Optimal. Leaf size=127 \[ \frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\sqrt {e} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} c^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1960, 288, 208} \[ \frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\sqrt {e} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 288
Rule 1960
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^3} \, dx &=((b c-a d) e) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a e+c x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {((b c-a d) e) \operatorname {Subst}\left (\int \frac {1}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 c}\\ &=\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 133, normalized size = 1.05 \[ \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} c^{3/2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{c x^2}\right )}{2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 333, normalized size = 2.62 \[ \left [-\frac {{\left (b c - a d\right )} x^{2} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) + 4 \, {\left (d x^{2} + c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{8 \, c x^{2}}, \frac {{\left (b c - a d\right )} x^{2} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) - 2 \, {\left (d x^{2} + c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{4 \, c x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.68, size = 269, normalized size = 2.12 \[ \frac {1}{2} \, {\left (\frac {{\left (b c e - a d e\right )} \arctan \left (-\frac {\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}}{\sqrt {-a c e}}\right )}{\sqrt {-a c e} c} - \frac {{\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} b c e + {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} a d e + 2 \, \sqrt {b d} a c e^{\frac {3}{2}}}{{\left (a c e - {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{2}\right )} c}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 326, normalized size = 2.57 \[ -\frac {\sqrt {\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-a^{2} c d \,x^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+a b \,c^{2} x^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b d \,x^{4}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a d \,x^{2}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b c \,x^{2}+2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\right )}{4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a \,c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.23, size = 145, normalized size = 1.14 \[ \frac {1}{4} \, e {\left (\frac {2 \, {\left (b c - a d\right )} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{a c e - \frac {{\left (b x^{2} + a\right )} c^{2} e}{d x^{2} + c}} + \frac {{\left (b c - a d\right )} \log \left (\frac {c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {a c e}}{c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {a c e}}\right )}{\sqrt {a c e} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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