\(\int x^4 (a+b x)^2 (A+B x) \, dx\) [91]

Optimal result

Integrand size = 16, antiderivative size = 55 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {1}{5} a^2 A x^5+\frac {1}{6} a (2 A b+a B) x^6+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{8} b^2 B x^8 \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {1}{5} a^2 A x^5+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{6} a x^6 (a B+2 A b)+\frac {1}{8} b^2 B x^8 \]

[In]

Int[x^4*(a + b*x)^2*(A + B*x),x]

[Out]

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

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^2 A x^4+a (2 A b+a B) x^5+b (A b+2 a B) x^6+b^2 B x^7\right ) \, dx \\ & = \frac {1}{5} a^2 A x^5+\frac {1}{6} a (2 A b+a B) x^6+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{8} b^2 B x^8 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {1}{5} a^2 A x^5+\frac {1}{6} a (2 A b+a B) x^6+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{8} b^2 B x^8 \]

[In]

Integrate[x^4*(a + b*x)^2*(A + B*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {b^{2} B \,x^{8}}{8}+\frac {\left (b^{2} A +2 a b B \right ) x^{7}}{7}+\frac {\left (2 a b A +a^{2} B \right ) x^{6}}{6}+\frac {a^{2} A \,x^{5}}{5}\)	\(52\)
norman	\(\frac {b^{2} B \,x^{8}}{8}+\left (\frac {1}{7} b^{2} A +\frac {2}{7} a b B \right ) x^{7}+\left (\frac {1}{3} a b A +\frac {1}{6} a^{2} B \right ) x^{6}+\frac {a^{2} A \,x^{5}}{5}\)	\(52\)
gosper	\(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {1}{3} x^{6} a b A +\frac {1}{6} x^{6} a^{2} B +\frac {1}{5} a^{2} A \,x^{5}\)	\(54\)
risch	\(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {1}{3} x^{6} a b A +\frac {1}{6} x^{6} a^{2} B +\frac {1}{5} a^{2} A \,x^{5}\)	\(54\)
parallelrisch	\(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {1}{3} x^{6} a b A +\frac {1}{6} x^{6} a^{2} B +\frac {1}{5} a^{2} A \,x^{5}\)	\(54\)

[In]

int(x^4*(b*x+a)^2*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {1}{8} \, B b^{2} x^{8} + \frac {1}{5} \, A a^{2} x^{5} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{2} + 2 \, A a b\right )} x^{6} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {A a^{2} x^{5}}{5} + \frac {B b^{2} x^{8}}{8} + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 B a b}{7}\right ) + x^{6} \left (\frac {A a b}{3} + \frac {B a^{2}}{6}\right ) \]

[In]

integrate(x**4*(b*x+a)**2*(B*x+A),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {1}{8} \, B b^{2} x^{8} + \frac {1}{5} \, A a^{2} x^{5} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{2} + 2 \, A a b\right )} x^{6} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=\frac {1}{8} \, B b^{2} x^{8} + \frac {2}{7} \, B a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{6} \, B a^{2} x^{6} + \frac {1}{3} \, A a b x^{6} + \frac {1}{5} \, A a^{2} x^{5} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^4 (a+b x)^2 (A+B x) \, dx=x^6\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+x^7\,\left (\frac {A\,b^2}{7}+\frac {2\,B\,a\,b}{7}\right )+\frac {A\,a^2\,x^5}{5}+\frac {B\,b^2\,x^8}{8} \]

[In]

int(x^4*(A + B*x)*(a + b*x)^2,x)

[Out]

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