3.2548 \(\int \frac{\left (a+b x^n\right )^5}{x} \, dx\)

Optimal. Leaf size=84 \[ a^5 \log (x)+\frac{5 a^4 b x^n}{n}+\frac{5 a^3 b^2 x^{2 n}}{n}+\frac{10 a^2 b^3 x^{3 n}}{3 n}+\frac{5 a b^4 x^{4 n}}{4 n}+\frac{b^5 x^{5 n}}{5 n} \]

[Out]

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

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Rubi [A]  time = 0.0894612, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ a^5 \log (x)+\frac{5 a^4 b x^n}{n}+\frac{5 a^3 b^2 x^{2 n}}{n}+\frac{10 a^2 b^3 x^{3 n}}{3 n}+\frac{5 a b^4 x^{4 n}}{4 n}+\frac{b^5 x^{5 n}}{5 n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^5/x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0574277, size = 67, normalized size = 0.8 \[ a^5 \log (x)+\frac{b x^n \left (300 a^4+300 a^3 b x^n+200 a^2 b^2 x^{2 n}+75 a b^3 x^{3 n}+12 b^4 x^{4 n}\right )}{60 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^5/x,x]

[Out]

(b*x^n*(300*a^4 + 300*a^3*b*x^n + 200*a^2*b^2*x^(2*n) + 75*a*b^3*x^(3*n) + 12*b^
4*x^(4*n)))/(60*n) + a^5*Log[x]

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Maple [A]  time = 0.003, size = 84, normalized size = 1. \[{\frac{{b}^{5} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{4}}{4\,n}}+{\frac{10\,{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{3}}{3\,n}}+5\,{\frac{{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{2}}{n}}+5\,{\frac{{a}^{4}b{x}^{n}}{n}}+{\frac{{a}^{5}\ln \left ({x}^{n} \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^5/x,x)

[Out]

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

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Maxima [A]  time = 1.42592, size = 100, normalized size = 1.19 \[ \frac{a^{5} \log \left (x^{n}\right )}{n} + \frac{12 \, b^{5} x^{5 \, n} + 75 \, a b^{4} x^{4 \, n} + 200 \, a^{2} b^{3} x^{3 \, n} + 300 \, a^{3} b^{2} x^{2 \, n} + 300 \, a^{4} b x^{n}}{60 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^5*log(x^n)/n + 1/60*(12*b^5*x^(5*n) + 75*a*b^4*x^(4*n) + 200*a^2*b^3*x^(3*n) +
300*a^3*b^2*x^(2*n) + 300*a^4*b*x^n)/n

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Fricas [A]  time = 0.227897, size = 95, normalized size = 1.13 \[ \frac{60 \, a^{5} n \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} + 75 \, a b^{4} x^{4 \, n} + 200 \, a^{2} b^{3} x^{3 \, n} + 300 \, a^{3} b^{2} x^{2 \, n} + 300 \, a^{4} b x^{n}}{60 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(60*a^5*n*log(x) + 12*b^5*x^(5*n) + 75*a*b^4*x^(4*n) + 200*a^2*b^3*x^(3*n)
+ 300*a^3*b^2*x^(2*n) + 300*a^4*b*x^n)/n

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Sympy [A]  time = 2.32435, size = 85, normalized size = 1.01 \[ \begin{cases} a^{5} \log{\left (x \right )} + \frac{5 a^{4} b x^{n}}{n} + \frac{5 a^{3} b^{2} x^{2 n}}{n} + \frac{10 a^{2} b^{3} x^{3 n}}{3 n} + \frac{5 a b^{4} x^{4 n}}{4 n} + \frac{b^{5} x^{5 n}}{5 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{5} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)**5/x,x)

[Out]

Piecewise((a**5*log(x) + 5*a**4*b*x**n/n + 5*a**3*b**2*x**(2*n)/n + 10*a**2*b**3
*x**(3*n)/(3*n) + 5*a*b**4*x**(4*n)/(4*n) + b**5*x**(5*n)/(5*n), Ne(n, 0)), ((a
+ b)**5*log(x), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{5}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^5/x, x)