Optimal. Leaf size=47 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {444, 63, 217, 203} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 217
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {a d}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {a-b x^2}\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a-b x^2}}{\sqrt {c+d x^2}}\right )}{b}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 108, normalized size = 2.30 \begin {gather*} \frac {\sqrt {-b} \sqrt {-a d-b c} \sqrt {\frac {b \left (c+d x^2\right )}{a d+b c}} \sin ^{-1}\left (\frac {\sqrt {-b} \sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {-a d-b c}}\right )}{b^{3/2} \sqrt {d} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 46, normalized size = 0.98 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a-b x^2}}\right )}{\sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.39, size = 201, normalized size = 4.28 \begin {gather*} \left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c - a d\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}\right )}{4 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c - a d\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}}{2 \, {\left (b^{2} d^{2} x^{4} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, b d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 57, normalized size = 1.21 \begin {gather*} \frac {b \log \left ({\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 |}\right )}{\sqrt {-b d} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 108, normalized size = 2.30 \begin {gather*} \frac {\sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \arctan \left (\frac {\sqrt {b d}\, \left (2 b d \,x^{2}-a d +b c \right )}{2 \sqrt {-x^{4} b d +a d \,x^{2}-b c \,x^{2}+a c}\, b d}\right )}{2 \sqrt {b d}\, \sqrt {-x^{4} b d +a d \,x^{2}-b c \,x^{2}+a c}} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 48, normalized size = 1.02 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}{\sqrt {b\,d}\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \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 {x}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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