Optimal. Leaf size=256 \[ -\frac {d (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^3}-\frac {a (b c-a d)^2 e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(5 b c-9 a d) (b c-a d) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 (b c-5 a d) (b c-a d) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 \sqrt {a} c^{7/2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1981, 1980,
466, 1171, 396, 214} \begin {gather*} -\frac {3 e^{3/2} (b c-5 a d) (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 \sqrt {a} c^{7/2}}-\frac {a e^3 (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {e^2 (5 b c-9 a d) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {d e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 396
Rule 466
Rule 1171
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^5} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {x^4 \left (b e-d x^2\right )}{\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 {a (b c-a d)^2 e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {((b c-a d) e) \text {Subst}\left (\int \frac {-a (b c-a d) e^2-4 c (b c-a d) e x^2+4 c^2 d x^4}{\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 c^3}\\ &=-\frac {a (b c-a d)^2 e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(5 b c-9 a d) (b c-a d) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \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 {-a (3 b c-7 a d) e^2+8 a c d e x^2}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{8 a c^3}\\ &=-\frac {d (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^3}-\frac {a (b c-a d)^2 e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(5 b c-9 a d) (b c-a d) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {\left (3 (b c-5 a d) (b c-a d) e^2\right ) \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 c^3}\\ &=-\frac {d (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^3}-\frac {a (b c-a d)^2 e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(5 b c-9 a d) (b c-a d) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 c^3 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 (b c-5 a d) (b c-a d) e^{3/2} \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 \sqrt {a} c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 4.47, size = 186, normalized size = 0.73 \begin {gather*} -\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (b c x^2 \left (5 c+13 d x^2\right )+a \left (2 c^2-5 c d x^2-15 d^2 x^4\right )\right )+3 \left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) x^4 \sqrt {c+d x^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{8 \sqrt {a} c^{7/2} x^4 \sqrt {a+b 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. \(1041\) vs.
\(2(230)=460\).
time = 0.15, size = 1042, normalized size = 4.07
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (-7 a d \,x^{2}+5 b c \,x^{2}+2 a c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{8 c^{3} x^{4}}+\frac {\left (-\frac {15 \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) a^{2} d^{2}}{16 c^{3} \sqrt {a c e}}+\frac {9 \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) a b d}{8 c^{2} \sqrt {a c e}}-\frac {3 \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b^{2}}{16 c \sqrt {a c e}}+\frac {d^{3} x^{2} a^{2} b}{c^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d^{2} x^{2} a \,b^{2}}{c^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {d \,x^{2} b^{3}}{c \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {d^{3} a^{3}}{c^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d^{2} a^{2} b}{c^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {d a \,b^{2}}{c \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{b \,x^{2}+a}\) | \(628\) |
default | \(-\frac {\left (18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b \,d^{3} x^{8}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} c \,d^{2} x^{8}+15 \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 ) a^{3} c \,d^{3} x^{6}-18 \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 ) a^{2} b \,c^{2} d^{2} x^{6}+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 ) a \,b^{2} c^{3} d \,x^{6}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} d^{3} x^{6}+26 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c \,d^{2} x^{6}-12 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} c^{2} d \,x^{6}+15 \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 ) a^{3} c^{2} d^{2} x^{4}-18 \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 ) a^{2} b \,c^{3} d \,x^{4}+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 ) a \,b^{2} c^{4} x^{4}-18 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a \,d^{2} x^{4}+6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b c d \,x^{4}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} c \,d^{2} x^{4}+8 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b \,c^{2} d \,x^{4}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} c^{3} x^{4}-16 \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} c \,d^{2} x^{4}+16 \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a b \,c^{2} d \,x^{4}-14 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a c d \,x^{2}+6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b \,c^{2} x^{2}+4 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a \,c^{2}\right ) \left (d \,x^{2}+c \right ) \left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}}}{16 a \sqrt {a c}\, x^{4} c^{4} \left (b \,x^{2}+a \right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}}\) | \(1042\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 282, normalized size = 1.10 \begin {gather*} -\frac {1}{16} \, {\left (\frac {2 \, {\left ({\left (5 \, b^{2} c^{3} - 14 \, a b c^{2} d + 9 \, a^{2} c d^{2}\right )} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (3 \, a b^{2} c^{2} - 10 \, a^{2} b c d + 7 \, a^{3} d^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )}}{a^{2} c^{3} - \frac {2 \, {\left (b x^{2} + a\right )} a c^{4}}{d x^{2} + c} + \frac {{\left (b x^{2} + a\right )}^{2} c^{5}}{{\left (d x^{2} + c\right )}^{2}}} + \frac {16 \, {\left (b c d - a d^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{c^{3}} - \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, 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} c^{3}}\right )} e^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.23, size = 422, normalized size = 1.65 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {a c} x^{4} e^{\frac {3}{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 (2 \, a^{2} c^{3} + {\left (13 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{4} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {3}{2}}}{32 \, a c^{4} x^{4}}, \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, 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 ) e^{\frac {3}{2}} - 2 \, {\left (2 \, a^{2} c^{3} + {\left (13 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{4} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {3}{2}}}{16 \, a c^{4} 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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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