\(\int x^4 (a+b x) (A+B x) \, dx\) [80]

Optimal result

Integrand size = 14, antiderivative size = 33 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {1}{5} a A x^5+\frac {1}{6} (A b+a B) x^6+\frac {1}{7} b B x^7 \]

[Out]

1/5*a*A*x^5+1/6*(A*b+B*a)*x^6+1/7*b*B*x^7

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, 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} \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {1}{6} x^6 (a B+A b)+\frac {1}{5} a A x^5+\frac {1}{7} b B x^7 \]

[In]

Int[x^4*(a + b*x)*(A + B*x),x]

[Out]

(a*A*x^5)/5 + ((A*b + a*B)*x^6)/6 + (b*B*x^7)/7

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 (a A x^4+(A b+a B) x^5+b B x^6\right ) \, dx \\ & = \frac {1}{5} a A x^5+\frac {1}{6} (A b+a B) x^6+\frac {1}{7} b B x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {1}{5} a A x^5+\frac {1}{6} (A b+a B) x^6+\frac {1}{7} b B x^7 \]

[In]

Integrate[x^4*(a + b*x)*(A + B*x),x]

[Out]

(a*A*x^5)/5 + ((A*b + a*B)*x^6)/6 + (b*B*x^7)/7

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {a A \,x^{5}}{5}+\frac {\left (A b +B a \right ) x^{6}}{6}+\frac {b B \,x^{7}}{7}\)	\(28\)
norman	\(\frac {b B \,x^{7}}{7}+\left (\frac {A b}{6}+\frac {B a}{6}\right ) x^{6}+\frac {a A \,x^{5}}{5}\)	\(29\)
gosper	\(\frac {1}{7} b B \,x^{7}+\frac {1}{6} x^{6} A b +\frac {1}{6} x^{6} B a +\frac {1}{5} a A \,x^{5}\)	\(30\)
risch	\(\frac {1}{7} b B \,x^{7}+\frac {1}{6} x^{6} A b +\frac {1}{6} x^{6} B a +\frac {1}{5} a A \,x^{5}\)	\(30\)
parallelrisch	\(\frac {1}{7} b B \,x^{7}+\frac {1}{6} x^{6} A b +\frac {1}{6} x^{6} B a +\frac {1}{5} a A \,x^{5}\)	\(30\)

[In]

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

[Out]

1/5*a*A*x^5+1/6*(A*b+B*a)*x^6+1/7*b*B*x^7

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {1}{7} \, B b x^{7} + \frac {1}{5} \, A a x^{5} + \frac {1}{6} \, {\left (B a + A b\right )} x^{6} \]

[In]

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

[Out]

1/7*B*b*x^7 + 1/5*A*a*x^5 + 1/6*(B*a + A*b)*x^6

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {A a x^{5}}{5} + \frac {B b x^{7}}{7} + x^{6} \left (\frac {A b}{6} + \frac {B a}{6}\right ) \]

[In]

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

[Out]

A*a*x**5/5 + B*b*x**7/7 + x**6*(A*b/6 + B*a/6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {1}{7} \, B b x^{7} + \frac {1}{5} \, A a x^{5} + \frac {1}{6} \, {\left (B a + A b\right )} x^{6} \]

[In]

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

[Out]

1/7*B*b*x^7 + 1/5*A*a*x^5 + 1/6*(B*a + A*b)*x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {1}{7} \, B b x^{7} + \frac {1}{6} \, B a x^{6} + \frac {1}{6} \, A b x^{6} + \frac {1}{5} \, A a x^{5} \]

[In]

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

[Out]

1/7*B*b*x^7 + 1/6*B*a*x^6 + 1/6*A*b*x^6 + 1/5*A*a*x^5

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int x^4 (a+b x) (A+B x) \, dx=\frac {B\,b\,x^7}{7}+\left (\frac {A\,b}{6}+\frac {B\,a}{6}\right )\,x^6+\frac {A\,a\,x^5}{5} \]

[In]

int(x^4*(A + B*x)*(a + b*x),x)

[Out]

x^6*((A*b)/6 + (B*a)/6) + (A*a*x^5)/5 + (B*b*x^7)/7