac {8}{5} a \,b^{3} d e +\frac {1}{5} b^{4} d^{2}\right ) x^{5}+\left (e^{2} a^{3} b +3 d e \,a^{2} b^{2}+d^{2} a \,b^{3}\right ) x^{4}+\left (\frac {1}{3} a^{4} e^{2}+\frac {8}{3} d e \,a^{3} b +2 b^{2} d^{2} a^{2}\right ) x^{3}+\left (d e \,a^{4}+2 a^{3} b \,d^{2}\right ) x^{2}+a^{4} d^{2} x\)	\(157\)
default	\(\frac {e^{2} b^{4} x^{7}}{7}+\frac {\left (4 e^{2} a \,b^{3}+2 b^{4} d e \right ) x^{6}}{6}+\frac {\left (6 a^{2} b^{2} e^{2}+8 a \,b^{3} d e +b^{4} d^{2}\right ) x^{5}}{5}+\frac {\left (4 e^{2} a^{3} b +12 d e \,a^{2} b^{2}+4 d^{2} a \,b^{3}\right ) x^{4}}{4}+\frac {\left (a^{4} e^{2}+8 d e \,a^{3} b +6 b^{2} d^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{4}+4 a^{3} b \,d^{2}\right ) x^{2}}{2}+a^{4} d^{2} x\)	\(163\)
risch	\(\frac {1}{7} e^{2} b^{4} x^{7}+\frac {2}{3} x^{6} e^{2} a \,b^{3}+\frac {1}{3} x^{6} b^{4} d e +\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+\frac {8}{5} x^{5} a \,b^{3} d e +\frac {1}{5} d^{2} x^{5} b^{4}+a^{3} b \,e^{2} x^{4}+3 a^{2} b^{2} d e \,x^{4}+a \,b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} b^{2} d^{2} x^{3}+d e \,a^{4} x^{2}+2 a^{3} b \,d^{2} x^{2}+a^{4} d^{2} x\)	\(171\)
parallelrisch	\(\frac {1}{7} e^{2} b^{4} x^{7}+\frac {2}{3} x^{6} e^{2} a \,b^{3}+\frac {1}{3} x^{6} b^{4} d e +\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+\frac {8}{5} x^{5} a \,b^{3} d e +\frac {1}{5} d^{2} x^{5} b^{4}+a^{3} b \,e^{2} x^{4}+3 a^{2} b^{2} d e \,x^{4}+a \,b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} b^{2} d^{2} x^{3}+d e \,a^{4} x^{2}+2 a^{3} b \,d^{2} x^{2}+a^{4} d^{2} x\)	\(171\)
gosper	\(\frac {x \left (15 e^{2} b^{4} x^{6}+70 x^{5} e^{2} a \,b^{3}+35 x^{5} b^{4} d e +126 x^{4} a^{2} b^{2} e^{2}+168 x^{4} a \,b^{3} d e +21 x^{4} b^{4} d^{2}+105 a^{3} b \,e^{2} x^{3}+315 a^{2} b^{2} d e \,x^{3}+105 a \,b^{3} d^{2} x^{3}+35 x^{2} a^{4} e^{2}+280 x^{2} d e \,a^{3} b +210 x^{2} b^{2} d^{2} a^{2}+105 a^{4} d e x +210 a^{3} b \,d^{2} x +105 a^{4} d^{2}\right )}{105}\)	\(173\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).

Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} + {\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (54) = 108\).

Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.58 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d^{2} x + \frac {b^{4} e^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} e^{2}}{3} + \frac {b^{4} d e}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} e^{2}}{5} + \frac {8 a b^{3} d e}{5} + \frac {b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 3 a^{2} b^{2} d e + a b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {8 a^{3} b d e}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).

Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} + {\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (59) = 118\).

Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.62 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} e^{2} x^{7} + \frac {1}{3} \, b^{4} d e x^{6} + \frac {2}{3} \, a b^{3} e^{2} x^{6} + \frac {1}{5} \, b^{4} d^{2} x^{5} + \frac {8}{5} \, a b^{3} d e x^{5} + \frac {6}{5} \, a^{2} b^{2} e^{2} x^{5} + a b^{3} d^{2} x^{4} + 3 \, a^{2} b^{2} d e x^{4} + a^{3} b e^{2} x^{4} + 2 \, a^{2} b^{2} d^{2} x^{3} + \frac {8}{3} \, a^{3} b d e x^{3} + \frac {1}{3} \, a^{4} e^{2} x^{3} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d e x^{2} + a^{4} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.65 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.22 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^3\,\left (\frac {a^4\,e^2}{3}+\frac {8\,a^3\,b\,d\,e}{3}+2\,a^2\,b^2\,d^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,e^2}{5}+\frac {8\,a\,b^3\,d\,e}{5}+\frac {b^4\,d^2}{5}\right )+a^4\,d^2\,x+\frac {b^4\,e^2\,x^7}{7}+a^3\,d\,x^2\,\left (a\,e+2\,b\,d\right )+\frac {b^3\,e\,x^6\,\left (2\,a\,e+b\,d\right )}{3}+a\,b\,x^4\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right ) \]

[In]

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

[Out]

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