3.6.64 \(\int \frac {A+B x^2}{x^4 \sqrt {a+b x^2}} \, dx\) [564]

Optimal. Leaf size=53 \[ -\frac {A \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 A b-3 a B) \sqrt {a+b x^2}}{3 a^2 x} \]

[Out]

-1/3*A*(b*x^2+a)^(1/2)/a/x^3+1/3*(2*A*b-3*B*a)*(b*x^2+a)^(1/2)/a^2/x

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Rubi [A]
time = 0.01, antiderivative size = 53, 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 = {464, 270} \begin {gather*} \frac {\sqrt {a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac {A \sqrt {a+b x^2}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^4*Sqrt[a + b*x^2]),x]

[Out]

-1/3*(A*Sqrt[a + b*x^2])/(a*x^3) + ((2*A*b - 3*a*B)*Sqrt[a + b*x^2])/(3*a^2*x)

Rule 270

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

Rule 464

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 {A+B x^2}{x^4 \sqrt {a+b x^2}} \, dx &=-\frac {A \sqrt {a+b x^2}}{3 a x^3}-\frac {(2 A b-3 a B) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{3 a}\\ &=-\frac {A \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 A b-3 a B) \sqrt {a+b x^2}}{3 a^2 x}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.75 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-a A+2 A b x^2-3 a B x^2\right )}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^4*Sqrt[a + b*x^2]),x]

[Out]

(Sqrt[a + b*x^2]*(-(a*A) + 2*A*b*x^2 - 3*a*B*x^2))/(3*a^2*x^3)

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Maple [A]
time = 0.08, size = 58, normalized size = 1.09

method	result	size
gosper	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{2} x^{3}}\)	\(36\)
trager	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{2} x^{3}}\)	\(36\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{2} x^{3}}\)	\(36\)
default	\(A \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )-\frac {B \sqrt {b \,x^{2}+a}}{a x}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

A*(-1/3/a/x^3*(b*x^2+a)^(1/2)+2/3*b/a^2*(b*x^2+a)^(1/2)/x)-B*(b*x^2+a)^(1/2)/a/x

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Maxima [A]
time = 0.29, size = 56, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {b x^{2} + a} B}{a x} + \frac {2 \, \sqrt {b x^{2} + a} A b}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a} A}{3 \, a x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(b*x^2 + a)*B/(a*x) + 2/3*sqrt(b*x^2 + a)*A*b/(a^2*x) - 1/3*sqrt(b*x^2 + a)*A/(a*x^3)

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Fricas [A]
time = 1.86, size = 34, normalized size = 0.64 \begin {gather*} -\frac {{\left ({\left (3 \, B a - 2 \, A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a}}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

-1/3*((3*B*a - 2*A*b)*x^2 + A*a)*sqrt(b*x^2 + a)/(a^2*x^3)

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Sympy [A]
time = 0.96, size = 70, normalized size = 1.32 \begin {gather*} - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**4/(b*x**2+a)**(1/2),x)

[Out]

-A*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*a*x**2) + 2*A*b**(3/2)*sqrt(a/(b*x**2) + 1)/(3*a**2) - B*sqrt(b)*sqrt(a/(b*
x**2) + 1)/a

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (45) = 90\).
time = 1.23, size = 120, normalized size = 2.26 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B \sqrt {b} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {3}{2}} + 3 \, B a^{2} \sqrt {b} - 2 \, A a b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*sqrt(b) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b) + 6*(sqrt(b)*
x - sqrt(b*x^2 + a))^2*A*b^(3/2) + 3*B*a^2*sqrt(b) - 2*A*a*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3

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Mupad [B]
time = 0.29, size = 35, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {b\,x^2+a}\,\left (A\,a-2\,A\,b\,x^2+3\,B\,a\,x^2\right )}{3\,a^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^4*(a + b*x^2)^(1/2)),x)

[Out]

-((a + b*x^2)^(1/2)*(A*a - 2*A*b*x^2 + 3*B*a*x^2))/(3*a^2*x^3)

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