Optimal. Leaf size=136 \[ \frac {5}{16} a (6 A b+a B) x \sqrt {a+b x^2}+\frac {5}{24} (6 A b+a B) x \left (a+b x^2\right )^{3/2}+\frac {(6 A b+a B) x \left (a+b x^2\right )^{5/2}}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{a x}+\frac {5 a^2 (6 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 \sqrt {b}} \]
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Rubi [A]
time = 0.04, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 \sqrt {b}}+\frac {x \left (a+b x^2\right )^{5/2} (a B+6 A b)}{6 a}+\frac {5}{24} x \left (a+b x^2\right )^{3/2} (a B+6 A b)+\frac {5}{16} a x \sqrt {a+b x^2} (a B+6 A b)-\frac {A \left (a+b x^2\right )^{7/2}}{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 )^{5/2} \left (A+B x^2\right )}{x^2} \, dx &=-\frac {A \left (a+b x^2\right )^{7/2}}{a x}-\frac {(-6 A b-a B) \int \left (a+b x^2\right )^{5/2} \, dx}{a}\\ &=\frac {(6 A b+a B) x \left (a+b x^2\right )^{5/2}}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{a x}+\frac {1}{6} (5 (6 A b+a B)) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} (6 A b+a B) x \left (a+b x^2\right )^{3/2}+\frac {(6 A b+a B) x \left (a+b x^2\right )^{5/2}}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{a x}+\frac {1}{8} (5 a (6 A b+a B)) \int \sqrt {a+b x^2} \, dx\\ &=\frac {5}{16} a (6 A b+a B) x \sqrt {a+b x^2}+\frac {5}{24} (6 A b+a B) x \left (a+b x^2\right )^{3/2}+\frac {(6 A b+a B) x \left (a+b x^2\right )^{5/2}}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{a x}+\frac {1}{16} \left (5 a^2 (6 A b+a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {5}{16} a (6 A b+a B) x \sqrt {a+b x^2}+\frac {5}{24} (6 A b+a B) x \left (a+b x^2\right )^{3/2}+\frac {(6 A b+a B) x \left (a+b x^2\right )^{5/2}}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{a x}+\frac {1}{16} \left (5 a^2 (6 A b+a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {5}{16} a (6 A b+a B) x \sqrt {a+b x^2}+\frac {5}{24} (6 A b+a B) x \left (a+b x^2\right )^{3/2}+\frac {(6 A b+a B) x \left (a+b x^2\right )^{5/2}}{6 a}-\frac {A \left (a+b x^2\right )^{7/2}}{a x}+\frac {5 a^2 (6 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 110, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-48 a^2 A+54 a A b x^2+33 a^2 B x^2+12 A b^2 x^4+26 a b B x^4+8 b^2 B x^6\right )}{48 x}-\frac {5 a^2 (6 A b+a B) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{16 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 164, normalized size = 1.21
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-8 b^{2} B \,x^{6}-12 A \,b^{2} x^{4}-26 B a b \,x^{4}-54 a A b \,x^{2}-33 B \,a^{2} x^{2}+48 a^{2} A \right )}{48 x}+\frac {15 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) \sqrt {b}\, A}{8}+\frac {5 a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) B}{16 \sqrt {b}}\) | \(116\) |
default | \(B \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\right )\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 124, normalized size = 0.91 \begin {gather*} \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a x + \frac {5}{16} \, \sqrt {b x^{2} + a} B a^{2} x + \frac {5}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b x + \frac {15}{8} \, \sqrt {b x^{2} + a} A a b x + \frac {5 \, B a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} + \frac {15}{8} \, A a^{2} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.87, size = 236, normalized size = 1.74 \begin {gather*} \left [\frac {15 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (8 \, B b^{3} x^{6} + 2 \, {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{4} - 48 \, A a^{2} b + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, b x}, -\frac {15 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, B b^{3} x^{6} + 2 \, {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{4} - 48 \, A a^{2} b + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, 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. 306 vs.
\(2 (126) = 252\).
time = 12.62, size = 306, normalized size = 2.25 \begin {gather*} - \frac {A a^{\frac {5}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + A a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x^{2}}{a}} - \frac {7 A a^{\frac {3}{2}} b x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A \sqrt {a} b^{2} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 A a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8} + \frac {A b^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {B a^{\frac {5}{2}} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {3 B a^{\frac {5}{2}} x}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 B a^{\frac {3}{2}} b x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 B \sqrt {a} b^{2} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 B a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {B b^{3} x^{7}}{6 \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 = 0.89, size = 146, normalized size = 1.07 \begin {gather*} \frac {2 \, A a^{3} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{48} \, {\left (2 \, {\left (4 \, B b^{2} x^{2} + \frac {13 \, B a b^{5} + 6 \, A b^{6}}{b^{4}}\right )} x^{2} + \frac {3 \, {\left (11 \, B a^{2} b^{4} + 18 \, A a b^{5}\right )}}{b^{4}}\right )} \sqrt {b x^{2} + a} x - \frac {5 \, {\left (B a^{3} \sqrt {b} + 6 \, A a^{2} b^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 80, normalized size = 0.59 \begin {gather*} \frac {B\,x\,{\left (b\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{5/2}}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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