Optimal. Leaf size=34 \[ -\frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {444, 37} \[ -\frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^2}}{(b c-a d) \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.97 \[ \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 48, normalized size = 1.41 \[ -\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 70, normalized size = 2.06 \[ -\frac {2 \, \sqrt {b d} b}{{\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.88 \[ \frac {\sqrt {d \,x^{2}+c}}{\sqrt {b \,x^{2}+a}\, \left (a d -b c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 45, normalized size = 1.32 \[ \frac {d\,x^2+c}{\left (a\,d\,\sqrt {d\,x^2+c}-b\,c\,\sqrt {d\,x^2+c}\right )\,\sqrt {b\,x^2+a}} \]
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 \[ \int \frac {x}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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