3.96 \(\int \frac{(a+b x)^5 (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=108 \[ -\frac{a^5 A}{2 x^2}-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b \log (x) (a B+2 A b)+10 a^2 b^2 x (a B+A b)+\frac{1}{3} b^4 x^3 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{4} b^5 B x^4 \]

[Out]

-(a^5*A)/(2*x^2) - (a^4*(5*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(
A*b + 2*a*B)*x^2)/2 + (b^4*(A*b + 5*a*B)*x^3)/3 + (b^5*B*x^4)/4 + 5*a^3*b*(2*A*b
+ a*B)*Log[x]

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Rubi [A]  time = 0.166551, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^5 A}{2 x^2}-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b \log (x) (a B+2 A b)+10 a^2 b^2 x (a B+A b)+\frac{1}{3} b^4 x^3 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^3,x]

[Out]

-(a^5*A)/(2*x^2) - (a^4*(5*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(
A*b + 2*a*B)*x^2)/2 + (b^4*(A*b + 5*a*B)*x^3)/3 + (b^5*B*x^4)/4 + 5*a^3*b*(2*A*b
+ a*B)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{2 x^{2}} + \frac{B b^{5} x^{4}}{4} - \frac{a^{4} \left (5 A b + B a\right )}{x} + 5 a^{3} b \left (2 A b + B a\right ) \log{\left (x \right )} + 10 a^{2} b^{2} x \left (A b + B a\right ) + 5 a b^{3} \left (A b + 2 B a\right ) \int x\, dx + \frac{b^{4} x^{3} \left (A b + 5 B a\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**3,x)

[Out]

-A*a**5/(2*x**2) + B*b**5*x**4/4 - a**4*(5*A*b + B*a)/x + 5*a**3*b*(2*A*b + B*a)
*log(x) + 10*a**2*b**2*x*(A*b + B*a) + 5*a*b**3*(A*b + 2*B*a)*Integral(x, x) + b
**4*x**3*(A*b + 5*B*a)/3

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Mathematica [A]  time = 0.0643415, size = 106, normalized size = 0.98 \[ -\frac{a^5 (A+2 B x)}{2 x^2}-\frac{5 a^4 A b}{x}+5 a^3 b \log (x) (a B+2 A b)+10 a^3 b^2 B x+5 a^2 b^3 x (2 A+B x)+\frac{5}{6} a b^4 x^2 (3 A+2 B x)+\frac{1}{12} b^5 x^3 (4 A+3 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^3,x]

[Out]

(-5*a^4*A*b)/x + 10*a^3*b^2*B*x + 5*a^2*b^3*x*(2*A + B*x) - (a^5*(A + 2*B*x))/(2
*x^2) + (5*a*b^4*x^2*(3*A + 2*B*x))/6 + (b^5*x^3*(4*A + 3*B*x))/12 + 5*a^3*b*(2*
A*b + a*B)*Log[x]

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Maple [A]  time = 0.01, size = 120, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{4}}{4}}+{\frac{A{x}^{3}{b}^{5}}{3}}+{\frac{5\,B{x}^{3}a{b}^{4}}{3}}+{\frac{5\,A{x}^{2}a{b}^{4}}{2}}+5\,B{x}^{2}{a}^{2}{b}^{3}+10\,Ax{a}^{2}{b}^{3}+10\,Bx{a}^{3}{b}^{2}+10\,A\ln \left ( x \right ){a}^{3}{b}^{2}+5\,B\ln \left ( x \right ){a}^{4}b-{\frac{A{a}^{5}}{2\,{x}^{2}}}-5\,{\frac{{a}^{4}bA}{x}}-{\frac{{a}^{5}B}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.35645, size = 157, normalized size = 1.45 \[ \frac{1}{4} \, B b^{5} x^{4} + \frac{1}{3} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{3} + \frac{5}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203217, size = 163, normalized size = 1.51 \[ \frac{3 \, B b^{5} x^{6} - 6 \, A a^{5} + 4 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 60 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} \log \left (x\right ) - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^3,x, algorithm="fricas")

[Out]

1/12*(3*B*b^5*x^6 - 6*A*a^5 + 4*(5*B*a*b^4 + A*b^5)*x^5 + 30*(2*B*a^2*b^3 + A*a*
b^4)*x^4 + 120*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 60*(B*a^4*b + 2*A*a^3*b^2)*x^2*log(
x) - 12*(B*a^5 + 5*A*a^4*b)*x)/x^2

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Sympy [A]  time = 2.67121, size = 121, normalized size = 1.12 \[ \frac{B b^{5} x^{4}}{4} + 5 a^{3} b \left (2 A b + B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{5}}{3} + \frac{5 B a b^{4}}{3}\right ) + x^{2} \left (\frac{5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) - \frac{A a^{5} + x \left (10 A a^{4} b + 2 B a^{5}\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**3,x)

[Out]

B*b**5*x**4/4 + 5*a**3*b*(2*A*b + B*a)*log(x) + x**3*(A*b**5/3 + 5*B*a*b**4/3) +
x**2*(5*A*a*b**4/2 + 5*B*a**2*b**3) + x*(10*A*a**2*b**3 + 10*B*a**3*b**2) - (A*
a**5 + x*(10*A*a**4*b + 2*B*a**5))/(2*x**2)

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GIAC/XCAS [A]  time = 0.255587, size = 161, normalized size = 1.49 \[ \frac{1}{4} \, B b^{5} x^{4} + \frac{5}{3} \, B a b^{4} x^{3} + \frac{1}{3} \, A b^{5} x^{3} + 5 \, B a^{2} b^{3} x^{2} + \frac{5}{2} \, A a b^{4} x^{2} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^3,x, algorithm="giac")

[Out]

1/4*B*b^5*x^4 + 5/3*B*a*b^4*x^3 + 1/3*A*b^5*x^3 + 5*B*a^2*b^3*x^2 + 5/2*A*a*b^4*
x^2 + 10*B*a^3*b^2*x + 10*A*a^2*b^3*x + 5*(B*a^4*b + 2*A*a^3*b^2)*ln(abs(x)) - 1
/2*(A*a^5 + 2*(B*a^5 + 5*A*a^4*b)*x)/x^2