Optimal. Leaf size=694 \[ -\frac {\sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} (2 e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-d g)}+\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (a e^2 g+c d (2 e f-d g)\right ) \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) (e f-d g)}+\frac {\sqrt {-a} \sqrt {c} f \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e \sqrt {a+c x^2} \sqrt {f+g x} (e f-d g)}-\frac {\sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e \sqrt {a+c x^2} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
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Rubi [A] time = 1.72, antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {925, 6742, 719, 419, 844, 424, 933, 168, 538, 537} \[ -\frac {\sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} (2 e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} (e f-d g)}+\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (a e^2 g+c d (2 e f-d g)\right ) \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)}-\frac {\sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) (e f-d g)}+\frac {\sqrt {-a} \sqrt {c} f \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e \sqrt {a+c x^2} \sqrt {f+g x} (e f-d g)}-\frac {\sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e \sqrt {a+c x^2} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 925
Rule 933
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{(d+e x)^2 \sqrt {f+g x}} \, dx &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}+\frac {\int \frac {-a g+2 c f x+c g x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 (e f-d g)}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}+\frac {\int \left (\frac {c (2 e f-d g)}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {c g x}{e \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {-a e^2 g-c d (2 e f-d g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{2 (e f-d g)}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}+\frac {(c g) \int \frac {x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 e (e f-d g)}+\frac {(c (2 e f-d g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 e^2 (e f-d g)}-\frac {\left (a g+\frac {c d (2 e f-d g)}{e^2}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 (e f-d g)}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}+\frac {c \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{2 e (e f-d g)}-\frac {(c f) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 e (e f-d g)}-\frac {\left (\left (a g+\frac {c d (2 e f-d g)}{e^2}\right ) \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}} \sqrt {1+\frac {\sqrt {c} x}{\sqrt {-a}}} (d+e x) \sqrt {f+g x}} \, dx}{2 (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} (2 e f-d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} (2 e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (\left (a g+\frac {c d (2 e f-d g)}{e^2}\right ) \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {f+\frac {\sqrt {-a} g}{\sqrt {c}}-\frac {\sqrt {-a} g x^2}{\sqrt {c}}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{(e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} e (e f-d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (a \sqrt {c} f \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} e (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} (2 e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (\left (a g+\frac {c d (2 e f-d g)}{e^2}\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {1-\frac {\sqrt {-a} g x^2}{\sqrt {c} \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{(e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} (2 e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (a g+\frac {c d (2 e f-d g)}{e^2}\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 6.41, size = 1336, normalized size = 1.93 \[ \frac {\sqrt {f+g x} \left (\frac {c x^2+a}{d+e x}-\frac {c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^3-2 c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x) f^2-c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f^2+c e^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2 f+2 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x) f-2 i c d e g \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right ) f+a e^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} f-c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+i \sqrt {c} e \left (\sqrt {c} f+i \sqrt {a} g\right ) (d g-e f) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+e \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (\sqrt {a} e g+i \sqrt {c} (2 e f-d g)\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+i c d^2 g^2 \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-i a e^2 g^2 \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-a d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{e^2 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g) (f+g x)}\right )}{(d g-e f) \sqrt {c x^2+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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{2} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 6034, normalized size = 8.69 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{2} \sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+a}}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{2} \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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