Optimal. Leaf size=318 \[ \frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{5/2} c^{7/2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 318, 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,
474, 466, 393, 214} \begin {gather*} -\frac {\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\sqrt {e} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) (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 )}{16 a^{5/2} c^{7/2}}+\frac {(3 a d+b c) (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {e^2 (b c-a d)^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 393
Rule 466
Rule 474
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {x^2 \left (b e-d x^2\right )^2}{\left (-a e+c x^2\right )^4} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {(b c-a d) \text {Subst}\left (\int \frac {x^2 \left (-3 \left (2 b^2 c^2 e^2-(b c e-a d e)^2\right )+6 a c d^2 e x^2\right )}{\left (-a e+c x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{6 a c^2}\\ &=\frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}-\frac {(b c-a d) \text {Subst}\left (\int \frac {3 c (b c-a d) (b c+3 a d) e^2-24 a c^2 d^2 e x^2}{\left (-a e+c x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{24 a c^4}\\ &=\frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\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}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{16 a^2 c^3}\\ &=\frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{5/2} c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.50, size = 222, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \sqrt {c+d x^2} \left (3 b^2 c^2 x^4-2 a b c x^2 \left (c-2 d x^2\right )+a^2 \left (-8 c^2+10 c d x^2-15 d^2 x^4\right )\right )-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 ) x^6 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{48 a^{5/2} c^{7/2} x^6 \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. \(848\) vs.
\(2(290)=580\).
time = 0.11, size = 849, normalized size = 2.67
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (15 d^{2} a^{2} x^{4}-4 a b c d \,x^{4}-3 b^{2} c^{2} x^{4}-10 a^{2} d c \,x^{2}+2 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 c^{3} x^{6} a^{2}}+\frac {\left (\frac {5 a \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d^{3}}{32 c^{3} \sqrt {a c e}}-\frac {3 \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b \,d^{2}}{32 c^{2} \sqrt {a c e}}-\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d \,b^{2}}{32 a c \sqrt {a c e}}-\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b^{3}}{32 a^{2} \sqrt {a c e}}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{b \,x^{2}+a}\) | \(447\) |
default | \(\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (66 b \,d^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{8} a^{2} \sqrt {a c}+24 b^{2} d^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{8} a c \sqrt {a c}+6 b^{3} d \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{8} c^{2} \sqrt {a c}+15 a^{4} \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 ) d^{3} c \,x^{6}-9 \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 ) d^{2} b \,a^{3} c^{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 ) d \,b^{2} a^{2} c^{3} x^{6}-3 c^{4} \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 ) b^{3} a \,x^{6}+66 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{3} a^{3} x^{6} \sqrt {a c}+54 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{2} b \,a^{2} c \,x^{6} \sqrt {a c}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{2} d a \,c^{2} x^{6} \sqrt {a c}+6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{3} c^{3} x^{6} \sqrt {a c}-66 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d^{2} a^{2} x^{4} \sqrt {a c}-24 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d b a c \,x^{4} \sqrt {a c}-6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b^{2} c^{2} x^{4} \sqrt {a c}+36 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d \,a^{2} c \,x^{2} \sqrt {a c}+12 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b a \,c^{2} x^{2} \sqrt {a c}-16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{2} c^{2} \sqrt {a c}\right )}{96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c^{4} a^{3} x^{6} \sqrt {a c}}\) | \(849\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 387, normalized size = 1.22 \begin {gather*} -\frac {1}{96} \, {\left (\frac {2 \, {\left (3 \, {\left (b^{3} c^{5} + a b^{2} c^{4} d - 13 \, a^{2} b c^{3} d^{2} + 11 \, a^{3} c^{2} d^{3}\right )} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} + 5 \, a^{4} c d^{3}\right )} \left (\frac {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} - 3 \, {\left (a^{2} b^{3} c^{3} + a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )}}{a^{5} c^{3} - \frac {3 \, {\left (b x^{2} + a\right )} a^{4} c^{4}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a^{3} c^{5}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} a^{2} c^{6}}{{\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 {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} a^{2} c^{3}}\right )} e^{\frac {1}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.24, size = 554, normalized size = 1.74 \begin {gather*} \left [-\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 )} \sqrt {a c} x^{6} e^{\frac {1}{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 d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (8 \, a^{3} c^{4} - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{6} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {1}{2}}}{192 \, a^{3} c^{4} x^{6}}, \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 )} \sqrt {-a c} x^{6} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {-a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{2 \, {\left (a b c x^{2} + a^{2} c\right )}}\right ) e^{\frac {1}{2}} - 2 \, {\left (8 \, a^{3} c^{4} - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{6} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {1}{2}}}{96 \, a^{3} c^{4} x^{6}}\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. 871 vs.
\(2 (287) = 574\).
time = 6.96, size = 871, normalized size = 2.74 \begin {gather*} \frac {1}{48} \, {\left (\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 )} \arctan \left (-\frac {\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{2} c^{3}} - \frac {3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} b^{3} c^{3} - 8 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a b^{3} c^{4} - 3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{2} b^{3} c^{5} + 3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} a b^{2} c^{2} d - 72 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{2} b^{2} c^{3} d - 51 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{3} b^{2} c^{4} d + 9 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} a^{2} b c d^{2} - 24 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{3} b c^{2} d^{2} - 105 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{4} b c^{3} d^{2} - 15 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} a^{3} d^{3} + 40 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{4} c d^{3} - 33 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{5} c^{2} d^{3} - 48 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{2} b^{2} c^{4} - 144 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{3} b c^{3} d - 16 \, \sqrt {b d} a^{4} b c^{4} d - 48 \, \sqrt {b d} a^{5} c^{3} d^{2}}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} - a c\right )}^{3} a^{2} c^{3}}\right )} e^{\frac {1}{2}} \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 {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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