\(\int x^2 (a+b x)^3 (A+B x) \, dx\) [106]

Optimal result

Integrand size = 16, antiderivative size = 75 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {3}{5} a b (A b+a B) x^5+\frac {1}{6} b^2 (A b+3 a B) x^6+\frac {1}{7} b^3 B x^7 \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 x^4 (a B+3 A b)+\frac {1}{6} b^2 x^6 (3 a B+A b)+\frac {3}{5} a b x^5 (a B+A b)+\frac {1}{7} b^3 B x^7 \]

[In]

Int[x^2*(a + b*x)^3*(A + B*x),x]

[Out]

(a^3*A*x^3)/3 + (a^2*(3*A*b + a*B)*x^4)/4 + (3*a*b*(A*b + a*B)*x^5)/5 + (b^2*(A*b + 3*a*B)*x^6)/6 + (b^3*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^3 A x^2+a^2 (3 A b+a B) x^3+3 a b (A b+a B) x^4+b^2 (A b+3 a B) x^5+b^3 B x^6\right ) \, dx \\ & = \frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {3}{5} a b (A b+a B) x^5+\frac {1}{6} b^2 (A b+3 a B) x^6+\frac {1}{7} b^3 B x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {3}{5} a b (A b+a B) x^5+\frac {1}{6} b^2 (A b+3 a B) x^6+\frac {1}{7} b^3 B x^7 \]

[In]

Integrate[x^2*(a + b*x)^3*(A + B*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {b^{3} B \,x^{7}}{7}+\left (\frac {1}{6} b^{3} A +\frac {1}{2} a \,b^{2} B \right ) x^{6}+\left (\frac {3}{5} a \,b^{2} A +\frac {3}{5} a^{2} b B \right ) x^{5}+\left (\frac {3}{4} a^{2} b A +\frac {1}{4} a^{3} B \right ) x^{4}+\frac {a^{3} A \,x^{3}}{3}\)	\(75\)
default	\(\frac {b^{3} B \,x^{7}}{7}+\frac {\left (b^{3} A +3 a \,b^{2} B \right ) x^{6}}{6}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{5}}{5}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{4}}{4}+\frac {a^{3} A \,x^{3}}{3}\)	\(76\)
gosper	\(\frac {1}{7} b^{3} B \,x^{7}+\frac {1}{6} x^{6} b^{3} A +\frac {1}{2} x^{6} a \,b^{2} B +\frac {3}{5} x^{5} a \,b^{2} A +\frac {3}{5} x^{5} a^{2} b B +\frac {3}{4} x^{4} a^{2} b A +\frac {1}{4} x^{4} a^{3} B +\frac {1}{3} a^{3} A \,x^{3}\)	\(78\)
risch	\(\frac {1}{7} b^{3} B \,x^{7}+\frac {1}{6} x^{6} b^{3} A +\frac {1}{2} x^{6} a \,b^{2} B +\frac {3}{5} x^{5} a \,b^{2} A +\frac {3}{5} x^{5} a^{2} b B +\frac {3}{4} x^{4} a^{2} b A +\frac {1}{4} x^{4} a^{3} B +\frac {1}{3} a^{3} A \,x^{3}\)	\(78\)
parallelrisch	\(\frac {1}{7} b^{3} B \,x^{7}+\frac {1}{6} x^{6} b^{3} A +\frac {1}{2} x^{6} a \,b^{2} B +\frac {3}{5} x^{5} a \,b^{2} A +\frac {3}{5} x^{5} a^{2} b B +\frac {3}{4} x^{4} a^{2} b A +\frac {1}{4} x^{4} a^{3} B +\frac {1}{3} a^{3} A \,x^{3}\)	\(78\)

[In]

int(x^2*(b*x+a)^3*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {1}{7} \, B b^{3} x^{7} + \frac {1}{3} \, A a^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \]

[In]

integrate(x^2*(b*x+a)^3*(B*x+A),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {A a^{3} x^{3}}{3} + \frac {B b^{3} x^{7}}{7} + x^{6} \left (\frac {A b^{3}}{6} + \frac {B a b^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 A a b^{2}}{5} + \frac {3 B a^{2} b}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b}{4} + \frac {B a^{3}}{4}\right ) \]

[In]

integrate(x**2*(b*x+a)**3*(B*x+A),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {1}{7} \, B b^{3} x^{7} + \frac {1}{3} \, A a^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \]

[In]

integrate(x^2*(b*x+a)^3*(B*x+A),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=\frac {1}{7} \, B b^{3} x^{7} + \frac {1}{2} \, B a b^{2} x^{6} + \frac {1}{6} \, A b^{3} x^{6} + \frac {3}{5} \, B a^{2} b x^{5} + \frac {3}{5} \, A a b^{2} x^{5} + \frac {1}{4} \, B a^{3} x^{4} + \frac {3}{4} \, A a^{2} b x^{4} + \frac {1}{3} \, A a^{3} x^{3} \]

[In]

integrate(x^2*(b*x+a)^3*(B*x+A),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int x^2 (a+b x)^3 (A+B x) \, dx=x^4\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+x^6\,\left (\frac {A\,b^3}{6}+\frac {B\,a\,b^2}{2}\right )+\frac {A\,a^3\,x^3}{3}+\frac {B\,b^3\,x^7}{7}+\frac {3\,a\,b\,x^5\,\left (A\,b+B\,a\right )}{5} \]

[In]

int(x^2*(A + B*x)*(a + b*x)^3,x)

[Out]

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