\(\int x (a+b x)^5 (A+B x) \, dx\) [123]

Optimal result

Integrand size = 14, antiderivative size = 61 \[ \int x (a+b x)^5 (A+B x) \, dx=-\frac {a (A b-a B) (a+b x)^6}{6 b^3}+\frac {(A b-2 a B) (a+b x)^7}{7 b^3}+\frac {B (a+b x)^8}{8 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \[ \int x (a+b x)^5 (A+B x) \, dx=\frac {(a+b x)^7 (A b-2 a B)}{7 b^3}-\frac {a (a+b x)^6 (A b-a B)}{6 b^3}+\frac {B (a+b x)^8}{8 b^3} \]

[In]

Int[x*(a + b*x)^5*(A + B*x),x]

[Out]

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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B) (a+b x)^5}{b^2}+\frac {(A b-2 a B) (a+b x)^6}{b^2}+\frac {B (a+b x)^7}{b^2}\right ) \, dx \\ & = -\frac {a (A b-a B) (a+b x)^6}{6 b^3}+\frac {(A b-2 a B) (a+b x)^7}{7 b^3}+\frac {B (a+b x)^8}{8 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.89 \[ \int x (a+b x)^5 (A+B x) \, dx=\frac {1}{2} a^5 A x^2+\frac {1}{3} a^4 (5 A b+a B) x^3+\frac {5}{4} a^3 b (2 A b+a B) x^4+2 a^2 b^2 (A b+a B) x^5+\frac {5}{6} a b^3 (A b+2 a B) x^6+\frac {1}{7} b^4 (A b+5 a B) x^7+\frac {1}{8} b^5 B x^8 \]

[In]

Integrate[x*(a + b*x)^5*(A + B*x),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs. \(2(55)=110\).

Time = 0.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.98

method	result	size
norman	\(\frac {b^{5} B \,x^{8}}{8}+\left (\frac {1}{7} b^{5} A +\frac {5}{7} a \,b^{4} B \right ) x^{7}+\left (\frac {5}{6} a \,b^{4} A +\frac {5}{3} a^{2} b^{3} B \right ) x^{6}+\left (2 a^{2} b^{3} A +2 a^{3} b^{2} B \right ) x^{5}+\left (\frac {5}{2} a^{3} b^{2} A +\frac {5}{4} a^{4} b B \right ) x^{4}+\left (\frac {5}{3} a^{4} b A +\frac {1}{3} a^{5} B \right ) x^{3}+\frac {a^{5} A \,x^{2}}{2}\)	\(121\)
default	\(\frac {b^{5} B \,x^{8}}{8}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{7}}{7}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{6}}{6}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{5}}{5}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{4}}{4}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{3}}{3}+\frac {a^{5} A \,x^{2}}{2}\)	\(124\)
gosper	\(\frac {1}{8} b^{5} B \,x^{8}+\frac {1}{7} x^{7} b^{5} A +\frac {5}{7} x^{7} a \,b^{4} B +\frac {5}{6} x^{6} a \,b^{4} A +\frac {5}{3} x^{6} a^{2} b^{3} B +2 A \,a^{2} b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+\frac {5}{2} x^{4} a^{3} b^{2} A +\frac {5}{4} x^{4} a^{4} b B +\frac {5}{3} x^{3} a^{4} b A +\frac {1}{3} x^{3} a^{5} B +\frac {1}{2} a^{5} A \,x^{2}\)	\(126\)
risch	\(\frac {1}{8} b^{5} B \,x^{8}+\frac {1}{7} x^{7} b^{5} A +\frac {5}{7} x^{7} a \,b^{4} B +\frac {5}{6} x^{6} a \,b^{4} A +\frac {5}{3} x^{6} a^{2} b^{3} B +2 A \,a^{2} b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+\frac {5}{2} x^{4} a^{3} b^{2} A +\frac {5}{4} x^{4} a^{4} b B +\frac {5}{3} x^{3} a^{4} b A +\frac {1}{3} x^{3} a^{5} B +\frac {1}{2} a^{5} A \,x^{2}\)	\(126\)
parallelrisch	\(\frac {1}{8} b^{5} B \,x^{8}+\frac {1}{7} x^{7} b^{5} A +\frac {5}{7} x^{7} a \,b^{4} B +\frac {5}{6} x^{6} a \,b^{4} A +\frac {5}{3} x^{6} a^{2} b^{3} B +2 A \,a^{2} b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+\frac {5}{2} x^{4} a^{3} b^{2} A +\frac {5}{4} x^{4} a^{4} b B +\frac {5}{3} x^{3} a^{4} b A +\frac {1}{3} x^{3} a^{5} B +\frac {1}{2} a^{5} A \,x^{2}\)	\(126\)

[In]

int(x*(b*x+a)^5*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

1/8*b^5*B*x^8+(1/7*b^5*A+5/7*a*b^4*B)*x^7+(5/6*a*b^4*A+5/3*a^2*b^3*B)*x^6+(2*A*a^2*b^3+2*B*a^3*b^2)*x^5+(5/2*a
^3*b^2*A+5/4*a^4*b*B)*x^4+(5/3*a^4*b*A+1/3*a^5*B)*x^3+1/2*a^5*A*x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (56) = 112\).

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int x (a+b x)^5 (A+B x) \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {1}{2} \, A a^{5} x^{2} + \frac {1}{7} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \]

[In]

integrate(x*(b*x+a)^5*(B*x+A),x, algorithm="fricas")

[Out]

1/8*B*b^5*x^8 + 1/2*A*a^5*x^2 + 1/7*(5*B*a*b^4 + A*b^5)*x^7 + 5/6*(2*B*a^2*b^3 + A*a*b^4)*x^6 + 2*(B*a^3*b^2 +
A*a^2*b^3)*x^5 + 5/4*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (54) = 108\).

Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.20 \[ \int x (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{2}}{2} + \frac {B b^{5} x^{8}}{8} + x^{7} \left (\frac {A b^{5}}{7} + \frac {5 B a b^{4}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a b^{4}}{6} + \frac {5 B a^{2} b^{3}}{3}\right ) + x^{5} \cdot \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{4} \cdot \left (\frac {5 A a^{3} b^{2}}{2} + \frac {5 B a^{4} b}{4}\right ) + x^{3} \cdot \left (\frac {5 A a^{4} b}{3} + \frac {B a^{5}}{3}\right ) \]

[In]

integrate(x*(b*x+a)**5*(B*x+A),x)

[Out]

A*a**5*x**2/2 + B*b**5*x**8/8 + x**7*(A*b**5/7 + 5*B*a*b**4/7) + x**6*(5*A*a*b**4/6 + 5*B*a**2*b**3/3) + x**5*
(2*A*a**2*b**3 + 2*B*a**3*b**2) + x**4*(5*A*a**3*b**2/2 + 5*B*a**4*b/4) + x**3*(5*A*a**4*b/3 + B*a**5/3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (56) = 112\).

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int x (a+b x)^5 (A+B x) \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {1}{2} \, A a^{5} x^{2} + \frac {1}{7} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \]

[In]

integrate(x*(b*x+a)^5*(B*x+A),x, algorithm="maxima")

[Out]

1/8*B*b^5*x^8 + 1/2*A*a^5*x^2 + 1/7*(5*B*a*b^4 + A*b^5)*x^7 + 5/6*(2*B*a^2*b^3 + A*a*b^4)*x^6 + 2*(B*a^3*b^2 +
A*a^2*b^3)*x^5 + 5/4*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1/3*(B*a^5 + 5*A*a^4*b)*x^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (56) = 112\).

Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.05 \[ \int x (a+b x)^5 (A+B x) \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {5}{7} \, B a b^{4} x^{7} + \frac {1}{7} \, A b^{5} x^{7} + \frac {5}{3} \, B a^{2} b^{3} x^{6} + \frac {5}{6} \, A a b^{4} x^{6} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac {5}{4} \, B a^{4} b x^{4} + \frac {5}{2} \, A a^{3} b^{2} x^{4} + \frac {1}{3} \, B a^{5} x^{3} + \frac {5}{3} \, A a^{4} b x^{3} + \frac {1}{2} \, A a^{5} x^{2} \]

[In]

integrate(x*(b*x+a)^5*(B*x+A),x, algorithm="giac")

[Out]

1/8*B*b^5*x^8 + 5/7*B*a*b^4*x^7 + 1/7*A*b^5*x^7 + 5/3*B*a^2*b^3*x^6 + 5/6*A*a*b^4*x^6 + 2*B*a^3*b^2*x^5 + 2*A*
a^2*b^3*x^5 + 5/4*B*a^4*b*x^4 + 5/2*A*a^3*b^2*x^4 + 1/3*B*a^5*x^3 + 5/3*A*a^4*b*x^3 + 1/2*A*a^5*x^2

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.75 \[ \int x (a+b x)^5 (A+B x) \, dx=x^3\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )+x^7\,\left (\frac {A\,b^5}{7}+\frac {5\,B\,a\,b^4}{7}\right )+\frac {A\,a^5\,x^2}{2}+\frac {B\,b^5\,x^8}{8}+2\,a^2\,b^2\,x^5\,\left (A\,b+B\,a\right )+\frac {5\,a^3\,b\,x^4\,\left (2\,A\,b+B\,a\right )}{4}+\frac {5\,a\,b^3\,x^6\,\left (A\,b+2\,B\,a\right )}{6} \]

[In]

int(x*(A + B*x)*(a + b*x)^5,x)

[Out]

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