3.100 \(\int \frac{(a+b x)^5 (A+B x)}{x^7} \, dx\)

Optimal. Leaf size=85 \[ -\frac{a^5 B}{5 x^5}-\frac{5 a^4 b B}{4 x^4}-\frac{10 a^3 b^2 B}{3 x^3}-\frac{5 a^2 b^3 B}{x^2}-\frac{A (a+b x)^6}{6 a x^6}-\frac{5 a b^4 B}{x}+b^5 B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0900029, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^5 B}{5 x^5}-\frac{5 a^4 b B}{4 x^4}-\frac{10 a^3 b^2 B}{3 x^3}-\frac{5 a^2 b^3 B}{x^2}-\frac{A (a+b x)^6}{6 a x^6}-\frac{5 a b^4 B}{x}+b^5 B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^5*(A + B*x))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 24.7194, size = 85, normalized size = 1. \[ - \frac{A \left (a + b x\right )^{6}}{6 a x^{6}} - \frac{B a^{5}}{5 x^{5}} - \frac{5 B a^{4} b}{4 x^{4}} - \frac{10 B a^{3} b^{2}}{3 x^{3}} - \frac{5 B a^{2} b^{3}}{x^{2}} - \frac{5 B a b^{4}}{x} + B b^{5} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(B*x+A)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0667776, size = 109, normalized size = 1.28 \[ -\frac{2 a^5 (5 A+6 B x)+15 a^4 b x (4 A+5 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+100 a^2 b^3 x^3 (2 A+3 B x)+150 a b^4 x^4 (A+2 B x)+60 A b^5 x^5-60 b^5 B x^6 \log (x)}{60 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^5*(A + B*x))/x^7,x]

[Out]

-(60*A*b^5*x^5 + 150*a*b^4*x^4*(A + 2*B*x) + 100*a^2*b^3*x^3*(2*A + 3*B*x) + 50*
a^3*b^2*x^2*(3*A + 4*B*x) + 15*a^4*b*x*(4*A + 5*B*x) + 2*a^5*(5*A + 6*B*x) - 60*
b^5*B*x^6*Log[x])/(60*x^6)

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Maple [A]  time = 0.012, size = 124, normalized size = 1.5 \[{b}^{5}B\ln \left ( x \right ) -{\frac{5\,a{b}^{4}A}{2\,{x}^{2}}}-5\,{\frac{{a}^{2}{b}^{3}B}{{x}^{2}}}-{\frac{{a}^{4}bA}{{x}^{5}}}-{\frac{{a}^{5}B}{5\,{x}^{5}}}-{\frac{{b}^{5}A}{x}}-5\,{\frac{a{b}^{4}B}{x}}-{\frac{10\,{a}^{2}{b}^{3}A}{3\,{x}^{3}}}-{\frac{10\,{a}^{3}{b}^{2}B}{3\,{x}^{3}}}-{\frac{5\,{a}^{3}{b}^{2}A}{2\,{x}^{4}}}-{\frac{5\,{a}^{4}bB}{4\,{x}^{4}}}-{\frac{A{a}^{5}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(B*x+A)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.36306, size = 159, normalized size = 1.87 \[ B b^{5} \log \left (x\right ) - \frac{10 \, A a^{5} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^7,x, algorithm="maxima")

[Out]

B*b^5*log(x) - 1/60*(10*A*a^5 + 60*(5*B*a*b^4 + A*b^5)*x^5 + 150*(2*B*a^2*b^3 +
A*a*b^4)*x^4 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^2
+ 12*(B*a^5 + 5*A*a^4*b)*x)/x^6

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Fricas [A]  time = 0.203301, size = 163, normalized size = 1.92 \[ \frac{60 \, B b^{5} x^{6} \log \left (x\right ) - 10 \, A a^{5} - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^7,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*x^6*log(x) - 10*A*a^5 - 60*(5*B*a*b^4 + A*b^5)*x^5 - 150*(2*B*a^2
*b^3 + A*a*b^4)*x^4 - 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 75*(B*a^4*b + 2*A*a^3*b^
2)*x^2 - 12*(B*a^5 + 5*A*a^4*b)*x)/x^6

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Sympy [A]  time = 12.5409, size = 122, normalized size = 1.44 \[ B b^{5} \log{\left (x \right )} - \frac{10 A a^{5} + x^{5} \left (60 A b^{5} + 300 B a b^{4}\right ) + x^{4} \left (150 A a b^{4} + 300 B a^{2} b^{3}\right ) + x^{3} \left (200 A a^{2} b^{3} + 200 B a^{3} b^{2}\right ) + x^{2} \left (150 A a^{3} b^{2} + 75 B a^{4} b\right ) + x \left (60 A a^{4} b + 12 B a^{5}\right )}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(B*x+A)/x**7,x)

[Out]

B*b**5*log(x) - (10*A*a**5 + x**5*(60*A*b**5 + 300*B*a*b**4) + x**4*(150*A*a*b**
4 + 300*B*a**2*b**3) + x**3*(200*A*a**2*b**3 + 200*B*a**3*b**2) + x**2*(150*A*a*
*3*b**2 + 75*B*a**4*b) + x*(60*A*a**4*b + 12*B*a**5))/(60*x**6)

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GIAC/XCAS [A]  time = 0.29547, size = 161, normalized size = 1.89 \[ B b^{5}{\rm ln}\left ({\left | x \right |}\right ) - \frac{10 \, A a^{5} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^5/x^7,x, algorithm="giac")

[Out]

B*b^5*ln(abs(x)) - 1/60*(10*A*a^5 + 60*(5*B*a*b^4 + A*b^5)*x^5 + 150*(2*B*a^2*b^
3 + A*a*b^4)*x^4 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 75*(B*a^4*b + 2*A*a^3*b^2)*
x^2 + 12*(B*a^5 + 5*A*a^4*b)*x)/x^6