Optimal. Leaf size=354 \[ -\frac {e \sqrt {f+g x}}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right ) (e f-d g)}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \sqrt {\sqrt {-a} g+\sqrt {c} f}} \]
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Rubi [A] time = 0.61, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {912, 96, 93, 208} \[ -\frac {e \sqrt {f+g x}}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right ) (e f-d g)}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \sqrt {\sqrt {-a} g+\sqrt {c} f}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 208
Rule 912
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx &=\int \left (\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}}+\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx}{2 \sqrt {-a}}\\ &=-\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}-\frac {\sqrt {c} \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \left (\sqrt {-a} \sqrt {c} d-a e\right )}-\frac {\sqrt {c} \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \left (\sqrt {-a} \sqrt {c} d+a e\right )}\\ &=-\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}-\frac {\sqrt {c} \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} \sqrt {c} d-a e}-\frac {\sqrt {c} \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} \sqrt {c} d+a e}\\ &=-\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {c} \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} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {c} \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} \left (\sqrt {c} d+\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f+\sqrt {-a} g}}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 287, normalized size = 0.81 \[ \frac {\frac {2 \sqrt {-a} e^2 \sqrt {f+g x}}{\sqrt {d+e x} \left (a e^2+c d^2\right ) (d g-e f)}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g-\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e-\sqrt {c} d}}\right )}{\left (\sqrt {-a} e-\sqrt {c} d\right )^{3/2} \sqrt {\sqrt {-a} g-\sqrt {c} f}}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \sqrt {\sqrt {-a} g+\sqrt {c} f}}}{\sqrt {-a}} \]
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 = 0.09, size = 10977, normalized size = 31.01 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {f+g\,x}\,\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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