3.1.36 \(\int \frac {(a+b x^2)^3}{x^5} \, dx\) [36]

Optimal. Leaf size=40 \[ -\frac {a^3}{4 x^4}-\frac {3 a^2 b}{2 x^2}+\frac {b^3 x^2}{2}+3 a b^2 \log (x) \]

[Out]

-1/4*a^3/x^4-3/2*a^2*b/x^2+1/2*b^3*x^2+3*a*b^2*ln(x)

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^3}{4 x^4}-\frac {3 a^2 b}{2 x^2}+3 a b^2 \log (x)+\frac {b^3 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^5,x]

[Out]

-1/4*a^3/x^4 - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^3}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^3}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (b^3+\frac {a^3}{x^3}+\frac {3 a^2 b}{x^2}+\frac {3 a b^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3}{4 x^4}-\frac {3 a^2 b}{2 x^2}+\frac {b^3 x^2}{2}+3 a b^2 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} -\frac {a^3}{4 x^4}-\frac {3 a^2 b}{2 x^2}+\frac {b^3 x^2}{2}+3 a b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^5,x]

[Out]

-1/4*a^3/x^4 - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*Log[x]

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Maple [A]
time = 0.02, size = 35, normalized size = 0.88

method	result	size
default	\(-\frac {a^{3}}{4 x^{4}}-\frac {3 a^{2} b}{2 x^{2}}+\frac {b^{3} x^{2}}{2}+3 a \,b^{2} \ln \left (x \right )\)	\(35\)
norman	\(\frac {-\frac {1}{4} a^{3}+\frac {1}{2} b^{3} x^{6}-\frac {3}{2} a^{2} b \,x^{2}}{x^{4}}+3 a \,b^{2} \ln \left (x \right )\)	\(37\)
risch	\(\frac {b^{3} x^{2}}{2}+\frac {-\frac {3}{2} a^{2} b \,x^{2}-\frac {1}{4} a^{3}}{x^{4}}+3 a \,b^{2} \ln \left (x \right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^3/x^4-3/2*a^2*b/x^2+1/2*b^3*x^2+3*a*b^2*ln(x)

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Maxima [A]
time = 0.29, size = 37, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} + \frac {3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac {6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^5,x, algorithm="maxima")

[Out]

1/2*b^3*x^2 + 3/2*a*b^2*log(x^2) - 1/4*(6*a^2*b*x^2 + a^3)/x^4

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Fricas [A]
time = 0.95, size = 39, normalized size = 0.98 \begin {gather*} \frac {2 \, b^{3} x^{6} + 12 \, a b^{2} x^{4} \log \left (x\right ) - 6 \, a^{2} b x^{2} - a^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^5,x, algorithm="fricas")

[Out]

1/4*(2*b^3*x^6 + 12*a*b^2*x^4*log(x) - 6*a^2*b*x^2 - a^3)/x^4

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Sympy [A]
time = 0.07, size = 37, normalized size = 0.92 \begin {gather*} 3 a b^{2} \log {\left (x \right )} + \frac {b^{3} x^{2}}{2} + \frac {- a^{3} - 6 a^{2} b x^{2}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**5,x)

[Out]

3*a*b**2*log(x) + b**3*x**2/2 + (-a**3 - 6*a**2*b*x**2)/(4*x**4)

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Giac [A]
time = 0.51, size = 46, normalized size = 1.15 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} + \frac {3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac {9 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^5,x, algorithm="giac")

[Out]

1/2*b^3*x^2 + 3/2*a*b^2*log(x^2) - 1/4*(9*a*b^2*x^4 + 6*a^2*b*x^2 + a^3)/x^4

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Mupad [B]
time = 4.90, size = 37, normalized size = 0.92 \begin {gather*} \frac {b^3\,x^2}{2}-\frac {\frac {a^3}{4}+\frac {3\,b\,a^2\,x^2}{2}}{x^4}+3\,a\,b^2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^5,x)

[Out]

(b^3*x^2)/2 - (a^3/4 + (3*a^2*b*x^2)/2)/x^4 + 3*a*b^2*log(x)

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