3.3.24 \(\int \frac {(a+b x^3)^2}{x^2} \, dx\) [224]

Optimal. Leaf size=25 \[ -\frac {a^2}{x}+a b x^2+\frac {b^2 x^5}{5} \]

[Out]

-a^2/x+a*b*x^2+1/5*b^2*x^5

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Rubi [A]
time = 0.01, antiderivative size = 25, 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^2}{x}+a b x^2+\frac {b^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^2,x]

[Out]

-(a^2/x) + a*b*x^2 + (b^2*x^5)/5

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

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} -\frac {a^2}{x}+a b x^2+\frac {b^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^2,x]

[Out]

-(a^2/x) + a*b*x^2 + (b^2*x^5)/5

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Maple [A]
time = 0.11, size = 24, normalized size = 0.96

method	result	size
default	\(-\frac {a^{2}}{x}+a b \,x^{2}+\frac {b^{2} x^{5}}{5}\)	\(24\)
risch	\(-\frac {a^{2}}{x}+a b \,x^{2}+\frac {b^{2} x^{5}}{5}\)	\(24\)
norman	\(\frac {\frac {1}{5} b^{2} x^{6}+a b \,x^{3}-a^{2}}{x}\)	\(25\)
gosper	\(-\frac {-b^{2} x^{6}-5 a b \,x^{3}+5 a^{2}}{5 x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

-a^2/x+a*b*x^2+1/5*b^2*x^5

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} + a b x^{2} - \frac {a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^2,x, algorithm="maxima")

[Out]

1/5*b^2*x^5 + a*b*x^2 - a^2/x

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Fricas [A]
time = 0.34, size = 25, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{6} + 5 \, a b x^{3} - 5 \, a^{2}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^2,x, algorithm="fricas")

[Out]

1/5*(b^2*x^6 + 5*a*b*x^3 - 5*a^2)/x

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Sympy [A]
time = 0.02, size = 19, normalized size = 0.76 \begin {gather*} - \frac {a^{2}}{x} + a b x^{2} + \frac {b^{2} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**2,x)

[Out]

-a**2/x + a*b*x**2 + b**2*x**5/5

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Giac [A]
time = 2.25, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} + a b x^{2} - \frac {a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^2,x, algorithm="giac")

[Out]

1/5*b^2*x^5 + a*b*x^2 - a^2/x

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Mupad [B]
time = 0.03, size = 24, normalized size = 0.96 \begin {gather*} \frac {-a^2+a\,b\,x^3+\frac {b^2\,x^6}{5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^2,x)

[Out]

((b^2*x^6)/5 - a^2 + a*b*x^3)/x

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