Optimal. Leaf size=165 \[ \frac {3 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c^2}+\frac {(b c-a d) \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 \sqrt {a} (b c-a d) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 c^{5/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 165, 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,
294, 327, 214} \begin {gather*} -\frac {3 \sqrt {a} e^{3/2} (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 c^{5/2}}+\frac {3 e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c^2}+\frac {(b c-a d) \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 294
Rule 327
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^3} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {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 )\\ &=\frac {(b c-a d) \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(3 (b c-a d) e) \text {Subst}\left (\int \frac {x^2}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 c}\\ &=\frac {3 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c^2}+\frac {(b c-a d) \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {\left (3 a (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 )}{2 c^2}\\ &=\frac {3 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 c^2}+\frac {(b c-a d) \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{2 c \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 \sqrt {a} (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 )}{2 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 4.02, size = 146, normalized size = 0.88 \begin {gather*} \frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {c} \sqrt {a+b x^2} \left (2 b c x^2-a \left (c+3 d x^2\right )\right )-3 \sqrt {a} (b c-a d) 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 )\right )}{2 c^{5/2} x^2 \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. \(640\) vs.
\(2(141)=282\).
time = 0.13, size = 641, normalized size = 3.88
method | result | size |
risch | \(-\frac {a \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{2 c^{2} x^{2}}+\frac {\left (\frac {3 a^{2} \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d}{4 c^{2} \sqrt {a c e}}-\frac {3 a \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b}{4 c \sqrt {a c e}}-\frac {x^{2} a^{2} b \,d^{2}}{c^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 x^{2} a \,b^{2} d}{c \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {x^{2} b^{3}}{\left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a^{3} d^{2}}{c^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 a^{2} b d}{c \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a \,b^{2}}{\left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{b \,x^{2}+a}\) | \(526\) |
default | \(-\frac {\left (-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b \,d^{2} x^{6}-3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a^{2} c \,d^{2} x^{4}+3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a b \,c^{2} d \,x^{4}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,d^{2} x^{4}-4 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b c d \,x^{4}-3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a^{2} c^{2} d \,x^{2}+3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a b \,c^{3} x^{2}+2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, d \,x^{2}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a c d \,x^{2}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b \,c^{2} x^{2}+4 \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a c d \,x^{2}-4 \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b \,c^{2} x^{2}+2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, c \right ) \left (d \,x^{2}+c \right ) \left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}}}{4 \sqrt {a c}\, x^{2} c^{3} \left (b \,x^{2}+a \right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}}\) | \(641\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 177, normalized size = 1.07 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, {\left (a b c - a^{2} d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{a c^{2} - \frac {{\left (b x^{2} + a\right )} c^{3}}{d x^{2} + c}} + \frac {3 \, {\left (b c - a d\right )} a \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^{2}} + \frac {4 \, {\left (b c - a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{c^{2}}\right )} e^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 330, normalized size = 2.00 \begin {gather*} \left [-\frac {3 \, {\left (b c - a d\right )} x^{2} \sqrt {\frac {a}{c}} 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^{2} d + a c d^{2}\right )} x^{4} + 2 \, a c^{3} + {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {\frac {a}{c}}}{x^{4}}\right ) - 4 \, {\left ({\left (2 \, b c - 3 \, a d\right )} x^{2} - a c\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {3}{2}}}{8 \, c^{2} x^{2}}, \frac {3 \, {\left (b c - a d\right )} x^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b x^{2} + a^{2}\right )}}\right ) e^{\frac {3}{2}} + 2 \, {\left ({\left (2 \, b c - 3 \, a d\right )} x^{2} - a c\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {3}{2}}}{4 \, c^{2} 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 [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 = -0.01 \begin {gather*} \int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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