Optimal. Leaf size=283 \[ \frac {x \sqrt {c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} (3 a d f-2 b c f+b d e) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 d^2 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {e^{3/2} \sqrt {c+d x^2} (b c-3 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 d} \]
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Rubi [A] time = 0.18, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ \frac {x \sqrt {c+d x^2} (3 a d f-2 b c f+b d e)}{3 d^2 \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} (3 a d f-2 b c f+b d e) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 d^2 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {e^{3/2} \sqrt {c+d x^2} (b c-3 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 492
Rule 528
Rule 531
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx &=\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 d}+\frac {\int \frac {-(b c-3 a d) e+(b d e-2 b c f+3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 d}\\ &=\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 d}-\frac {((b c-3 a d) e) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 d}+\frac {(b d e-2 b c f+3 a d f) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 d}\\ &=\frac {(b d e-2 b c f+3 a d f) x \sqrt {c+d x^2}}{3 d^2 \sqrt {e+f x^2}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 d}-\frac {(b c-3 a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {(e (b d e-2 b c f+3 a d f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac {(b d e-2 b c f+3 a d f) x \sqrt {c+d x^2}}{3 d^2 \sqrt {e+f x^2}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 d}-\frac {\sqrt {e} (b d e-2 b c f+3 a d f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 d^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {(b c-3 a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}
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Mathematica [C] time = 0.43, size = 212, normalized size = 0.75 \[ \frac {i e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (-3 a d f+2 b c f-b d e) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+b f x \sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right )-i b e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (c f-d e) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{3 d f \sqrt {\frac {d}{c}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}}{\sqrt {d x^{2} + c}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}}{\sqrt {d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 394, normalized size = 1.39 \[ \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \left (\sqrt {-\frac {d}{c}}\, b d \,f^{2} x^{5}+\sqrt {-\frac {d}{c}}\, b c \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, b d e f \,x^{3}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a d e f \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+\sqrt {-\frac {d}{c}}\, b c e f x -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b c e f \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b c e f \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b d \,e^{2} \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b d \,e^{2} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )\right )}{3 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) \sqrt {-\frac {d}{c}}\, d f} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}}{\sqrt {d x^{2} + c}}\,{d x} \]
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 \[ \int \frac {\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}}{\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}{\sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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