Optimal. Leaf size=202 \[ \frac {a (b c-a d)}{b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(3 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{8 b^3 e^2}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 b^2 e^2}+\frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{8 b^{7/2} \sqrt {d} e^{3/2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1981, 1980,
467, 464, 214} \begin {gather*} \frac {3 (b c-5 a d) (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 )}{8 b^{7/2} \sqrt {d} e^{3/2}}+\frac {\left (c+d x^2\right ) (3 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 b^3 e^2}+\frac {a (b c-a d)}{b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b^2 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 464
Rule 467
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int \frac {x^3}{\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 {-a e+c x^2}{x^2 \left (b e-d x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 b^2 e^2}-\frac {1}{4} ((b c-a d) e) \text {Subst}\left (\int \frac {\frac {4 a}{b}-\frac {3 (b c-a d) x^2}{b^2 e}}{x^2 \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 {(3 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{8 b^3 e^2}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 b^2 e^2}+\frac {1}{8} ((b c-a d) e) \text {Subst}\left (\int \frac {-\frac {8 a}{b^2 e}+\frac {(3 b c-7 a d) x^2}{b^3 e^2}}{x^2 \left (b e-d x^2\right )} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {a (b c-a d)}{b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(3 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{8 b^3 e^2}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 b^2 e^2}+\frac {(3 (b c-5 a d) (b c-a d)) \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 )}{8 b^3 e}\\ &=\frac {a (b c-a d)}{b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(3 b c-7 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{8 b^3 e^2}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{4 b^2 e^2}+\frac {3 (b c-5 a d) (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 )}{8 b^{7/2} \sqrt {d} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.21, size = 179, normalized size = 0.89 \begin {gather*} \frac {\sqrt {b} \sqrt {d} \sqrt {c+d x^2} \left (-15 a^2 d+a b \left (13 c-5 d x^2\right )+b^2 x^2 \left (5 c+2 d x^2\right )\right )+3 \left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 b^{7/2} \sqrt {d} e \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. \(678\) vs.
\(2(176)=352\).
time = 0.11, size = 679, normalized size = 3.36
method | result | size |
risch | \(-\frac {\left (-2 b d \,x^{2}+7 a d -5 b c \right ) \left (b \,x^{2}+a \right )}{8 b^{3} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (\frac {15 \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a^{2} d^{2}}{16 b^{3} \sqrt {d e b}}-\frac {9 \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a c d}{8 b^{2} \sqrt {d e b}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) c^{2}}{16 b \sqrt {d e b}}-\frac {a^{3} x^{2} d^{3}}{b^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 a^{2} x^{2} c \,d^{2}}{b^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a \,x^{2} c^{2} d}{b \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a^{3} c \,d^{2}}{b^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 a^{2} c^{2} d}{b^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a \,c^{3}}{b \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(615\) |
default | \(-\frac {\left (-4 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} d \,x^{4}-15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,d^{2} x^{2}+18 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c d \,x^{2}-3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{2}+10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a b d \,x^{2}-10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} c \,x^{2}-15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{2}+18 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c d -3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2}+14 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} d -10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a b c +16 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} d -16 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a b c \right ) \left (b \,x^{2}+a \right )}{16 b^{3} \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (d \,x^{2}+c \right ) \left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}}}\) | \(679\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 281, normalized size = 1.39 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, {\left (8 \, a b^{3} c - 8 \, a^{2} b^{2} d - \frac {3 \, {\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} {\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} + \frac {5 \, {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} {\left (b x^{2} + a\right )}}{d x^{2} + c}\right )}}{b^{3} d^{2} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {5}{2}} - 2 \, b^{4} d \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} + b^{5} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}} - \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \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} b^{3}}\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 554, normalized size = 2.74 \begin {gather*} \left [\frac {{\left (3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {b d} \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^{2} x^{4} + b c^{2} + a c d + {\left (3 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {b d} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right ) + 4 \, {\left (2 \, b^{3} d^{3} x^{6} + 13 \, a b^{2} c^{2} d - 15 \, a^{2} b c d^{2} + {\left (7 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{4} + {\left (5 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} - 15 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{32 \, {\left (b^{5} d x^{2} + a b^{4} d\right )}}, -\frac {{\left (3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {-b d} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{2 \, {\left (b^{2} d x^{2} + a b d\right )}}\right ) - 2 \, {\left (2 \, b^{3} d^{3} x^{6} + 13 \, a b^{2} c^{2} d - 15 \, a^{2} b c d^{2} + {\left (7 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{4} + {\left (5 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} - 15 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{16 \, {\left (b^{5} d x^{2} + a b^{4} d\right )}}\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 {x^3}{{\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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