Optimal. Leaf size=342 \[ \frac {\left (-\sqrt {-a} \sqrt {c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (\sqrt {-a} \sqrt {c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} c \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}+\frac {2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c} \]
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Rubi [A] time = 2.09, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {906, 63, 217, 206, 6725, 93, 208} \[ \frac {\left (-\sqrt {-a} \sqrt {c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (\sqrt {-a} \sqrt {c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} c \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}+\frac {2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 906
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{a+c x^2} \, dx &=\frac {\int \frac {c d f-a e g+c (e f+d g) x}{\sqrt {d+e x} \sqrt {f+g x} \left (a+c x^2\right )} \, dx}{c}+\frac {(e g) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{c}\\ &=\frac {\int \left (\frac {-a \sqrt {c} (e f+d g)+\sqrt {-a} (c d f-a e g)}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {a \sqrt {c} (e f+d g)+\sqrt {-a} (c d f-a e g)}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx}{c}+\frac {(2 g) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{c}\\ &=\frac {(2 g) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{c}-\frac {\left (c d f-a e g-\sqrt {-a} \sqrt {c} (e f+d g)\right ) \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} c}-\frac {\left (c d f-a e g+\sqrt {-a} \sqrt {c} (e f+d g)\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \sqrt {-a} c}\\ &=\frac {2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c}-\frac {\left (c d f-a e g-\sqrt {-a} \sqrt {c} (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {c} d+\sqrt {-a} e-\left (-\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} c}-\frac {\left (c d f-a e g+\sqrt {-a} \sqrt {c} (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} d+\sqrt {-a} e-\left (\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} c}\\ &=\frac {2 \sqrt {e} \sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c}+\frac {\left (c d f-a e g-\sqrt {-a} \sqrt {c} (e f+d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (c d f-a e g+\sqrt {-a} \sqrt {c} (e f+d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g}}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 339, normalized size = 0.99 \[ \frac {\frac {\frac {\left (\sqrt {-a} \sqrt {c} d-a e\right ) \sqrt {\sqrt {-a} g+\sqrt {c} f} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}-\frac {\left (\sqrt {-a} \sqrt {c} d+a e\right ) \sqrt {\sqrt {-a} g-\sqrt {c} f} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g-\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e-\sqrt {c} d}}\right )}{\sqrt {\sqrt {-a} e-\sqrt {c} d}}}{a}+\frac {2 \sqrt {g} \sqrt {e f-d g} \sqrt {\frac {e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{\sqrt {f+g x}}}{c} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1569, normalized size = 4.59 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {e x + d} \sqrt {g x + f}}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \frac {\sqrt {d + e x} \sqrt {f + g x}}{a + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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