3.7.23 \(\int \frac {(a+\frac {b}{x^2}) x^2}{\sqrt {c+\frac {d}{x^2}}} \, dx\)

Optimal. Leaf size=51 \[ \frac {x \sqrt {c+\frac {d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac {a x^3 \sqrt {c+\frac {d}{x^2}}}{3 c} \]

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Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {453, 191} \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} (3 b c-2 a d)}{3 c^2}+\frac {a x^3 \sqrt {c+\frac {d}{x^2}}}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]

[Out]

((3*b*c - 2*a*d)*Sqrt[c + d/x^2]*x)/(3*c^2) + (a*Sqrt[c + d/x^2]*x^3)/(3*c)

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) x^2}{\sqrt {c+\frac {d}{x^2}}} \, dx &=\frac {a \sqrt {c+\frac {d}{x^2}} x^3}{3 c}+\frac {(3 b c-2 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}}} \, dx}{3 c}\\ &=\frac {(3 b c-2 a d) \sqrt {c+\frac {d}{x^2}} x}{3 c^2}+\frac {a \sqrt {c+\frac {d}{x^2}} x^3}{3 c}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 34, normalized size = 0.67 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (a c x^2-2 a d+3 b c\right )}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]

[Out]

(Sqrt[c + d/x^2]*x*(3*b*c - 2*a*d + a*c*x^2))/(3*c^2)

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IntegrateAlgebraic [A]  time = 0.06, size = 34, normalized size = 0.67 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (a c x^2-2 a d+3 b c\right )}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b/x^2)*x^2)/Sqrt[c + d/x^2],x]

[Out]

(Sqrt[c + d/x^2]*x*(3*b*c - 2*a*d + a*c*x^2))/(3*c^2)

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fricas [A]  time = 0.42, size = 36, normalized size = 0.71 \begin {gather*} \frac {{\left (a c x^{3} + {\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^2/(c+d/x^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*(a*c*x^3 + (3*b*c - 2*a*d)*x)*sqrt((c*x^2 + d)/x^2)/c^2

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^2/(c+d/x^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [sign(t_nostep),sign(t_nostep+sqrt(d)/c*sign(t_nostep))]sym2poly/r2sym(const gen & e,const index_
m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 44, normalized size = 0.86 \begin {gather*} \frac {\left (a \,x^{2} c -2 a d +3 b c \right ) \left (c \,x^{2}+d \right )}{3 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)*x^2/(c+d/x^2)^(1/2),x)

[Out]

1/3/x*(a*c*x^2-2*a*d+3*b*c)*(c*x^2+d)/((c*x^2+d)/x^2)^(1/2)/c^2

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maxima [A]  time = 0.59, size = 49, normalized size = 0.96 \begin {gather*} \frac {b \sqrt {c + \frac {d}{x^{2}}} x}{c} + \frac {{\left ({\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} d x\right )} a}{3 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^2/(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

b*sqrt(c + d/x^2)*x/c + 1/3*((c + d/x^2)^(3/2)*x^3 - 3*sqrt(c + d/x^2)*d*x)*a/c^2

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mupad [B]  time = 4.92, size = 67, normalized size = 1.31 \begin {gather*} \frac {a\,x^3\,\sqrt {c+\frac {d}{x^2}}\,\left (c-\frac {2\,d}{x^2}\right )}{3\,c^2}+\frac {b\,x\,\sqrt {\frac {c\,x^2}{d}+1}}{\sqrt {c+\frac {d}{x^2}}\,\left (\sqrt {\frac {c\,x^2}{d}+1}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b/x^2))/(c + d/x^2)^(1/2),x)

[Out]

(a*x^3*(c + d/x^2)^(1/2)*(c - (2*d)/x^2))/(3*c^2) + (b*x*((c*x^2)/d + 1)^(1/2))/((c + d/x^2)^(1/2)*(((c*x^2)/d
+ 1)^(1/2) + 1))

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sympy [A]  time = 2.76, size = 70, normalized size = 1.37 \begin {gather*} \frac {a \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c} - \frac {2 a d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c^{2}} + \frac {b \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**2/(c+d/x**2)**(1/2),x)

[Out]

a*sqrt(d)*x**2*sqrt(c*x**2/d + 1)/(3*c) - 2*a*d**(3/2)*sqrt(c*x**2/d + 1)/(3*c**2) + b*sqrt(d)*sqrt(c*x**2/d +
1)/c

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