Optimal. Leaf size=109 \[ \frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {3 a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}} \]
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Rubi [A]
time = 0.03, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {464, 201, 223,
212} \begin {gather*} \frac {x \left (a+b x^2\right )^{3/2} (a B+4 A b)}{4 a}+\frac {3}{8} x \sqrt {a+b x^2} (a B+4 A b)+\frac {3 a (a B+4 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}-\frac {A \left (a+b x^2\right )^{5/2}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 464
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx &=-\frac {A \left (a+b x^2\right )^{5/2}}{a x}-\frac {(-4 A b-a B) \int \left (a+b x^2\right )^{3/2} \, dx}{a}\\ &=\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {1}{4} (3 (4 A b+a B)) \int \sqrt {a+b x^2} \, dx\\ &=\frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {1}{8} (3 a (4 A b+a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {1}{8} (3 a (4 A b+a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {3 a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 84, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-8 a A+4 A b x^2+5 a B x^2+2 b B x^4\right )}{8 x}-\frac {3 a (4 A b+a B) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 132, normalized size = 1.21
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b B \,x^{4}-4 A b \,x^{2}-5 B a \,x^{2}+8 A a \right )}{8 x}+\frac {3 a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) \sqrt {b}\, A}{2}+\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) B}{8 \sqrt {b}}\) | \(90\) |
default | \(B \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 91, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B x + \frac {3}{8} \, \sqrt {b x^{2} + a} B a x + \frac {3}{2} \, \sqrt {b x^{2} + a} A b x + \frac {3 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {3}{2} \, A a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.31, size = 182, normalized size = 1.67 \begin {gather*} \left [\frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, B b^{2} x^{4} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{16 \, b x}, -\frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b^{2} x^{4} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{8 \, b x}\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. 216 vs.
\(2 (99) = 198\).
time = 5.18, size = 216, normalized size = 1.98 \begin {gather*} - \frac {A a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {A \sqrt {a} b x \sqrt {1 + \frac {b x^{2}}{a}}}{2} - \frac {A \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2} + \frac {B a^{\frac {3}{2}} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {B a^{\frac {3}{2}} x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B \sqrt {a} b x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {B b^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.90, size = 114, normalized size = 1.05 \begin {gather*} \frac {2 \, A a^{2} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{8} \, {\left (2 \, B b x^{2} + \frac {5 \, B a b^{2} + 4 \, A b^{3}}{b^{2}}\right )} \sqrt {b x^{2} + a} x - \frac {3 \, {\left (B a^{2} \sqrt {b} + 4 \, A a b^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 80, normalized size = 0.73 \begin {gather*} \frac {B\,x\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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