Optimal. Leaf size=281 \[ \frac {\left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^3 d^2 e}-\frac {(3 b c+5 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^2 d^2 e}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}{6 b d (b c-a d) e}+\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{7/2} d^{5/2} \sqrt {e}} \]
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Rubi [A]
time = 0.29, antiderivative size = 281, 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,
424, 393, 205, 214} \begin {gather*} \frac {(b c-a d) \left (5 a^2 d^2+2 a b c d+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^{7/2} d^{5/2} \sqrt {e}}+\frac {\left (c+d x^2\right ) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 b^3 d^2 e}-\frac {\left (c+d x^2\right )^2 (5 a d+3 b c) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 b^2 d^2 e}-\frac {\left (c+d x^2\right )^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 b d e (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 393
Rule 424
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {\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 {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}{6 b d (b c-a d) e}-\frac {(b c-a d) \text {Subst}\left (\int \frac {-a (b c+5 a d) e^2+3 c (b c+a d) e 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 )}{6 b d}\\ &=-\frac {(3 b c+5 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^2 d^2 e}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}{6 b d (b c-a d) e}+\frac {\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\left (b e-d x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{8 b^2 d^2}\\ &=\frac {\left (b^2 c^2+2 a b c d+5 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^3 d^2 e}-\frac {(3 b c+5 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^2 d^2 e}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}{6 b d (b c-a d) e}+\frac {\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right )\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 )}{16 b^3 d^2}\\ &=\frac {\left (b^2 c^2+2 a b c d+5 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^3 d^2 e}-\frac {(3 b c+5 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^2 d^2 e}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}{6 b d (b c-a d) e}+\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^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^{7/2} d^{5/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 1.54, size = 224, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a+b x^2} \left (\sqrt {d} \sqrt {a+b x^2} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \left (15 a^2 d^2-2 a b d \left (2 c+5 d x^2\right )+b^2 \left (-3 c^2+2 c d x^2+8 d^2 x^4\right )\right )+3 \sqrt {b c-a d} \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )\right )}{48 b^3 d^{5/2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}} \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. \(526\) vs.
\(2(253)=506\).
time = 0.09, size = 527, normalized size = 1.88
method | result | size |
risch | \(\frac {\left (8 b^{2} d^{2} x^{4}-10 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+15 a^{2} d^{2}-4 a b c d -3 b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{48 b^{3} d^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (-\frac {5 d \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a^{3}}{32 b^{3} \sqrt {d e b}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) a^{2} c}{32 b^{2} \sqrt {d e b}}+\frac {\ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) c^{2} a}{32 b d \sqrt {d e b}}+\frac {\ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +d e b \,x^{2}}{\sqrt {d e b}}+\sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) c^{3}}{32 d^{2} \sqrt {d e b}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(418\) |
default | \(\frac {\left (b \,x^{2}+a \right ) \left (-36 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{2} a b \,d^{2} \sqrt {b d}-12 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{2} c \,b^{2} d \sqrt {b d}-15 d^{3} \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3}+9 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c b \,d^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,c^{2} b^{2} d +3 b^{3} \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) c^{3}+16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b d \sqrt {b d}+30 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} d^{2} \sqrt {b d}-24 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a c b d \sqrt {b d}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, c^{2} b^{2} \sqrt {b d}\right )}{96 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{3} d^{2} \sqrt {b d}}\) | \(527\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 387, normalized size = 1.38 \begin {gather*} \frac {1}{96} \, {\left (\frac {2 \, {\left (3 \, {\left (b^{3} c^{3} d^{2} + a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - 5 \, a^{3} d^{5}\right )} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {5}{2}} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} - 3 \, {\left (b^{5} c^{3} + a b^{4} c^{2} d - 13 \, a^{2} b^{3} c d^{2} + 11 \, a^{3} b^{2} d^{3}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )}}{b^{6} d^{2} - \frac {3 \, {\left (b x^{2} + a\right )} b^{5} d^{3}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} b^{4} d^{4}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} b^{3} d^{5}}{{\left (d x^{2} + c\right )}^{3}}} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\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} d^{2}}\right )} e^{\left (-\frac {1}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 520, normalized size = 1.85 \begin {gather*} \left [-\frac {{\left (3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\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 (8 \, b^{3} d^{4} x^{6} - 3 \, b^{3} c^{3} d - 4 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} + 10 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{4} - {\left (b^{3} c^{2} d^{2} + 14 \, a b^{2} c d^{3} - 15 \, a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{192 \, b^{4} d^{3}}, -\frac {{\left (3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\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 (8 \, b^{3} d^{4} x^{6} - 3 \, b^{3} c^{3} d - 4 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} + 10 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{4} - {\left (b^{3} c^{2} d^{2} + 14 \, a b^{2} c d^{3} - 15 \, a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{96 \, b^{4} d^{3}}\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 [A]
time = 3.76, size = 210, normalized size = 0.75 \begin {gather*} \frac {{\left (2 \, \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{b} + \frac {b^{2} c d - 5 \, a b d^{2}}{b^{3} d^{2}}\right )} - \frac {3 \, b^{2} c^{2} + 4 \, a b c d - 15 \, a^{2} d^{2}}{b^{3} d^{2}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -b c - a d - 2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} \sqrt {b d} \right |}\right )}{\sqrt {b d} b^{3} d^{2}}\right )} e^{\left (-\frac {1}{2}\right )}}{96 \, \mathrm {sgn}\left (d x^{2} + c\right )} \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^5}{\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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