Optimal. Leaf size=130 \[ \frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {c} \sqrt {e}} \]
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Rubi [A]
time = 0.11, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1981, 1980,
205, 214} \begin {gather*} \frac {(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 a^{3/2} \sqrt {c} \sqrt {e}}+\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {1}{\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 a \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) \text {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 a}\\ &=\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(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 a^{3/2} \sqrt {c} \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 143, normalized size = 1.10 \begin {gather*} \frac {-\sqrt {a} \sqrt {c} \left (a+b x^2\right ) \left (c+d x^2\right )+(b c-a d) x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} \sqrt {c} x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(325\) vs.
\(2(110)=220\).
time = 0.08, size = 326, normalized size = 2.51
method | result | size |
risch | \(-\frac {b \,x^{2}+a}{2 a \,x^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (-\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d}{4 \sqrt {a c e}}+\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b c}{4 a \sqrt {a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(226\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-2 b d \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{4} \sqrt {a c}+a^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d c \,x^{2}-c^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b a \,x^{2}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d a \,x^{2} \sqrt {a c}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b c \,x^{2} \sqrt {a c}+2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\right )}{4 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} c \,x^{2} \sqrt {a c}}\) | \(326\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 138, normalized size = 1.06 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, {\left (b c - a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{a^{2} - \frac {{\left (b x^{2} + a\right )} a c}{d x^{2} + c}} - \frac {{\left (b c - a d\right )} \log \left (\frac {c \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} - \sqrt {a c}}{c \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} + \sqrt {a c}}\right )}{\sqrt {a c} a}\right )} e^{\left (-\frac {1}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 310, normalized size = 2.38 \begin {gather*} \left [-\frac {{\left (\sqrt {a c} {\left (b c - a d\right )} x^{2} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (a c d x^{2} + a c^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{8 \, a^{2} c x^{2}}, -\frac {{\left (\sqrt {-a c} {\left (b c - a d\right )} x^{2} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {-a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{2 \, {\left (a b c x^{2} + a^{2} c\right )}}\right ) + 2 \, {\left (a c d x^{2} + a c^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{4 \, a^{2} c x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.01, size = 216, normalized size = 1.66 \begin {gather*} -\frac {{\left (\frac {{\left (b c - a d\right )} \arctan \left (-\frac {\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a} - \frac {{\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} b c + {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a d + 2 \, \sqrt {b d} a c}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} - a c\right )} a}\right )} e^{\left (-\frac {1}{2}\right )}}{2 \, \mathrm {sgn}\left (d x^{2} + c\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^3\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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