Optimal. Leaf size=91 \[ -\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {x \left (b^2 c^2-2 a d (b c-a d)\right )}{c^2 d \sqrt {c+d x^2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 385, 217, 206} \begin {gather*} -\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {x \left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right )}{\sqrt {c+d x^2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 385
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{3/2}} \, dx &=-\frac {a^2}{c x \sqrt {c+d x^2}}+\frac {\int \frac {2 a (b c-a d)+b^2 c x^2}{\left (c+d x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right ) x}{\sqrt {c+d x^2}}+\frac {b^2 \int \frac {1}{\sqrt {c+d x^2}} \, dx}{d}\\ &=-\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right ) x}{\sqrt {c+d x^2}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{d}\\ &=-\frac {a^2}{c x \sqrt {c+d x^2}}-\frac {\left (\frac {b^2}{d}-\frac {2 a (b c-a d)}{c^2}\right ) x}{\sqrt {c+d x^2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 81, normalized size = 0.89 \begin {gather*} \frac {b^2 \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{d^{3/2}}-\frac {\sqrt {c+d x^2} \left (a^2+\frac {x^2 (b c-a d)^2}{d \left (c+d x^2\right )}\right )}{c^2 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 92, normalized size = 1.01 \begin {gather*} \frac {-a^2 c d-2 a^2 d^2 x^2+2 a b c d x^2-b^2 c^2 x^2}{c^2 d x \sqrt {c+d x^2}}-\frac {b^2 \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.55, size = 239, normalized size = 2.63 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left (c^{2} d^{3} x^{3} + c^{3} d^{2} x\right )}}, -\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{c^{2} d^{3} x^{3} + c^{3} d^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 104, normalized size = 1.14 \begin {gather*} -\frac {b^{2} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, d^{\frac {3}{2}}} + \frac {2 \, a^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} c} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt {d x^{2} + c} c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 99, normalized size = 1.09 \begin {gather*} -\frac {2 a^{2} d x}{\sqrt {d \,x^{2}+c}\, c^{2}}+\frac {2 a b x}{\sqrt {d \,x^{2}+c}\, c}-\frac {b^{2} x}{\sqrt {d \,x^{2}+c}\, d}+\frac {b^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}-\frac {a^{2}}{\sqrt {d \,x^{2}+c}\, c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 91, normalized size = 1.00 \begin {gather*} \frac {2 \, a b x}{\sqrt {d x^{2} + c} c} - \frac {b^{2} x}{\sqrt {d x^{2} + c} d} - \frac {2 \, a^{2} d x}{\sqrt {d x^{2} + c} c^{2}} + \frac {b^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {3}{2}}} - \frac {a^{2}}{\sqrt {d x^{2} + c} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^2}{x^2\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{2}}{x^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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