Optimal. Leaf size=75 \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {(b c-a d)^2}{c d^2 \sqrt {c+d x^2}}+\frac {b^2 \sqrt {c+d x^2}}{d^2} \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {446, 87, 63, 208} \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {(b c-a d)^2}{c d^2 \sqrt {c+d x^2}}+\frac {b^2 \sqrt {c+d x^2}}{d^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 87
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {(b c-a d)^2}{c d (c+d x)^{3/2}}+\frac {b^2}{d \sqrt {c+d x}}+\frac {a^2}{c x \sqrt {c+d x}}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2}{c d^2 \sqrt {c+d x^2}}+\frac {b^2 \sqrt {c+d x^2}}{d^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {(b c-a d)^2}{c d^2 \sqrt {c+d x^2}}+\frac {b^2 \sqrt {c+d x^2}}{d^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{c d}\\ &=\frac {(b c-a d)^2}{c d^2 \sqrt {c+d x^2}}+\frac {b^2 \sqrt {c+d x^2}}{d^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 62, normalized size = 0.83 \[ \frac {a^2 d^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^2}{c}+1\right )+b c \left (-2 a d+2 b c+b d x^2\right )}{c d^2 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 232, normalized size = 3.09 \[ \left [\frac {{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}}, \frac {{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (b^{2} c^{2} d x^{2} + 2 \, b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {d x^{2} + c}}{c^{2} d^{3} x^{2} + c^{3} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 82, normalized size = 1.09 \[ \frac {a^{2} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {\sqrt {d x^{2} + c} b^{2}}{d^{2}} + \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{\sqrt {d x^{2} + c} c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 102, normalized size = 1.36 \[ \frac {b^{2} x^{2}}{\sqrt {d \,x^{2}+c}\, d}-\frac {a^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{c^{\frac {3}{2}}}+\frac {a^{2}}{\sqrt {d \,x^{2}+c}\, c}-\frac {2 a b}{\sqrt {d \,x^{2}+c}\, d}+\frac {2 b^{2} c}{\sqrt {d \,x^{2}+c}\, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 90, normalized size = 1.20 \[ \frac {b^{2} x^{2}}{\sqrt {d x^{2} + c} d} - \frac {a^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{c^{\frac {3}{2}}} + \frac {a^{2}}{\sqrt {d x^{2} + c} c} + \frac {2 \, b^{2} c}{\sqrt {d x^{2} + c} d^{2}} - \frac {2 \, a b}{\sqrt {d x^{2} + c} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.86, size = 76, normalized size = 1.01 \[ \frac {b^2\,\sqrt {d\,x^2+c}}{d^2}-\frac {a^2\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{c\,d^2\,\sqrt {d\,x^2+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.05, size = 70, normalized size = 0.93 \[ \frac {a^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{c \sqrt {- c}} + \frac {b^{2} \sqrt {c + d x^{2}}}{d^{2}} + \frac {\left (a d - b c\right )^{2}}{c d^{2} \sqrt {c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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