Optimal. Leaf size=84 \[ -\frac {\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (A d+B c)-3 b B d x^2\right )}{15 b^2}+A c \sqrt {a+b x^2}-\sqrt {a} A c \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {573, 147, 50, 63, 208} \[ -\frac {\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (A d+B c)-3 b B d x^2\right )}{15 b^2}+A c \sqrt {a+b x^2}-\sqrt {a} A c \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 147
Rule 208
Rule 573
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right ) \left (c+d x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x) (c+d x)}{x} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}+\frac {1}{2} (A c) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=A c \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}+\frac {1}{2} (a A c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=A c \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}+\frac {(a A c) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=A c \sqrt {a+b x^2}-\frac {\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}-\sqrt {a} A c \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 91, normalized size = 1.08 \[ \frac {\sqrt {a+b x^2} \left (5 A b \left (a d+3 b c+b d x^2\right )-B \left (a+b x^2\right ) \left (2 a d-5 b c-3 b d x^2\right )\right )}{15 b^2}-\sqrt {a} A c \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 231, normalized size = 2.75 \[ \left [\frac {15 \, A \sqrt {a} b^{2} c \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, B b^{2} d x^{4} + {\left (5 \, B b^{2} c + {\left (B a b + 5 \, A b^{2}\right )} d\right )} x^{2} + 5 \, {\left (B a b + 3 \, A b^{2}\right )} c - {\left (2 \, B a^{2} - 5 \, A a b\right )} d\right )} \sqrt {b x^{2} + a}}{30 \, b^{2}}, \frac {15 \, A \sqrt {-a} b^{2} c \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, B b^{2} d x^{4} + {\left (5 \, B b^{2} c + {\left (B a b + 5 \, A b^{2}\right )} d\right )} x^{2} + 5 \, {\left (B a b + 3 \, A b^{2}\right )} c - {\left (2 \, B a^{2} - 5 \, A a b\right )} d\right )} \sqrt {b x^{2} + a}}{15 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 113, normalized size = 1.35 \[ \frac {A a c \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{9} c + 15 \, \sqrt {b x^{2} + a} A b^{10} c + 3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{8} d - 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b^{8} d + 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{9} d}{15 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 112, normalized size = 1.33 \[ -A \sqrt {a}\, c \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B d \,x^{2}}{5 b}+\sqrt {b \,x^{2}+a}\, A c +\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A d}{3 b}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B a d}{15 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B c}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 100, normalized size = 1.19 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B d x^{2}}{5 \, b} - A \sqrt {a} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \sqrt {b x^{2} + a} A c + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B c}{3 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a d}{15 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A d}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 101, normalized size = 1.20 \[ \sqrt {b\,x^2+a}\,\left (\frac {B\,d\,x^4}{5}-\frac {B\,a\,\left (2\,a\,d-5\,b\,c\right )}{15\,b^2}+\frac {B\,x^2\,\left (a\,d+5\,b\,c\right )}{15\,b}\right )+A\,c\,\sqrt {b\,x^2+a}+\frac {A\,d\,{\left (b\,x^2+a\right )}^{3/2}}{3\,b}-A\,\sqrt {a}\,c\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 59.29, size = 97, normalized size = 1.15 \[ \frac {A a c \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + A c \sqrt {a + b x^{2}} + \frac {B d \left (a + b x^{2}\right )^{\frac {5}{2}}}{5 b^{2}} + \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (2 A b d - 2 B a d + 2 B b c\right )}{6 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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