Optimal. Leaf size=82 \[ -\frac {A}{3 a x^3 \sqrt {a+b x^2}}+\frac {4 A b-3 a B}{3 a^2 x \sqrt {a+b x^2}}+\frac {2 b (4 A b-3 a B) x}{3 a^3 \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {464, 277, 197}
\begin {gather*} \frac {2 b x (4 A b-3 a B)}{3 a^3 \sqrt {a+b x^2}}+\frac {4 A b-3 a B}{3 a^2 x \sqrt {a+b x^2}}-\frac {A}{3 a x^3 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 277
Rule 464
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {A}{3 a x^3 \sqrt {a+b x^2}}-\frac {(4 A b-3 a B) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {A}{3 a x^3 \sqrt {a+b x^2}}+\frac {4 A b-3 a B}{3 a^2 x \sqrt {a+b x^2}}+\frac {(2 b (4 A b-3 a B)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a^2}\\ &=-\frac {A}{3 a x^3 \sqrt {a+b x^2}}+\frac {4 A b-3 a B}{3 a^2 x \sqrt {a+b x^2}}+\frac {2 b (4 A b-3 a B) x}{3 a^3 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 62, normalized size = 0.76 \begin {gather*} \frac {-a^2 A+4 a A b x^2-3 a^2 B x^2+8 A b^2 x^4-6 a b B x^4}{3 a^3 x^3 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 98, normalized size = 1.20
method | result | size |
gosper | \(-\frac {-8 A \,b^{2} x^{4}+6 B a b \,x^{4}-4 a A b \,x^{2}+3 B \,a^{2} x^{2}+a^{2} A}{3 x^{3} \sqrt {b \,x^{2}+a}\, a^{3}}\) | \(58\) |
trager | \(-\frac {-8 A \,b^{2} x^{4}+6 B a b \,x^{4}-4 a A b \,x^{2}+3 B \,a^{2} x^{2}+a^{2} A}{3 x^{3} \sqrt {b \,x^{2}+a}\, a^{3}}\) | \(58\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-5 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{3} x^{3}}+\frac {x \left (A b -B a \right ) b}{\sqrt {b \,x^{2}+a}\, a^{3}}\) | \(60\) |
default | \(A \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )+B \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 92, normalized size = 1.12 \begin {gather*} -\frac {2 \, B b x}{\sqrt {b x^{2} + a} a^{2}} + \frac {8 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {B}{\sqrt {b x^{2} + a} a x} + \frac {4 \, A b}{3 \, \sqrt {b x^{2} + a} a^{2} x} - \frac {A}{3 \, \sqrt {b x^{2} + a} a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.73, size = 68, normalized size = 0.83 \begin {gather*} -\frac {{\left (2 \, {\left (3 \, B a b - 4 \, A b^{2}\right )} x^{4} + A a^{2} + {\left (3 \, B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs.
\(2 (73) = 146\).
time = 3.76, size = 284, normalized size = 3.46 \begin {gather*} A \left (- \frac {a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {3 a^{2} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {12 a b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {8 b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}}\right ) + B \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs.
\(2 (70) = 140\).
time = 1.09, size = 181, normalized size = 2.21 \begin {gather*} -\frac {{\left (B a b - A b^{2}\right )} x}{\sqrt {b x^{2} + a} a^{3}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} + 3 \, B a^{3} \sqrt {b} - 5 \, A a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 57, normalized size = 0.70 \begin {gather*} -\frac {3\,B\,a^2\,x^2+A\,a^2+6\,B\,a\,b\,x^4-4\,A\,a\,b\,x^2-8\,A\,b^2\,x^4}{3\,a^3\,x^3\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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