Optimal. Leaf size=319 \[ \frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt {c+d x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{e \sqrt {c+d x^2} \sqrt {e+f x^2} (b e-a f)}-\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {d e-c f} E\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e \sqrt {c+d x^2} (b e-a f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.47, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {554, 422, 418, 492, 411} \[ \frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt {c+d x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} (d e-c f)}{e \sqrt {c+d x^2} \sqrt {e+f x^2} (b e-a f)}-\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {d e-c f} E\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e \sqrt {c+d x^2} (b e-a f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 422
Rule 492
Rule 554
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {(-d e+c f) x^2}{c}}}{\sqrt {1-\frac {(-b e+a f) x^2}{a}}} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {(-b e+a f) x^2}{a}} \sqrt {1-\frac {(-d e+c f) x^2}{c}}} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\left ((-d e+c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {(-b e+a f) x^2}{a}} \sqrt {1-\frac {(-d e+c f) x^2}{c}}} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{c e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac {(d e-c f) x \sqrt {a+b x^2}}{e (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (a (-d e+c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {(-b e+a f) x^2}{a}}}{\left (1-\frac {(-d e+c f) x^2}{c}\right )^{3/2}} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{c e (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac {(d e-c f) x \sqrt {a+b x^2}}{e (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {\sqrt {c} \sqrt {d e-c f} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e (b e-a f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 148, normalized size = 0.46 \[ \frac {\sqrt {a} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} E\left (\sin ^{-1}\left (\frac {\sqrt {a f-b e} x}{\sqrt {a} \sqrt {f x^2+e}}\right )|\frac {a (c f-d e)}{c (a f-b e)}\right )}{e \sqrt {a+b x^2} \sqrt {a f-b e} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{b f^{2} x^{6} + {\left (2 \, b e f + a f^{2}\right )} x^{4} + a e^{2} + {\left (b e^{2} + 2 \, a e f\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \,x^{2}+c}}{\sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}}{\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2}}}{\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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