Optimal. Leaf size=255 \[ -\frac {3 (5 b c-a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{7/2} \sqrt {c} e^{3/2}}-\frac {(7 b c-3 a d) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac {c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {b (b c-a d)}{a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \]
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Rubi [A] time = 0.23, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1960, 456, 453, 208} \[ -\frac {(7 b c-3 a d) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac {c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 (5 b c-a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{7/2} \sqrt {c} e^{3/2}}-\frac {(b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {b (b c-a d)}{a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 453
Rule 456
Rule 1960
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=((b c-a d) e) \operatorname {Subst}\left (\int \frac {b e-d x^2}{x^2 \left (-a e+c x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=-\frac {(b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {1}{4} ((b c-a d) e) \operatorname {Subst}\left (\int \frac {\frac {4 b}{a}+\frac {3 (b c-a d) x^2}{a^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 )\\ &=-\frac {(b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(7 b c-3 a d) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac {c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {1}{8} ((b c-a d) e) \operatorname {Subst}\left (\int \frac {\frac {8 b}{a^2 e}+\frac {(7 b c-3 a d) x^2}{a^3 e^2}}{x^2 \left (-a e+c x^2\right )} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {b (b c-a d)}{a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(7 b c-3 a d) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac {c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(3 (b c-a d) (5 b c-a d)) \operatorname {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 )}{8 a^3 e}\\ &=\frac {b (b c-a d)}{a^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(7 b c-3 a d) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac {c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {3 (b c-a d) (5 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 )}{8 a^{7/2} \sqrt {c} e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 189, normalized size = 0.74 \[ \frac {\sqrt {a} \sqrt {c} \sqrt {c+d x^2} \left (-a^2 \left (2 c+5 d x^2\right )+a b x^2 \left (5 c-13 d x^2\right )+15 b^2 c x^4\right )-3 x^4 \sqrt {a+b x^2} \left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{7/2} \sqrt {c} e x^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 7.60, size = 613, normalized size = 2.40 \[ \left [\frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{6} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4}\right )} \sqrt {a c e} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} - 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left ({\left (15 \, a b^{2} c^{2} d - 13 \, a^{2} b c d^{2}\right )} x^{6} - 2 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{4} + {\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, {\left (a^{4} b c e^{2} x^{6} + a^{5} c e^{2} x^{4}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{6} + {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4}\right )} \sqrt {-a c e} \arctan \left (\frac {\sqrt {-a c e} {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (a b c e x^{2} + a^{2} c e\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{2} c^{2} d - 13 \, a^{2} b c d^{2}\right )} x^{6} - 2 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{4} + {\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, {\left (a^{4} b c e^{2} x^{6} + a^{5} c e^{2} x^{4}\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1042, normalized size = 4.09 \[ -\frac {\left (3 a^{3} b c \,d^{2} x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-18 a^{2} b^{2} c^{2} d \,x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+15 a \,b^{3} c^{3} x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} d^{2} x^{8}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{3} c d \,x^{8}+3 a^{4} c \,d^{2} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-18 a^{3} b \,c^{2} d \,x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+15 a^{2} b^{2} c^{3} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-12 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} b \,d^{2} x^{6}+26 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} c d \,x^{6}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{3} c^{2} x^{6}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{3} d^{2} x^{4}+16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a^{2} b c d \,x^{4}+8 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} b c d \,x^{4}-16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} x^{4}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} c^{2} x^{4}+6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a b d \,x^{4}-18 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b^{2} c \,x^{4}+6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} d \,x^{2}-14 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a b c \,x^{2}+4 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} c \right ) \left (b \,x^{2}+a \right )}{16 \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (d \,x^{2}+c \right ) \left (\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}\right )^{\frac {3}{2}} a^{4} c \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 311, normalized size = 1.22 \[ \frac {1}{16} \, e {\left (\frac {2 \, {\left (8 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} e^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left (b x^{2} + a\right )}^{2} e^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {5 \, {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} {\left (b x^{2} + a\right )} e^{2}}{d x^{2} + c}\right )}}{a^{3} c^{2} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {5}{2}} e^{2} - 2 \, a^{4} c \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} e^{3} + a^{5} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} e^{4}} + \frac {3 \, {\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac {c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {a c e}}{c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {a c e}}\right )}{\sqrt {a c e} a^{3} e^{2}}\right )} \]
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 \[ \int \frac {1}{x^5\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \]
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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