\(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2) \, dx\) [1451]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(b d-a e)^2 (d+e x)^5}{5 e^3}-\frac {b (b d-a e) (d+e x)^6}{3 e^3}+\frac {b^2 (d+e x)^7}{7 e^3} \]

[Out]

1/5*(-a*e+b*d)^2*(e*x+d)^5/e^3-1/3*b*(-a*e+b*d)*(e*x+d)^6/e^3+1/7*b^2*(e*x+d)^7/e^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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {b (d+e x)^6 (b d-a e)}{3 e^3}+\frac {(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac {b^2 (d+e x)^7}{7 e^3} \]

[In]

Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

Mathematica [B] (verified)

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

Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.28 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 d^4 x+a d^3 (b d+2 a e) x^2+\frac {1}{3} d^2 \left (b^2 d^2+8 a b d e+6 a^2 e^2\right ) x^3+d e \left (b^2 d^2+3 a b d e+a^2 e^2\right ) x^4+\frac {1}{5} e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right ) x^5+\frac {1}{3} b e^3 (2 b d+a e) x^6+\frac {1}{7} b^2 e^4 x^7 \]

[In]

Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

a^2*d^4*x + a*d^3*(b*d + 2*a*e)*x^2 + (d^2*(b^2*d^2 + 8*a*b*d*e + 6*a^2*e^2)*x^3)/3 + d*e*(b^2*d^2 + 3*a*b*d*e
+ a^2*e^2)*x^4 + (e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)*x^5)/5 + (b*e^3*(2*b*d + a*e)*x^6)/3 + (b^2*e^4*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 = 2.35 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.42

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

[In]

int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

1/7*b^2*e^4*x^7+(1/3*a*b*e^4+2/3*b^2*d*e^3)*x^6+(1/5*a^2*e^4+8/5*a*b*d*e^3+6/5*b^2*d^2*e^2)*x^5+(a^2*d*e^3+3*a
*b*d^2*e^2+b^2*d^3*e)*x^4+(2*a^2*d^2*e^2+8/3*a*b*d^3*e+1/3*b^2*d^4)*x^3+(2*a^2*d^3*e+a*b*d^4)*x^2+a^2*d^4*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 = 0.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} + {\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

1/7*b^2*e^4*x^7 + a^2*d^4*x + 1/3*(2*b^2*d*e^3 + a*b*e^4)*x^6 + 1/5*(6*b^2*d^2*e^2 + 8*a*b*d*e^3 + a^2*e^4)*x^
5 + (b^2*d^3*e + 3*a*b*d^2*e^2 + a^2*d*e^3)*x^4 + 1/3*(b^2*d^4 + 8*a*b*d^3*e + 6*a^2*d^2*e^2)*x^3 + (a*b*d^4 +
2*a^2*d^3*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 = 0.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.58 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^{2} d^{4} x + \frac {b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac {a b e^{4}}{3} + \frac {2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \cdot \left (2 a^{2} d^{3} e + a b d^{4}\right ) \]

[In]

integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

Maxima [B] (verification not implemented)

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

Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} + {\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

1/7*b^2*e^4*x^7 + a^2*d^4*x + 1/3*(2*b^2*d*e^3 + a*b*e^4)*x^6 + 1/5*(6*b^2*d^2*e^2 + 8*a*b*d*e^3 + a^2*e^4)*x^
5 + (b^2*d^3*e + 3*a*b*d^2*e^2 + a^2*d*e^3)*x^4 + 1/3*(b^2*d^4 + 8*a*b*d^3*e + 6*a^2*d^2*e^2)*x^3 + (a*b*d^4 +
2*a^2*d^3*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 = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.62 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{7} \, b^{2} e^{4} x^{7} + \frac {2}{3} \, b^{2} d e^{3} x^{6} + \frac {1}{3} \, a b e^{4} x^{6} + \frac {6}{5} \, b^{2} d^{2} e^{2} x^{5} + \frac {8}{5} \, a b d e^{3} x^{5} + \frac {1}{5} \, a^{2} e^{4} x^{5} + b^{2} d^{3} e x^{4} + 3 \, a b d^{2} e^{2} x^{4} + a^{2} d e^{3} x^{4} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {8}{3} \, a b d^{3} e x^{3} + 2 \, a^{2} d^{2} e^{2} x^{3} + a b d^{4} x^{2} + 2 \, a^{2} d^{3} e x^{2} + a^{2} d^{4} x \]

[In]

integrate((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.22 \[ \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^3\,\left (2\,a^2\,d^2\,e^2+\frac {8\,a\,b\,d^3\,e}{3}+\frac {b^2\,d^4}{3}\right )+x^5\,\left (\frac {a^2\,e^4}{5}+\frac {8\,a\,b\,d\,e^3}{5}+\frac {6\,b^2\,d^2\,e^2}{5}\right )+a^2\,d^4\,x+\frac {b^2\,e^4\,x^7}{7}+a\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )+\frac {b\,e^3\,x^6\,\left (a\,e+2\,b\,d\right )}{3}+d\,e\,x^4\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right ) \]

[In]

int((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

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