\(\int (a+b x) (A+B x) (d+e x)^4 \, dx\) [1011]

Optimal result

Integrand size = 18, antiderivative size = 77 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^5}{5 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^6}{6 e^3}+\frac {b B (d+e x)^7}{7 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=-\frac {(d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3}+\frac {(d+e x)^5 (b d-a e) (B d-A e)}{5 e^3}+\frac {b B (d+e x)^7}{7 e^3} \]

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^4}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^5}{e^2}+\frac {b B (d+e x)^6}{e^2}\right ) \, dx \\ & = \frac {(b d-a e) (B d-A e) (d+e x)^5}{5 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^6}{6 e^3}+\frac {b B (d+e x)^7}{7 e^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(77)=154\).

Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.23 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=a A d^4 x+\frac {1}{2} d^3 (A b d+a B d+4 a A e) x^2+\frac {1}{3} d^2 (2 a e (2 B d+3 A e)+b d (B d+4 A e)) x^3+\frac {1}{2} d e (a e (3 B d+2 A e)+b d (2 B d+3 A e)) x^4+\frac {1}{5} e^2 (a e (4 B d+A e)+2 b d (3 B d+2 A e)) x^5+\frac {1}{6} e^3 (4 b B d+A b e+a B e) x^6+\frac {1}{7} b B e^4 x^7 \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(175\) vs. \(2(71)=142\).

Time = 0.65 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.29

method	result	size
default	\(\frac {b B \,e^{4} x^{7}}{7}+\frac {\left (\left (A b +B a \right ) e^{4}+4 b B d \,e^{3}\right ) x^{6}}{6}+\frac {\left (A a \,e^{4}+4 \left (A b +B a \right ) d \,e^{3}+6 b B \,d^{2} e^{2}\right ) x^{5}}{5}+\frac {\left (4 A a d \,e^{3}+6 \left (A b +B a \right ) d^{2} e^{2}+4 b B \,d^{3} e \right ) x^{4}}{4}+\frac {\left (6 A a \,d^{2} e^{2}+4 \left (A b +B a \right ) d^{3} e +b B \,d^{4}\right ) x^{3}}{3}+\frac {\left (4 A a \,d^{3} e +\left (A b +B a \right ) d^{4}\right ) x^{2}}{2}+A a \,d^{4} x\)	\(176\)
norman	\(\frac {b B \,e^{4} x^{7}}{7}+\left (\frac {1}{6} A b \,e^{4}+\frac {1}{6} B a \,e^{4}+\frac {2}{3} b B d \,e^{3}\right ) x^{6}+\left (\frac {1}{5} A a \,e^{4}+\frac {4}{5} A b d \,e^{3}+\frac {4}{5} B a d \,e^{3}+\frac {6}{5} b B \,d^{2} e^{2}\right ) x^{5}+\left (A a d \,e^{3}+\frac {3}{2} A b \,d^{2} e^{2}+\frac {3}{2} B a \,d^{2} e^{2}+b B \,d^{3} e \right ) x^{4}+\left (2 A a \,d^{2} e^{2}+\frac {4}{3} A b \,d^{3} e +\frac {4}{3} B a \,d^{3} e +\frac {1}{3} b B \,d^{4}\right ) x^{3}+\left (2 A a \,d^{3} e +\frac {1}{2} A b \,d^{4}+\frac {1}{2} B a \,d^{4}\right ) x^{2}+A a \,d^{4} x\)	\(188\)
gosper	\(\frac {1}{7} b B \,e^{4} x^{7}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {1}{6} x^{6} B a \,e^{4}+\frac {2}{3} x^{6} b B d \,e^{3}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {4}{5} x^{5} A b d \,e^{3}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {6}{5} x^{5} b B \,d^{2} e^{2}+x^{4} A a d \,e^{3}+\frac {3}{2} x^{4} A b \,d^{2} e^{2}+\frac {3}{2} x^{4} B a \,d^{2} e^{2}+x^{4} b B \,d^{3} e +2 x^{3} A a \,d^{2} e^{2}+\frac {4}{3} x^{3} A b \,d^{3} e +\frac {4}{3} x^{3} B a \,d^{3} e +\frac {1}{3} x^{3} b B \,d^{4}+2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} A b \,d^{4}+\frac {1}{2} x^{2} B a \,d^{4}+A a \,d^{4} x\)	\(217\)
risch	\(\frac {1}{7} b B \,e^{4} x^{7}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {1}{6} x^{6} B a \,e^{4}+\frac {2}{3} x^{6} b B d \,e^{3}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {4}{5} x^{5} A b d \,e^{3}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {6}{5} x^{5} b B \,d^{2} e^{2}+x^{4} A a d \,e^{3}+\frac {3}{2} x^{4} A b \,d^{2} e^{2}+\frac {3}{2} x^{4} B a \,d^{2} e^{2}+x^{4} b B \,d^{3} e +2 x^{3} A a \,d^{2} e^{2}+\frac {4}{3} x^{3} A b \,d^{3} e +\frac {4}{3} x^{3} B a \,d^{3} e +\frac {1}{3} x^{3} b B \,d^{4}+2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} A b \,d^{4}+\frac {1}{2} x^{2} B a \,d^{4}+A a \,d^{4} x\)	\(217\)
parallelrisch	\(\frac {1}{7} b B \,e^{4} x^{7}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {1}{6} x^{6} B a \,e^{4}+\frac {2}{3} x^{6} b B d \,e^{3}+\frac {1}{5} x^{5} A a \,e^{4}+\frac {4}{5} x^{5} A b d \,e^{3}+\frac {4}{5} x^{5} B a d \,e^{3}+\frac {6}{5} x^{5} b B \,d^{2} e^{2}+x^{4} A a d \,e^{3}+\frac {3}{2} x^{4} A b \,d^{2} e^{2}+\frac {3}{2} x^{4} B a \,d^{2} e^{2}+x^{4} b B \,d^{3} e +2 x^{3} A a \,d^{2} e^{2}+\frac {4}{3} x^{3} A b \,d^{3} e +\frac {4}{3} x^{3} B a \,d^{3} e +\frac {1}{3} x^{3} b B \,d^{4}+2 x^{2} A a \,d^{3} e +\frac {1}{2} x^{2} A b \,d^{4}+\frac {1}{2} x^{2} B a \,d^{4}+A a \,d^{4} x\)	\(217\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (71) = 142\).

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.27 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=\frac {1}{7} \, B b e^{4} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (4 \, B b d e^{3} + {\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, B b d^{2} e^{2} + A a e^{4} + 4 \, {\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, B b d^{3} e + 2 \, A a d e^{3} + 3 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{4} + 6 \, A a d^{2} e^{2} + 4 \, {\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a d^{3} e + {\left (B a + A b\right )} d^{4}\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (71) = 142\).

Time = 0.03 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.94 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=A a d^{4} x + \frac {B b e^{4} x^{7}}{7} + x^{6} \left (\frac {A b e^{4}}{6} + \frac {B a e^{4}}{6} + \frac {2 B b d e^{3}}{3}\right ) + x^{5} \left (\frac {A a e^{4}}{5} + \frac {4 A b d e^{3}}{5} + \frac {4 B a d e^{3}}{5} + \frac {6 B b d^{2} e^{2}}{5}\right ) + x^{4} \left (A a d e^{3} + \frac {3 A b d^{2} e^{2}}{2} + \frac {3 B a d^{2} e^{2}}{2} + B b d^{3} e\right ) + x^{3} \cdot \left (2 A a d^{2} e^{2} + \frac {4 A b d^{3} e}{3} + \frac {4 B a d^{3} e}{3} + \frac {B b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a d^{3} e + \frac {A b d^{4}}{2} + \frac {B a d^{4}}{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (71) = 142\).

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.27 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=\frac {1}{7} \, B b e^{4} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (4 \, B b d e^{3} + {\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, B b d^{2} e^{2} + A a e^{4} + 4 \, {\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, B b d^{3} e + 2 \, A a d e^{3} + 3 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{4} + 6 \, A a d^{2} e^{2} + 4 \, {\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a d^{3} e + {\left (B a + A b\right )} d^{4}\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (71) = 142\).

Time = 0.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.81 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=\frac {1}{7} \, B b e^{4} x^{7} + \frac {2}{3} \, B b d e^{3} x^{6} + \frac {1}{6} \, B a e^{4} x^{6} + \frac {1}{6} \, A b e^{4} x^{6} + \frac {6}{5} \, B b d^{2} e^{2} x^{5} + \frac {4}{5} \, B a d e^{3} x^{5} + \frac {4}{5} \, A b d e^{3} x^{5} + \frac {1}{5} \, A a e^{4} x^{5} + B b d^{3} e x^{4} + \frac {3}{2} \, B a d^{2} e^{2} x^{4} + \frac {3}{2} \, A b d^{2} e^{2} x^{4} + A a d e^{3} x^{4} + \frac {1}{3} \, B b d^{4} x^{3} + \frac {4}{3} \, B a d^{3} e x^{3} + \frac {4}{3} \, A b d^{3} e x^{3} + 2 \, A a d^{2} e^{2} x^{3} + \frac {1}{2} \, B a d^{4} x^{2} + \frac {1}{2} \, A b d^{4} x^{2} + 2 \, A a d^{3} e x^{2} + A a d^{4} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.36 \[ \int (a+b x) (A+B x) (d+e x)^4 \, dx=x^3\,\left (\frac {B\,b\,d^4}{3}+\frac {4\,A\,b\,d^3\,e}{3}+\frac {4\,B\,a\,d^3\,e}{3}+2\,A\,a\,d^2\,e^2\right )+x^5\,\left (\frac {A\,a\,e^4}{5}+\frac {4\,A\,b\,d\,e^3}{5}+\frac {4\,B\,a\,d\,e^3}{5}+\frac {6\,B\,b\,d^2\,e^2}{5}\right )+x^2\,\left (\frac {A\,b\,d^4}{2}+\frac {B\,a\,d^4}{2}+2\,A\,a\,d^3\,e\right )+x^6\,\left (\frac {A\,b\,e^4}{6}+\frac {B\,a\,e^4}{6}+\frac {2\,B\,b\,d\,e^3}{3}\right )+A\,a\,d^4\,x+\frac {d\,e\,x^4\,\left (2\,A\,a\,e^2+2\,B\,b\,d^2+3\,A\,b\,d\,e+3\,B\,a\,d\,e\right )}{2}+\frac {B\,b\,e^4\,x^7}{7} \]

[In]

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

[Out]

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