3.3.81 \(\int \frac {(a+b x^3)^5}{x^5} \, dx\) [281]

Optimal. Leaf size=63 \[ -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{11}}{11} \]

[Out]

-1/4*a^5/x^4-5*a^4*b/x+5*a^3*b^2*x^2+2*a^2*b^3*x^5+5/8*a*b^4*x^8+1/11*b^5*x^11

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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \begin {gather*} -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^5,x]

[Out]

-1/4*a^5/x^4 - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (b^5*x^11)/11

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^5}{x^5} \, dx &=\int \left (\frac {a^5}{x^5}+\frac {5 a^4 b}{x^2}+10 a^3 b^2 x+10 a^2 b^3 x^4+5 a b^4 x^7+b^5 x^{10}\right ) \, dx\\ &=-\frac {a^5}{4 x^4}-\frac {5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{11}}{11}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 63, normalized size = 1.00 \begin {gather*} -\frac {a^5}{4 x^4}-\frac {5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^5,x]

[Out]

-1/4*a^5/x^4 - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (b^5*x^11)/11

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Maple [A]
time = 0.12, size = 58, normalized size = 0.92

method	result	size
default	\(-\frac {a^{5}}{4 x^{4}}-\frac {5 a^{4} b}{x}+5 a^{3} x^{2} b^{2}+2 a^{2} b^{3} x^{5}+\frac {5 a \,b^{4} x^{8}}{8}+\frac {b^{5} x^{11}}{11}\)	\(58\)
norman	\(\frac {-\frac {1}{4} a^{5}-5 a^{4} b \,x^{3}+5 a^{3} b^{2} x^{6}+2 a^{2} b^{3} x^{9}+\frac {5}{8} a \,b^{4} x^{12}+\frac {1}{11} b^{5} x^{15}}{x^{4}}\)	\(59\)
gosper	\(-\frac {-8 b^{5} x^{15}-55 a \,b^{4} x^{12}-176 a^{2} b^{3} x^{9}-440 a^{3} b^{2} x^{6}+440 a^{4} b \,x^{3}+22 a^{5}}{88 x^{4}}\)	\(60\)
risch	\(\frac {b^{5} x^{11}}{11}+\frac {5 a \,b^{4} x^{8}}{8}+2 a^{2} b^{3} x^{5}+5 a^{3} x^{2} b^{2}+\frac {-5 a^{4} b \,x^{3}-\frac {1}{4} a^{5}}{x^{4}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^5/x^4-5*a^4*b/x+5*a^3*x^2*b^2+2*a^2*b^3*x^5+5/8*a*b^4*x^8+1/11*b^5*x^11

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Maxima [A]
time = 0.30, size = 58, normalized size = 0.92 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} + \frac {5}{8} \, a b^{4} x^{8} + 2 \, a^{2} b^{3} x^{5} + 5 \, a^{3} b^{2} x^{2} - \frac {20 \, a^{4} b x^{3} + a^{5}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^5,x, algorithm="maxima")

[Out]

1/11*b^5*x^11 + 5/8*a*b^4*x^8 + 2*a^2*b^3*x^5 + 5*a^3*b^2*x^2 - 1/4*(20*a^4*b*x^3 + a^5)/x^4

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Fricas [A]
time = 0.34, size = 59, normalized size = 0.94 \begin {gather*} \frac {8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^5,x, algorithm="fricas")

[Out]

1/88*(8*b^5*x^15 + 55*a*b^4*x^12 + 176*a^2*b^3*x^9 + 440*a^3*b^2*x^6 - 440*a^4*b*x^3 - 22*a^5)/x^4

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Sympy [A]
time = 0.07, size = 63, normalized size = 1.00 \begin {gather*} 5 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{5} + \frac {5 a b^{4} x^{8}}{8} + \frac {b^{5} x^{11}}{11} + \frac {- a^{5} - 20 a^{4} b x^{3}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**5,x)

[Out]

5*a**3*b**2*x**2 + 2*a**2*b**3*x**5 + 5*a*b**4*x**8/8 + b**5*x**11/11 + (-a**5 - 20*a**4*b*x**3)/(4*x**4)

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Giac [A]
time = 2.46, size = 58, normalized size = 0.92 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} + \frac {5}{8} \, a b^{4} x^{8} + 2 \, a^{2} b^{3} x^{5} + 5 \, a^{3} b^{2} x^{2} - \frac {20 \, a^{4} b x^{3} + a^{5}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^5,x, algorithm="giac")

[Out]

1/11*b^5*x^11 + 5/8*a*b^4*x^8 + 2*a^2*b^3*x^5 + 5*a^3*b^2*x^2 - 1/4*(20*a^4*b*x^3 + a^5)/x^4

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Mupad [B]
time = 0.02, size = 60, normalized size = 0.95 \begin {gather*} \frac {b^5\,x^{11}}{11}-\frac {\frac {a^5}{4}+5\,b\,a^4\,x^3}{x^4}+\frac {5\,a\,b^4\,x^8}{8}+5\,a^3\,b^2\,x^2+2\,a^2\,b^3\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^5/x^5,x)

[Out]

(b^5*x^11)/11 - (a^5/4 + 5*a^4*b*x^3)/x^4 + (5*a*b^4*x^8)/8 + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5

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