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

Optimal. Leaf size=107 \[ -\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{10 a^2 b^2 (a B+A b)}{x}+b^4 x (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{2} b^5 B x^2 \]

[Out]

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

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Rubi [A]  time = 0.180618, antiderivative size = 107, 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 \[ -\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{10 a^2 b^2 (a B+A b)}{x}+b^4 x (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{2} b^5 B x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0698497, size = 106, normalized size = 0.99 \[ -\frac{a^5 (3 A+4 B x)}{12 x^4}-\frac{5 a^4 b (2 A+3 B x)}{6 x^3}-\frac{5 a^3 b^2 (A+2 B x)}{x^2}-\frac{10 a^2 A b^3}{x}+5 a b^3 \log (x) (2 a B+A b)+5 a b^4 B x+\frac{1}{2} b^5 x (2 A+B x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 119, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{2}}{2}}+Ax{b}^{5}+5\,Bxa{b}^{4}+5\,A\ln \left ( x \right ) a{b}^{4}+10\,B\ln \left ( x \right ){a}^{2}{b}^{3}-5\,{\frac{{a}^{3}{b}^{2}A}{{x}^{2}}}-{\frac{5\,{a}^{4}bB}{2\,{x}^{2}}}-10\,{\frac{{a}^{2}{b}^{3}A}{x}}-10\,{\frac{{a}^{3}{b}^{2}B}{x}}-{\frac{5\,{a}^{4}bA}{3\,{x}^{3}}}-{\frac{{a}^{5}B}{3\,{x}^{3}}}-{\frac{A{a}^{5}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.34535, size = 157, normalized size = 1.47 \[ \frac{1}{2} \, B b^{5} x^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} x + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left (x\right ) - \frac{3 \, A a^{5} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208087, size = 163, normalized size = 1.52 \[ \frac{6 \, B b^{5} x^{6} - 3 \, A a^{5} + 12 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 60 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} \log \left (x\right ) - 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.69284, size = 117, normalized size = 1.09 \[ \frac{B b^{5} x^{2}}{2} + 5 a b^{3} \left (A b + 2 B a\right ) \log{\left (x \right )} + x \left (A b^{5} + 5 B a b^{4}\right ) - \frac{3 A a^{5} + x^{3} \left (120 A a^{2} b^{3} + 120 B a^{3} b^{2}\right ) + x^{2} \left (60 A a^{3} b^{2} + 30 B a^{4} b\right ) + x \left (20 A a^{4} b + 4 B a^{5}\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**5*x**2/2 + 5*a*b**3*(A*b + 2*B*a)*log(x) + x*(A*b**5 + 5*B*a*b**4) - (3*A*a
**5 + x**3*(120*A*a**2*b**3 + 120*B*a**3*b**2) + x**2*(60*A*a**3*b**2 + 30*B*a**
4*b) + x*(20*A*a**4*b + 4*B*a**5))/(12*x**4)

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GIAC/XCAS [A]  time = 0.313214, size = 157, normalized size = 1.47 \[ \frac{1}{2} \, B b^{5} x^{2} + 5 \, B a b^{4} x + A b^{5} x + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, A a^{5} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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