\(\int (a+b x)^3 (A+B x) (d+e x) \, dx\) [1039]

Optimal result

Integrand size = 18, antiderivative size = 75 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^4}{4 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^5}{5 b^3}+\frac {B e (a+b x)^6}{6 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, 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} \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=\frac {(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac {(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac {B e (a+b x)^6}{6 b^3} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.73 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a (3 A b (b d+a e)+a B (3 b d+a e)) x^3+\frac {1}{4} b (3 a B (b d+a e)+A b (b d+3 a e)) x^4+\frac {1}{5} b^2 (b B d+A b e+3 a B e) x^5+\frac {1}{6} b^3 B e x^6 \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(69)=138\).

Time = 0.66 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.89

method	result	size
norman	\(\frac {b^{3} B e \,x^{6}}{6}+\left (\frac {1}{5} A \,b^{3} e +\frac {3}{5} B a \,b^{2} e +\frac {1}{5} b^{3} B d \right ) x^{5}+\left (\frac {3}{4} A a \,b^{2} e +\frac {1}{4} A \,b^{3} d +\frac {3}{4} B \,a^{2} b e +\frac {3}{4} B a \,b^{2} d \right ) x^{4}+\left (A \,a^{2} b e +A a \,b^{2} d +\frac {1}{3} B \,a^{3} e +B \,a^{2} b d \right ) x^{3}+\left (\frac {1}{2} a^{3} A e +\frac {3}{2} A \,a^{2} b d +\frac {1}{2} B \,a^{3} d \right ) x^{2}+a^{3} A d x\)	\(142\)
default	\(\frac {b^{3} B e \,x^{6}}{6}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) e +b^{3} B d \right ) x^{5}}{5}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) e +\left (b^{3} A +3 a \,b^{2} B \right ) d \right ) x^{4}}{4}+\frac {\left (\left (3 a^{2} b A +a^{3} B \right ) e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) d \right ) x^{3}}{3}+\frac {\left (a^{3} A e +\left (3 a^{2} b A +a^{3} B \right ) d \right ) x^{2}}{2}+a^{3} A d x\)	\(149\)
gosper	\(\frac {1}{6} b^{3} B e \,x^{6}+\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} B a \,b^{2} e +\frac {1}{5} x^{5} b^{3} B d +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {1}{4} x^{4} A \,b^{3} d +\frac {3}{4} x^{4} B \,a^{2} b e +\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} A a \,b^{2} d +\frac {1}{3} x^{3} B \,a^{3} e +x^{3} B \,a^{2} b d +\frac {1}{2} x^{2} a^{3} A e +\frac {3}{2} x^{2} A \,a^{2} b d +\frac {1}{2} x^{2} B \,a^{3} d +a^{3} A d x\)	\(164\)
risch	\(\frac {1}{6} b^{3} B e \,x^{6}+\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} B a \,b^{2} e +\frac {1}{5} x^{5} b^{3} B d +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {1}{4} x^{4} A \,b^{3} d +\frac {3}{4} x^{4} B \,a^{2} b e +\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} A a \,b^{2} d +\frac {1}{3} x^{3} B \,a^{3} e +x^{3} B \,a^{2} b d +\frac {1}{2} x^{2} a^{3} A e +\frac {3}{2} x^{2} A \,a^{2} b d +\frac {1}{2} x^{2} B \,a^{3} d +a^{3} A d x\)	\(164\)
parallelrisch	\(\frac {1}{6} b^{3} B e \,x^{6}+\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} B a \,b^{2} e +\frac {1}{5} x^{5} b^{3} B d +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {1}{4} x^{4} A \,b^{3} d +\frac {3}{4} x^{4} B \,a^{2} b e +\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} A a \,b^{2} d +\frac {1}{3} x^{3} B \,a^{3} e +x^{3} B \,a^{2} b d +\frac {1}{2} x^{2} a^{3} A e +\frac {3}{2} x^{2} A \,a^{2} b d +\frac {1}{2} x^{2} B \,a^{3} d +a^{3} A d x\)	\(164\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (69) = 138\).

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.95 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=\frac {1}{6} \, B b^{3} e x^{6} + A a^{3} d x + \frac {1}{5} \, {\left (B b^{3} d + {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (73) = 146\).

Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.24 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=A a^{3} d x + \frac {B b^{3} e x^{6}}{6} + x^{5} \left (\frac {A b^{3} e}{5} + \frac {3 B a b^{2} e}{5} + \frac {B b^{3} d}{5}\right ) + x^{4} \cdot \left (\frac {3 A a b^{2} e}{4} + \frac {A b^{3} d}{4} + \frac {3 B a^{2} b e}{4} + \frac {3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a b^{2} d + \frac {B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {3 A a^{2} b d}{2} + \frac {B a^{3} d}{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (69) = 138\).

Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.95 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=\frac {1}{6} \, B b^{3} e x^{6} + A a^{3} d x + \frac {1}{5} \, {\left (B b^{3} d + {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (69) = 138\).

Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.17 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=\frac {1}{6} \, B b^{3} e x^{6} + \frac {1}{5} \, B b^{3} d x^{5} + \frac {3}{5} \, B a b^{2} e x^{5} + \frac {1}{5} \, A b^{3} e x^{5} + \frac {3}{4} \, B a b^{2} d x^{4} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {3}{4} \, B a^{2} b e x^{4} + \frac {3}{4} \, A a b^{2} e x^{4} + B a^{2} b d x^{3} + A a b^{2} d x^{3} + \frac {1}{3} \, B a^{3} e x^{3} + A a^{2} b e x^{3} + \frac {1}{2} \, B a^{3} d x^{2} + \frac {3}{2} \, A a^{2} b d x^{2} + \frac {1}{2} \, A a^{3} e x^{2} + A a^{3} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.88 \[ \int (a+b x)^3 (A+B x) (d+e x) \, dx=x^2\,\left (\frac {A\,a^3\,e}{2}+\frac {B\,a^3\,d}{2}+\frac {3\,A\,a^2\,b\,d}{2}\right )+x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,B\,a\,b^2\,e}{5}\right )+x^3\,\left (\frac {B\,a^3\,e}{3}+A\,a\,b^2\,d+A\,a^2\,b\,e+B\,a^2\,b\,d\right )+x^4\,\left (\frac {A\,b^3\,d}{4}+\frac {3\,A\,a\,b^2\,e}{4}+\frac {3\,B\,a\,b^2\,d}{4}+\frac {3\,B\,a^2\,b\,e}{4}\right )+A\,a^3\,d\,x+\frac {B\,b^3\,e\,x^6}{6} \]

[In]

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

[Out]

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