3.93 \(\int (a+b x)^5 (A+B x) \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^6 (A b-a B)}{6 b^2}+\frac{B (a+b x)^7}{7 b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0500009, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x)^6 (A b-a B)}{6 b^2}+\frac{B (a+b x)^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5*(A + B*x),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.6282, size = 31, normalized size = 0.82 \[ \frac{B \left (a + b x\right )^{7}}{7 b^{2}} + \frac{\left (a + b x\right )^{6} \left (A b - B a\right )}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A),x)

[Out]

B*(a + b*x)**7/(7*b**2) + (a + b*x)**6*(A*b - B*a)/(6*b**2)

_______________________________________________________________________________________

Mathematica [B]  time = 0.0256271, size = 109, normalized size = 2.87 \[ a^5 A x+\frac{1}{2} a^4 x^2 (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5*(A + B*x),x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 121, normalized size = 3.2 \[{\frac{{b}^{5}B{x}^{7}}{7}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{2}}{2}}+{a}^{5}Ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A),x)

[Out]

1/7*b^5*B*x^7+1/6*(A*b^5+5*B*a*b^4)*x^6+1/5*(5*A*a*b^4+10*B*a^2*b^3)*x^5+1/4*(10
*A*a^2*b^3+10*B*a^3*b^2)*x^4+1/3*(10*A*a^3*b^2+5*B*a^4*b)*x^3+1/2*(5*A*a^4*b+B*a
^5)*x^2+a^5*A*x

_______________________________________________________________________________________

Maxima [A]  time = 1.34996, size = 155, normalized size = 4.08 \[ \frac{1}{7} \, B b^{5} x^{7} + A a^{5} x + \frac{1}{6} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} +{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac{5}{2} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.179545, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} b^{5} B + \frac{5}{6} x^{6} b^{4} a B + \frac{1}{6} x^{6} b^{5} A + 2 x^{5} b^{3} a^{2} B + x^{5} b^{4} a A + \frac{5}{2} x^{4} b^{2} a^{3} B + \frac{5}{2} x^{4} b^{3} a^{2} A + \frac{5}{3} x^{3} b a^{4} B + \frac{10}{3} x^{3} b^{2} a^{3} A + \frac{1}{2} x^{2} a^{5} B + \frac{5}{2} x^{2} b a^{4} A + x a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*b^5*B + 5/6*x^6*b^4*a*B + 1/6*x^6*b^5*A + 2*x^5*b^3*a^2*B + x^5*b^4*a*A
+ 5/2*x^4*b^2*a^3*B + 5/2*x^4*b^3*a^2*A + 5/3*x^3*b*a^4*B + 10/3*x^3*b^2*a^3*A +
1/2*x^2*a^5*B + 5/2*x^2*b*a^4*A + x*a^5*A

_______________________________________________________________________________________

Sympy [A]  time = 0.174539, size = 129, normalized size = 3.39 \[ A a^{5} x + \frac{B b^{5} x^{7}}{7} + x^{6} \left (\frac{A b^{5}}{6} + \frac{5 B a b^{4}}{6}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{4} \left (\frac{5 A a^{2} b^{3}}{2} + \frac{5 B a^{3} b^{2}}{2}\right ) + x^{3} \left (\frac{10 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{3}\right ) + x^{2} \left (\frac{5 A a^{4} b}{2} + \frac{B a^{5}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A),x)

[Out]

A*a**5*x + B*b**5*x**7/7 + x**6*(A*b**5/6 + 5*B*a*b**4/6) + x**5*(A*a*b**4 + 2*B
*a**2*b**3) + x**4*(5*A*a**2*b**3/2 + 5*B*a**3*b**2/2) + x**3*(10*A*a**3*b**2/3
+ 5*B*a**4*b/3) + x**2*(5*A*a**4*b/2 + B*a**5/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.257203, size = 163, normalized size = 4.29 \[ \frac{1}{7} \, B b^{5} x^{7} + \frac{5}{6} \, B a b^{4} x^{6} + \frac{1}{6} \, A b^{5} x^{6} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + \frac{5}{2} \, B a^{3} b^{2} x^{4} + \frac{5}{2} \, A a^{2} b^{3} x^{4} + \frac{5}{3} \, B a^{4} b x^{3} + \frac{10}{3} \, A a^{3} b^{2} x^{3} + \frac{1}{2} \, B a^{5} x^{2} + \frac{5}{2} \, A a^{4} b x^{2} + A a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b^5*x^7 + 5/6*B*a*b^4*x^6 + 1/6*A*b^5*x^6 + 2*B*a^2*b^3*x^5 + A*a*b^4*x^5
+ 5/2*B*a^3*b^2*x^4 + 5/2*A*a^2*b^3*x^4 + 5/3*B*a^4*b*x^3 + 10/3*A*a^3*b^2*x^3 +
1/2*B*a^5*x^2 + 5/2*A*a^4*b*x^2 + A*a^5*x