Optimal. Leaf size=473 \[ -\frac {2 \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right )}+\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} (d g+e f) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 g \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e g \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
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Rubi [A] time = 0.64, antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {923, 933, 168, 538, 537, 844, 719, 424, 419} \[ -\frac {2 \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right )}+\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} (d g+e f) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e^2 g \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{e g \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 923
Rule 933
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{(d+e x) \sqrt {f+g x}} \, dx &=\left (a+\frac {c d^2}{e^2}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx-\frac {\int \frac {c d-c e x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^2}\\ &=\frac {c \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{e g}-\frac {(c (e f+d g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^2 g}+\frac {\left (\left (a+\frac {c d^2}{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}{\sqrt {a+c x^2}}\\ &=-\frac {\left (2 \left (a+\frac {c d^2}{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 )}{\sqrt {a+c x^2}}+\frac {\left (2 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 g \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a \sqrt {c} (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 g \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {2 \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 g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} (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 g \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 \left (a+\frac {c d^2}{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 )}{\sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {2 \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 g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} (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 g \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 \left (a+\frac {c d^2}{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 ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 4.31, size = 1096, normalized size = 2.32 \[ -\frac {2 \left (-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-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+\sqrt {c} e \left (\sqrt {a} g-i \sqrt {c} f\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 (i \sqrt {c} d+\sqrt {a} e\right ) g \left (\sqrt {c} f+i \sqrt {a} 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}}}\right )}{e^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g) \sqrt {f+g x} \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 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1216, normalized size = 2.57 \[ \frac {2 \left (-a c d e \,g^{3} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+a c d e \,g^{3} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+a c \,e^{2} f \,g^{2} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-a c \,e^{2} f \,g^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+a c \,e^{2} f \,g^{2} \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-c^{2} d^{2} f \,g^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+c^{2} d^{2} f \,g^{2} \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-c^{2} d e \,f^{2} g \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+c^{2} d e \,f^{2} g \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+c^{2} e^{2} f^{3} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-\sqrt {-a c}\, a \,e^{2} g^{3} \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+\sqrt {-a c}\, c \,d^{2} g^{3} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-\sqrt {-a c}\, c \,d^{2} g^{3} \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-\sqrt {-a c}\, c \,e^{2} f^{2} g \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )\right ) \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{-c f +\sqrt {-a c}\, g}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{c f +\sqrt {-a c}\, g}}\, \sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{\left (d g -e f \right ) \left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right ) c \,e^{2} g^{2}} \]
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 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )} \sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+a}}{\sqrt {f+g\,x}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right ) \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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