Optimal. Leaf size=351 \[ -\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )}+\frac {\left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (a g^2+c f^2\right )}-\frac {\left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f} \left (a g^2+c f^2\right )} \]
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Rubi [A] time = 1.81, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {908, 37, 6725, 93, 208} \[ -\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )}+\frac {\left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (a g^2+c f^2\right )}-\frac {\left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f} \left (a g^2+c f^2\right )} \]
Antiderivative was successfully verified.
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Rule 37
Rule 93
Rule 208
Rule 908
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, 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 f^2+a g^2}-\frac {(g (e f-d g)) \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2}} \, dx}{c f^2+a g^2}\\ &=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}+\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 f^2+a g^2}\\ &=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}-\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} \left (c f^2+a g^2\right )}-\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} \left (c f^2+a g^2\right )}\\ &=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}-\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} \left (c f^2+a g^2\right )}-\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} \left (c f^2+a g^2\right )}\\ &=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}+\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} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (c f^2+a g^2\right )}-\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} \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c f^2+a g^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 265, normalized size = 0.75 \[ -\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )}+\frac {a \sqrt {\sqrt {-a} e-\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g-\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e-\sqrt {c} d}}\right )}{(-a)^{3/2} \left (\sqrt {-a} g-\sqrt {c} f\right )^{3/2}}+\frac {a \sqrt {\sqrt {-a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{(-a)^{3/2} \left (\sqrt {-a} g+\sqrt {c} f\right )^{3/2}} \]
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(-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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maple [B] time = 0.06, size = 5383, normalized size = 15.34 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )} {\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {d+e\,x}}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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