Optimal. Leaf size=153 \[ \frac {d x \sqrt {a+b x^2}}{a c \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}-\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {491, 12, 506,
422} \begin {gather*} -\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d x \sqrt {a+b x^2}}{a c \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 422
Rule 491
Rule 506
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}+\frac {\int \frac {b d x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a c}\\ &=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}+\frac {(b d) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a c}\\ &=\frac {d x \sqrt {a+b x^2}}{a c \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}-\frac {d \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{a}\\ &=\frac {d x \sqrt {a+b x^2}}{a c \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}-\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.22, size = 146, normalized size = 0.95 \begin {gather*} \frac {-\frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{c x}-i a \sqrt {\frac {b}{a}} \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 224, normalized size = 1.46
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{a c x}-\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(189\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{a c x}-\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(196\) |
default | \(\frac {\left (-\sqrt {-\frac {b}{a}}\, b d \,x^{4}-b c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )+b c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {b}{a}}\, a d \,x^{2}-\sqrt {-\frac {b}{a}}\, b c \,x^{2}-\sqrt {-\frac {b}{a}}\, a c \right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{x c \sqrt {-\frac {b}{a}}\, a \left (b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c \right )}\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {1}{x^{2} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{x^2\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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