Optimal. Leaf size=218 \[ -\frac {(b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) (3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{5/2} c^{3/2} \sqrt {e}} \]
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Rubi [A]
time = 0.17, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1981, 1980,
393, 205, 214} \begin {gather*} -\frac {(a d+3 b c) (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 )}{8 a^{5/2} c^{3/2} \sqrt {e}}-\frac {(a d+3 b c) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {e (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 393
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {b e-d x^2}{\left (-a e+c x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=-\frac {(b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {((b c-a d) (3 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 )}{4 a c}\\ &=-\frac {(b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) (3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {((b c-a d) (3 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 )}{8 a^2 c}\\ &=-\frac {(b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) (3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) (3 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 )}{8 a^{5/2} c^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 1.23, size = 173, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a} \sqrt {c} \left (a+b x^2\right ) \sqrt {c+d x^2} \left (3 b c x^2-a \left (2 c+d x^2\right )\right )-\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) x^4 \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{3/2} x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \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. \(558\) vs.
\(2(194)=388\).
time = 0.09, size = 559, normalized size = 2.56
method | result | size |
risch | \(-\frac {\left (b \,x^{2}+a \right ) \left (a d \,x^{2}-3 b c \,x^{2}+2 a c \right )}{8 a^{2} x^{4} c \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d^{2}}{16 c \sqrt {a c e}}+\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b d}{8 a \sqrt {a c e}}-\frac {3 c \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b^{2}}{16 a^{2} \sqrt {a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(326\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 b \,d^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} a \sqrt {a c}+10 b^{2} d \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} c \sqrt {a c}-a^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d^{2} c \,x^{4}-2 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d b \,a^{2} c^{2} x^{4}+3 c^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b^{2} a \,x^{4}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{2} a^{2} x^{4} \sqrt {a c}+8 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c d \,x^{4}+10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{2} c^{2} x^{4} \sqrt {a c}-2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d a \,x^{2} \sqrt {a c}-10 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b c \,x^{2} \sqrt {a c}+4 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a c \sqrt {a c}\right )}{16 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{3} c^{2} x^{4} \sqrt {a c}}\) | \(559\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 252, normalized size = 1.16 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, {\left ({\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )}}{a^{4} c - \frac {2 \, {\left (b x^{2} + a\right )} a^{3} c^{2}}{d x^{2} + c} + \frac {{\left (b x^{2} + a\right )}^{2} a^{2} c^{3}}{{\left (d x^{2} + c\right )}^{2}}} + \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\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^{2} c}\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 = 0.66, size = 420, normalized size = 1.93 \begin {gather*} \left [-\frac {{\left ({\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {a c} x^{4} \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 (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{32 \, a^{3} c^{2} x^{4}}, \frac {{\left ({\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-a c} x^{4} \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 (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{16 \, a^{3} c^{2} x^{4}}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 486 vs.
\(2 (194) = 388\).
time = 3.53, size = 486, normalized size = 2.23 \begin {gather*} \frac {{\left (\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\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^{2} c} - \frac {3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} b^{2} c^{2} - 5 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a b^{2} c^{3} - 2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a b c d - 10 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{2} b c^{2} d - {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{2} d^{2} - {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{3} c d^{2} - 8 \, \sqrt {b d} a^{2} b c^{3} - 8 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{2} c d}{{\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 )}^{2} a^{2} c}\right )} e^{\left (-\frac {1}{2}\right )}}{8 \, \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 = -0.00 \begin {gather*} \int \frac {1}{x^5\,\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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