Optimal. Leaf size=244 \[ \frac {\left (c+d x^2\right ) \left (-a^2 d^2-2 a b c d+11 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 b^2 d^3}-\frac {\sqrt {e} (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{5/2} d^{7/2}}-\frac {\left (c+d x^2\right )^2 (a d+3 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 b d^3}+\frac {\left (c+d x^2\right )^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 b d^2 e} \]
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Rubi [A] time = 0.33, antiderivative size = 244, 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 = {1960, 463, 455, 385, 208} \[ \frac {\left (c+d x^2\right ) \left (-a^2 d^2-2 a b c d+11 b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 b^2 d^3}-\frac {\sqrt {e} (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{5/2} d^{7/2}}+\frac {\left (c+d x^2\right )^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 b d^2 e}-\frac {\left (c+d x^2\right )^2 (a d+3 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 b d^3} \]
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 455
Rule 463
Rule 1960
Rubi steps
\begin {align*} \int x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx &=((b c-a d) e) \operatorname {Subst}\left (\int \frac {x^2 \left (-a e+c x^2\right )^2}{\left (b e-d x^2\right )^4} \, 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 )^3}{6 b d^2 e}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {x^2 \left (-3 \left (2 a^2 d^2 e^2-(b c e-a d e)^2\right )+6 b c^2 d e x^2\right )}{\left (b e-d x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{6 b d^2}\\ &=-\frac {(3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{8 b d^3}+\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )^3}{6 b d^2 e}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {3 d (b c-a d) (3 b c+a d) e^2+24 b c^2 d^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 )}{24 b d^4}\\ &=\frac {\left (11 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^2 d^3}-\frac {(3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{8 b d^3}+\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )^3}{6 b d^2 e}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) e\right ) \operatorname {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 )}{16 b^2 d^3}\\ &=\frac {\left (11 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^2 d^3}-\frac {(3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{8 b d^3}+\frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \left (c+d x^2\right )^3}{6 b d^2 e}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{5/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 198, normalized size = 0.81 \[ \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {3 \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) (b c-a d)^{3/2} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {a+b x^2}}-b \sqrt {d} \left (c+d x^2\right ) \left (3 a^2 d^2-2 a b d \left (d x^2-2 c\right )+b^2 \left (-15 c^2+10 c d x^2-8 d^2 x^4\right )\right )\right )}{48 b^3 d^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 541, normalized size = 2.22 \[ \left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {\frac {e}{b d}} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e + 4 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d + a b c d^{2} + {\left (3 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{b d}}\right ) - 4 \, {\left (8 \, b^{2} d^{3} x^{6} + 15 \, b^{2} c^{3} - 4 \, a b c^{2} d - 3 \, a^{2} c d^{2} - 2 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{4} + {\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, b^{2} d^{3}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-\frac {e}{b d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{b d}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left (8 \, b^{2} d^{3} x^{6} + 15 \, b^{2} c^{3} - 4 \, a b c^{2} d - 3 \, a^{2} c d^{2} - 2 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{4} + {\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, b^{2} d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 243, normalized size = 1.00 \[ \frac {1}{96} \, {\left (2 \, \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{d} - \frac {5 \, b^{2} c d - a b d^{2}}{b^{2} d^{3}}\right )} + \frac {15 \, b^{2} c^{2} - 4 \, a b c d - 3 \, a^{2} d^{2}}{b^{2} d^{3}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} e - 3 \, a b^{2} c^{2} d e - a^{2} b c d^{2} e - a^{3} d^{3} e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -b c e - a d e - 2 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} \sqrt {b d} e^{\frac {1}{2}} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 527, normalized size = 2.16 \[ \frac {\sqrt {\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (3 a^{3} d^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a^{2} b c \,d^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+9 a \,b^{2} c^{2} d \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-15 b^{3} c^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-12 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a b \,d^{2} x^{2}-36 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} c d \,x^{2}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} d^{2}-24 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a b c d +30 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{2} c^{2}+16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b d}\, b d \right )}{96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b d}\, b^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.28, size = 414, normalized size = 1.70 \[ \frac {1}{96} \, e {\left (\frac {2 \, {\left (3 \, {\left (11 \, b^{3} c^{3} d^{2} - 13 \, a b^{2} c^{2} d^{3} + a^{2} b c d^{4} + a^{3} d^{5}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (5 \, b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} e + 3 \, {\left (5 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d - a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} e^{2}\right )}}{b^{5} d^{3} e^{3} - \frac {3 \, {\left (b x^{2} + a\right )} b^{4} d^{4} e^{3}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} b^{3} d^{5} e^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} b^{2} d^{6} e^{3}}{{\left (d x^{2} + c\right )}^{3}}} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\frac {d \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {b d e}}{d \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {b d e}}\right )}{\sqrt {b d e} b^{2} d^{3}}\right )} \]
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 \[ \int x^5\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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