Integrand size = 46, antiderivative size = 24 \[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=x \sqrt {a+b x^2} \sqrt {c+d x^2} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {1604} \[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=x \sqrt {a+b x^2} \sqrt {c+d x^2} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = x \sqrt {a+b x^2} \sqrt {c+d x^2} \\ \end{align*}
Time = 7.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=x \sqrt {a+b x^2} \sqrt {c+d x^2} \]
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Time = 2.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\) | \(21\) |
default | \(x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\) | \(21\) |
risch | \(x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\) | \(21\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(62\) |
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none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x \]
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\[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {a c + 2 a d x^{2} + 2 b c x^{2} + 3 b d x^{4}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x \]
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\[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {3 \, b d x^{4} + 2 \, {\left (b c + a d\right )} x^{2} + a c}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \]
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Time = 9.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=x\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c} \]
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