Optimal. Leaf size=141 \[ \frac {3 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 d^2}+\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )}{2 d}-\frac {3 \sqrt {b} (b c-a d) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 d^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1981, 1979,
294, 327, 214} \begin {gather*} -\frac {3 \sqrt {b} e^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 d^{5/2}}+\frac {3 e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 d^2}+\frac {\left (c+d x^2\right ) \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 294
Rule 327
Rule 1979
Rule 1981
Rubi steps
\begin {align*} \int x \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {x^4}{\left (b e-d x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )}{2 d}-\frac {(3 (b c-a d) e) \text {Subst}\left (\int \frac {x^2}{b e-d x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 d}\\ &=\frac {3 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 d^2}+\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )}{2 d}-\frac {\left (3 b (b c-a d) e^2\right ) \text {Subst}\left (\int \frac {1}{b e-d x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 d^2}\\ &=\frac {3 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 d^2}+\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )}{2 d}-\frac {3 \sqrt {b} (b c-a d) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.40, size = 136, normalized size = 0.96 \begin {gather*} \frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {d} \sqrt {a+b x^2} \left (3 b c-2 a d+b d x^2\right )-3 \sqrt {b} (b c-a d) \sqrt {c+d x^2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )\right )}{2 d^{5/2} \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. \(431\) vs.
\(2(117)=234\).
time = 0.10, size = 432, normalized size = 3.06
method | result | size |
default | \(\frac {\left (3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,d^{2} x^{2}-3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c d \,x^{2}+2 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b d \,x^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a c b d -3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2}+2 \sqrt {b d}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b c -4 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a d +4 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b c \right ) \left (d \,x^{2}+c \right ) \left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}}}{4 d^{2} \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (b \,x^{2}+a \right )}\) | \(432\) |
risch | \(\frac {\left (d \,x^{2}+c \right ) b e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{2 d^{2}}+\frac {\left (\frac {3 b \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a}{4 d \sqrt {d e b}}-\frac {3 b^{2} \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) c}{4 d^{2} \sqrt {d e b}}-\frac {x^{2} a^{2} b}{\left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 x^{2} a \,b^{2} c}{d \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {x^{2} b^{3} c^{2}}{d^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a^{3}}{\left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 a^{2} b c}{d \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a \,b^{2} c^{2}}{d^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{b \,x^{2}+a}\) | \(513\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 177, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, {\left (b^{2} c - a b d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{b d^{2} - \frac {{\left (b x^{2} + a\right )} d^{3}}{d x^{2} + c}} + \frac {3 \, {\left (b c - a d\right )} b \log \left (\frac {d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} - \sqrt {b d}}{d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} + \sqrt {b d}}\right )}{\sqrt {b d} d^{2}} + \frac {4 \, {\left (b c - a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{d^{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 = 0.45, size = 303, normalized size = 2.15 \begin {gather*} \left [-\frac {3 \, {\left (b c - a d\right )} \sqrt {\frac {b}{d}} e^{\frac {3}{2}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{3} x^{4} + b c^{2} d + a c d^{2} + {\left (3 \, b c d^{2} + a d^{3}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {\frac {b}{d}}\right ) - 4 \, {\left (b d x^{2} + 3 \, b c - 2 \, a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {3}{2}}}{8 \, d^{2}}, \frac {3 \, {\left (b c - a d\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} x^{2} + a b\right )}}\right ) e^{\frac {3}{2}} + 2 \, {\left (b d x^{2} + 3 \, b c - 2 \, a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {3}{2}}}{4 \, d^{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 [B] Leaf count of result is larger than twice the leaf count of optimal. 247 vs.
\(2 (113) = 226\).
time = 5.49, size = 247, normalized size = 1.75 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c} b \mathrm {sgn}\left (d x^{2} + c\right )}{d^{2}} + \frac {4 \, {\left (b^{2} c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) - 2 \, a b c d \mathrm {sgn}\left (d x^{2} + c\right ) + a^{2} d^{2} \mathrm {sgn}\left (d x^{2} + c\right )\right )}}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} d + \sqrt {b d} c\right )} d^{2}} + \frac {3 \, {\left (\sqrt {b d} b^{2} c \mathrm {sgn}\left (d x^{2} + c\right ) - \sqrt {b d} a b d \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} b d - \sqrt {b d} b c - \sqrt {b d} a d \right |}\right )}{b d^{3}}\right )} e^{\frac {3}{2}} \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 x\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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