\(\int (d+e x)^3 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1466]

Optimal result

Integrand size = 26, antiderivative size = 92 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^3 (a+b x)^5}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^6}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^7}{7 b^4}+\frac {e^3 (a+b x)^8}{8 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 92, 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} \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac {e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac {(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac {e^3 (a+b x)^8}{8 b^4} \]

[In]

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

[Out]

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

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)^3 \, dx \\ & = \int \left (\frac {(b d-a e)^3 (a+b x)^4}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^5}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^6}{b^3}+\frac {e^3 (a+b x)^7}{b^3}\right ) \, dx \\ & = \frac {(b d-a e)^3 (a+b x)^5}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^6}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^7}{7 b^4}+\frac {e^3 (a+b x)^8}{8 b^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(92)=184\).

Time = 0.02 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.36 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 d^3 x+\frac {1}{2} a^3 d^2 (4 b d+3 a e) x^2+a^2 d \left (2 b^2 d^2+4 a b d e+a^2 e^2\right ) x^3+\frac {1}{4} a \left (4 b^3 d^3+18 a b^2 d^2 e+12 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {1}{5} b \left (b^3 d^3+12 a b^2 d^2 e+18 a^2 b d e^2+4 a^3 e^3\right ) x^5+\frac {1}{2} b^2 e \left (b^2 d^2+4 a b d e+2 a^2 e^2\right ) x^6+\frac {1}{7} b^3 e^2 (3 b d+4 a e) x^7+\frac {1}{8} b^4 e^3 x^8 \]

[In]

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

[Out]

a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + a^2*d*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^3 + (a*(4*b^3*d^3 + 18
*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^3)*x^4)/4 + (b*(b^3*d^3 + 12*a*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3)*x
^5)/5 + (b^2*e*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2)*x^6)/2 + (b^3*e^2*(3*b*d + 4*a*e)*x^7)/7 + (b^4*e^3*x^8)/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(84)=168\).

Time = 2.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.41

method	result	size
norman	\(\frac {e^{3} b^{4} x^{8}}{8}+\left (\frac {4}{7} a \,b^{3} e^{3}+\frac {3}{7} b^{4} d \,e^{2}\right ) x^{7}+\left (a^{2} b^{2} e^{3}+2 d \,e^{2} a \,b^{3}+\frac {1}{2} d^{2} e \,b^{4}\right ) x^{6}+\left (\frac {4}{5} a^{3} b \,e^{3}+\frac {18}{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} d^{2} e a \,b^{3}+\frac {1}{5} b^{4} d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{4}+3 d \,e^{2} a^{3} b +\frac {9}{2} d^{2} e \,a^{2} b^{2}+a \,b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{4}+4 d^{2} e \,a^{3} b +2 d^{3} a^{2} b^{2}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{4}+2 a^{3} b \,d^{3}\right ) x^{2}+a^{4} d^{3} x\)	\(222\)
default	\(\frac {e^{3} b^{4} x^{8}}{8}+\frac {\left (4 a \,b^{3} e^{3}+3 b^{4} d \,e^{2}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{2} e^{3}+12 d \,e^{2} a \,b^{3}+3 d^{2} e \,b^{4}\right ) x^{6}}{6}+\frac {\left (4 a^{3} b \,e^{3}+18 a^{2} b^{2} d \,e^{2}+12 d^{2} e a \,b^{3}+b^{4} d^{3}\right ) x^{5}}{5}+\frac {\left (e^{3} a^{4}+12 d \,e^{2} a^{3} b +18 d^{2} e \,a^{2} b^{2}+4 a \,b^{3} d^{3}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{4}+12 d^{2} e \,a^{3} b +6 d^{3} a^{2} b^{2}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{4}+4 a^{3} b \,d^{3}\right ) x^{2}}{2}+a^{4} d^{3} x\)	\(229\)
risch	\(\frac {1}{8} e^{3} b^{4} x^{8}+\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+x^{6} a^{2} b^{2} e^{3}+2 x^{6} d \,e^{2} a \,b^{3}+\frac {1}{2} x^{6} d^{2} e \,b^{4}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} d^{2} e a \,b^{3}+\frac {1}{5} d^{3} x^{5} b^{4}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d \,e^{2} a^{3} b +\frac {9}{2} x^{4} d^{2} e \,a^{2} b^{2}+a \,b^{3} d^{3} x^{4}+a^{4} d \,e^{2} x^{3}+4 a^{3} b \,d^{2} e \,x^{3}+2 a^{2} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+2 a^{3} b \,d^{3} x^{2}+a^{4} d^{3} x\)	\(246\)
parallelrisch	\(\frac {1}{8} e^{3} b^{4} x^{8}+\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+x^{6} a^{2} b^{2} e^{3}+2 x^{6} d \,e^{2} a \,b^{3}+\frac {1}{2} x^{6} d^{2} e \,b^{4}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} d^{2} e a \,b^{3}+\frac {1}{5} d^{3} x^{5} b^{4}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d \,e^{2} a^{3} b +\frac {9}{2} x^{4} d^{2} e \,a^{2} b^{2}+a \,b^{3} d^{3} x^{4}+a^{4} d \,e^{2} x^{3}+4 a^{3} b \,d^{2} e \,x^{3}+2 a^{2} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+2 a^{3} b \,d^{3} x^{2}+a^{4} d^{3} x\)	\(246\)
gosper	\(\frac {x \left (35 e^{3} b^{4} x^{7}+160 x^{6} a \,b^{3} e^{3}+120 x^{6} b^{4} d \,e^{2}+280 x^{5} a^{2} b^{2} e^{3}+560 x^{5} d \,e^{2} a \,b^{3}+140 x^{5} d^{2} e \,b^{4}+224 x^{4} a^{3} b \,e^{3}+1008 x^{4} a^{2} b^{2} d \,e^{2}+672 x^{4} d^{2} e a \,b^{3}+56 x^{4} b^{4} d^{3}+70 x^{3} e^{3} a^{4}+840 x^{3} d \,e^{2} a^{3} b +1260 x^{3} d^{2} e \,a^{2} b^{2}+280 x^{3} a \,b^{3} d^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 x^{2} a^{2} b^{2} d^{3}+420 x \,d^{2} e \,a^{4}+560 x \,a^{3} b \,d^{3}+280 a^{4} d^{3}\right )}{280}\)	\(248\)

[In]

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

[Out]

1/8*e^3*b^4*x^8+(4/7*a*b^3*e^3+3/7*b^4*d*e^2)*x^7+(a^2*b^2*e^3+2*d*e^2*a*b^3+1/2*d^2*e*b^4)*x^6+(4/5*a^3*b*e^3
+18/5*a^2*b^2*d*e^2+12/5*d^2*e*a*b^3+1/5*b^4*d^3)*x^5+(1/4*e^3*a^4+3*d*e^2*a^3*b+9/2*d^2*e*a^2*b^2+a*b^3*d^3)*
x^4+(a^4*d*e^2+4*a^3*b*d^2*e+2*a^2*b^2*d^3)*x^3+(3/2*d^2*e*a^4+2*a^3*b*d^3)*x^2+a^4*d^3*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).

Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.45 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]

[In]

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

[Out]

1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2
*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^
2*d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 +
3*a^4*d^2*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (80) = 160\).

Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.64 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d^{3} x + \frac {b^{4} e^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {4 a b^{3} e^{3}}{7} + \frac {3 b^{4} d e^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} e^{3} + 2 a b^{3} d e^{2} + \frac {b^{4} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b e^{3}}{5} + \frac {18 a^{2} b^{2} d e^{2}}{5} + \frac {12 a b^{3} d^{2} e}{5} + \frac {b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + \frac {9 a^{2} b^{2} d^{2} e}{2} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \]

[In]

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

[Out]

a**4*d**3*x + b**4*e**3*x**8/8 + x**7*(4*a*b**3*e**3/7 + 3*b**4*d*e**2/7) + x**6*(a**2*b**2*e**3 + 2*a*b**3*d*
e**2 + b**4*d**2*e/2) + x**5*(4*a**3*b*e**3/5 + 18*a**2*b**2*d*e**2/5 + 12*a*b**3*d**2*e/5 + b**4*d**3/5) + x*
*4*(a**4*e**3/4 + 3*a**3*b*d*e**2 + 9*a**2*b**2*d**2*e/2 + a*b**3*d**3) + x**3*(a**4*d*e**2 + 4*a**3*b*d**2*e
+ 2*a**2*b**2*d**3) + x**2*(3*a**4*d**2*e/2 + 2*a**3*b*d**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).

Time = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.45 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]

[In]

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

[Out]

1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2
*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^
2*d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 +
3*a^4*d^2*e)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (84) = 168\).

Time = 0.26 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.66 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + \frac {3}{7} \, b^{4} d e^{2} x^{7} + \frac {4}{7} \, a b^{3} e^{3} x^{7} + \frac {1}{2} \, b^{4} d^{2} e x^{6} + 2 \, a b^{3} d e^{2} x^{6} + a^{2} b^{2} e^{3} x^{6} + \frac {1}{5} \, b^{4} d^{3} x^{5} + \frac {12}{5} \, a b^{3} d^{2} e x^{5} + \frac {18}{5} \, a^{2} b^{2} d e^{2} x^{5} + \frac {4}{5} \, a^{3} b e^{3} x^{5} + a b^{3} d^{3} x^{4} + \frac {9}{2} \, a^{2} b^{2} d^{2} e x^{4} + 3 \, a^{3} b d e^{2} x^{4} + \frac {1}{4} \, a^{4} e^{3} x^{4} + 2 \, a^{2} b^{2} d^{3} x^{3} + 4 \, a^{3} b d^{2} e x^{3} + a^{4} d e^{2} x^{3} + 2 \, a^{3} b d^{3} x^{2} + \frac {3}{2} \, a^{4} d^{2} e x^{2} + a^{4} d^{3} x \]

[In]

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

[Out]

1/8*b^4*e^3*x^8 + 3/7*b^4*d*e^2*x^7 + 4/7*a*b^3*e^3*x^7 + 1/2*b^4*d^2*e*x^6 + 2*a*b^3*d*e^2*x^6 + a^2*b^2*e^3*
x^6 + 1/5*b^4*d^3*x^5 + 12/5*a*b^3*d^2*e*x^5 + 18/5*a^2*b^2*d*e^2*x^5 + 4/5*a^3*b*e^3*x^5 + a*b^3*d^3*x^4 + 9/
2*a^2*b^2*d^2*e*x^4 + 3*a^3*b*d*e^2*x^4 + 1/4*a^4*e^3*x^4 + 2*a^2*b^2*d^3*x^3 + 4*a^3*b*d^2*e*x^3 + a^4*d*e^2*
x^3 + 2*a^3*b*d^3*x^2 + 3/2*a^4*d^2*e*x^2 + a^4*d^3*x

Mupad [B] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.26 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^4\,\left (\frac {a^4\,e^3}{4}+3\,a^3\,b\,d\,e^2+\frac {9\,a^2\,b^2\,d^2\,e}{2}+a\,b^3\,d^3\right )+x^5\,\left (\frac {4\,a^3\,b\,e^3}{5}+\frac {18\,a^2\,b^2\,d\,e^2}{5}+\frac {12\,a\,b^3\,d^2\,e}{5}+\frac {b^4\,d^3}{5}\right )+a^4\,d^3\,x+\frac {b^4\,e^3\,x^8}{8}+\frac {a^3\,d^2\,x^2\,\left (3\,a\,e+4\,b\,d\right )}{2}+\frac {b^3\,e^2\,x^7\,\left (4\,a\,e+3\,b\,d\right )}{7}+a^2\,d\,x^3\,\left (a^2\,e^2+4\,a\,b\,d\,e+2\,b^2\,d^2\right )+\frac {b^2\,e\,x^6\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{2} \]

[In]

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

[Out]

x^4*((a^4*e^3)/4 + a*b^3*d^3 + (9*a^2*b^2*d^2*e)/2 + 3*a^3*b*d*e^2) + x^5*((b^4*d^3)/5 + (4*a^3*b*e^3)/5 + (18
*a^2*b^2*d*e^2)/5 + (12*a*b^3*d^2*e)/5) + a^4*d^3*x + (b^4*e^3*x^8)/8 + (a^3*d^2*x^2*(3*a*e + 4*b*d))/2 + (b^3
*e^2*x^7*(4*a*e + 3*b*d))/7 + a^2*d*x^3*(a^2*e^2 + 2*b^2*d^2 + 4*a*b*d*e) + (b^2*e*x^6*(2*a^2*e^2 + b^2*d^2 +
4*a*b*d*e))/2