\(\int (a+b x) (d+e x)^2 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1909]

   Optimal result
   Rubi [A] (verified)
   Mathematica [B] (verified)
   Maple [B] (verified)
   Fricas [B] (verification not implemented)
   Sympy [B] (verification not implemented)
   Maxima [B] (verification not implemented)
   Giac [B] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 31, antiderivative size = 65 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^2 (a+b x)^6}{6 b^3}+\frac {2 e (b d-a e) (a+b x)^7}{7 b^3}+\frac {e^2 (a+b x)^8}{8 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45} \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 e (a+b x)^7 (b d-a e)}{7 b^3}+\frac {(a+b x)^6 (b d-a e)^2}{6 b^3}+\frac {e^2 (a+b x)^8}{8 b^3} \]

[In]

Int[(a + b*x)*(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)^6)/(6*b^3) + (2*e*(b*d - a*e)*(a + b*x)^7)/(7*b^3) + (e^2*(a + b*x)^8)/(8*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)^5 (d+e x)^2 \, dx \\ & = \int \left (\frac {(b d-a e)^2 (a+b x)^5}{b^2}+\frac {2 e (b d-a e) (a+b x)^6}{b^2}+\frac {e^2 (a+b x)^7}{b^2}\right ) \, dx \\ & = \frac {(b d-a e)^2 (a+b x)^6}{6 b^3}+\frac {2 e (b d-a e) (a+b x)^7}{7 b^3}+\frac {e^2 (a+b x)^8}{8 b^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(189\) vs. \(2(65)=130\).

Time = 0.02 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.91 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^5 d^2 x+\frac {1}{2} a^4 d (5 b d+2 a e) x^2+\frac {1}{3} a^3 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right ) x^3+\frac {5}{4} a^2 b \left (2 b^2 d^2+4 a b d e+a^2 e^2\right ) x^4+a b^2 \left (b^2 d^2+4 a b d e+2 a^2 e^2\right ) x^5+\frac {1}{6} b^3 \left (b^2 d^2+10 a b d e+10 a^2 e^2\right ) x^6+\frac {1}{7} b^4 e (2 b d+5 a e) x^7+\frac {1}{8} b^5 e^2 x^8 \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(59)=118\).

Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.00

method result size
norman \(\frac {e^{2} b^{5} x^{8}}{8}+\left (\frac {5}{7} e^{2} b^{4} a +\frac {2}{7} d e \,b^{5}\right ) x^{7}+\left (\frac {5}{3} e^{2} a^{2} b^{3}+\frac {5}{3} d e \,b^{4} a +\frac {1}{6} d^{2} b^{5}\right ) x^{6}+\left (2 e^{2} a^{3} b^{2}+4 d e \,a^{2} b^{3}+d^{2} b^{4} a \right ) x^{5}+\left (\frac {5}{4} e^{2} a^{4} b +5 d e \,a^{3} b^{2}+\frac {5}{2} d^{2} a^{2} b^{3}\right ) x^{4}+\left (\frac {1}{3} e^{2} a^{5}+\frac {10}{3} d e \,a^{4} b +\frac {10}{3} d^{2} a^{3} b^{2}\right ) x^{3}+\left (d e \,a^{5}+\frac {5}{2} d^{2} a^{4} b \right ) x^{2}+d^{2} a^{5} x\) \(195\)
risch \(\frac {1}{8} e^{2} b^{5} x^{8}+\frac {5}{7} x^{7} e^{2} b^{4} a +\frac {2}{7} x^{7} d e \,b^{5}+\frac {5}{3} x^{6} e^{2} a^{2} b^{3}+\frac {5}{3} x^{6} d e \,b^{4} a +\frac {1}{6} x^{6} d^{2} b^{5}+2 a^{3} b^{2} e^{2} x^{5}+4 a^{2} b^{3} d e \,x^{5}+a \,b^{4} d^{2} x^{5}+\frac {5}{4} x^{4} e^{2} a^{4} b +5 x^{4} d e \,a^{3} b^{2}+\frac {5}{2} x^{4} d^{2} a^{2} b^{3}+\frac {1}{3} x^{3} e^{2} a^{5}+\frac {10}{3} x^{3} d e \,a^{4} b +\frac {10}{3} x^{3} d^{2} a^{3} b^{2}+x^{2} d e \,a^{5}+\frac {5}{2} x^{2} d^{2} a^{4} b +d^{2} a^{5} x\) \(213\)
parallelrisch \(\frac {1}{8} e^{2} b^{5} x^{8}+\frac {5}{7} x^{7} e^{2} b^{4} a +\frac {2}{7} x^{7} d e \,b^{5}+\frac {5}{3} x^{6} e^{2} a^{2} b^{3}+\frac {5}{3} x^{6} d e \,b^{4} a +\frac {1}{6} x^{6} d^{2} b^{5}+2 a^{3} b^{2} e^{2} x^{5}+4 a^{2} b^{3} d e \,x^{5}+a \,b^{4} d^{2} x^{5}+\frac {5}{4} x^{4} e^{2} a^{4} b +5 x^{4} d e \,a^{3} b^{2}+\frac {5}{2} x^{4} d^{2} a^{2} b^{3}+\frac {1}{3} x^{3} e^{2} a^{5}+\frac {10}{3} x^{3} d e \,a^{4} b +\frac {10}{3} x^{3} d^{2} a^{3} b^{2}+x^{2} d e \,a^{5}+\frac {5}{2} x^{2} d^{2} a^{4} b +d^{2} a^{5} x\) \(213\)
gosper \(\frac {x \left (21 e^{2} b^{5} x^{7}+120 x^{6} e^{2} b^{4} a +48 x^{6} d e \,b^{5}+280 x^{5} e^{2} a^{2} b^{3}+280 x^{5} d e \,b^{4} a +28 x^{5} d^{2} b^{5}+336 a^{3} b^{2} e^{2} x^{4}+672 a^{2} b^{3} d e \,x^{4}+168 a \,b^{4} d^{2} x^{4}+210 x^{3} e^{2} a^{4} b +840 x^{3} d e \,a^{3} b^{2}+420 x^{3} d^{2} a^{2} b^{3}+56 x^{2} e^{2} a^{5}+560 x^{2} d e \,a^{4} b +560 x^{2} d^{2} a^{3} b^{2}+168 x d e \,a^{5}+420 x \,d^{2} a^{4} b +168 d^{2} a^{5}\right )}{168}\) \(214\)
default \(\frac {e^{2} b^{5} x^{8}}{8}+\frac {\left (\left (e^{2} a +2 b d e \right ) b^{4}+4 e^{2} b^{4} a \right ) x^{7}}{7}+\frac {\left (\left (2 a d e +b \,d^{2}\right ) b^{4}+4 \left (e^{2} a +2 b d e \right ) b^{3} a +6 e^{2} a^{2} b^{3}\right ) x^{6}}{6}+\frac {\left (d^{2} b^{4} a +4 \left (2 a d e +b \,d^{2}\right ) b^{3} a +6 \left (e^{2} a +2 b d e \right ) b^{2} a^{2}+4 e^{2} a^{3} b^{2}\right ) x^{5}}{5}+\frac {\left (4 d^{2} a^{2} b^{3}+6 \left (2 a d e +b \,d^{2}\right ) b^{2} a^{2}+4 \left (e^{2} a +2 b d e \right ) b \,a^{3}+e^{2} a^{4} b \right ) x^{4}}{4}+\frac {\left (6 d^{2} a^{3} b^{2}+4 \left (2 a d e +b \,d^{2}\right ) b \,a^{3}+\left (e^{2} a +2 b d e \right ) a^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{2} a^{4} b +\left (2 a d e +b \,d^{2}\right ) a^{4}\right ) x^{2}}{2}+d^{2} a^{5} x\) \(301\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (59) = 118\).

Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.03 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} + {\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (56) = 112\).

Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.35 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{5} d^{2} x + \frac {b^{5} e^{2} x^{8}}{8} + x^{7} \cdot \left (\frac {5 a b^{4} e^{2}}{7} + \frac {2 b^{5} d e}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{2} b^{3} e^{2}}{3} + \frac {5 a b^{4} d e}{3} + \frac {b^{5} d^{2}}{6}\right ) + x^{5} \cdot \left (2 a^{3} b^{2} e^{2} + 4 a^{2} b^{3} d e + a b^{4} d^{2}\right ) + x^{4} \cdot \left (\frac {5 a^{4} b e^{2}}{4} + 5 a^{3} b^{2} d e + \frac {5 a^{2} b^{3} d^{2}}{2}\right ) + x^{3} \left (\frac {a^{5} e^{2}}{3} + \frac {10 a^{4} b d e}{3} + \frac {10 a^{3} b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{5} d e + \frac {5 a^{4} b d^{2}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (59) = 118\).

Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.03 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} + {\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (59) = 118\).

Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.26 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{5} e^{2} x^{8} + \frac {2}{7} \, b^{5} d e x^{7} + \frac {5}{7} \, a b^{4} e^{2} x^{7} + \frac {1}{6} \, b^{5} d^{2} x^{6} + \frac {5}{3} \, a b^{4} d e x^{6} + \frac {5}{3} \, a^{2} b^{3} e^{2} x^{6} + a b^{4} d^{2} x^{5} + 4 \, a^{2} b^{3} d e x^{5} + 2 \, a^{3} b^{2} e^{2} x^{5} + \frac {5}{2} \, a^{2} b^{3} d^{2} x^{4} + 5 \, a^{3} b^{2} d e x^{4} + \frac {5}{4} \, a^{4} b e^{2} x^{4} + \frac {10}{3} \, a^{3} b^{2} d^{2} x^{3} + \frac {10}{3} \, a^{4} b d e x^{3} + \frac {1}{3} \, a^{5} e^{2} x^{3} + \frac {5}{2} \, a^{4} b d^{2} x^{2} + a^{5} d e x^{2} + a^{5} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.78 \[ \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^3\,\left (\frac {a^5\,e^2}{3}+\frac {10\,a^4\,b\,d\,e}{3}+\frac {10\,a^3\,b^2\,d^2}{3}\right )+x^6\,\left (\frac {5\,a^2\,b^3\,e^2}{3}+\frac {5\,a\,b^4\,d\,e}{3}+\frac {b^5\,d^2}{6}\right )+a^5\,d^2\,x+\frac {b^5\,e^2\,x^8}{8}+\frac {a^4\,d\,x^2\,\left (2\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e\,x^7\,\left (5\,a\,e+2\,b\,d\right )}{7}+\frac {5\,a^2\,b\,x^4\,\left (a^2\,e^2+4\,a\,b\,d\,e+2\,b^2\,d^2\right )}{4}+a\,b^2\,x^5\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right ) \]

[In]

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

[Out]

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