Optimal. Leaf size=366 \[ -\frac {e^2 \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {e^{3/2} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{3/2} c^{9/2}}+\frac {d^2 e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {e^3 (11 a d+b c) (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {e^2 (b c-a d)^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3} \]
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Rubi [A] time = 0.37, antiderivative size = 366, 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 = {1960, 463, 455, 1157, 388, 208} \[ -\frac {e^2 \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {e^{3/2} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{3/2} c^{9/2}}+\frac {d^2 e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {e^2 (b c-a d)^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {e^3 (11 a d+b c) (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \]
Antiderivative was successfully verified.
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Rule 208
Rule 388
Rule 455
Rule 463
Rule 1157
Rule 1960
Rubi steps
\begin {align*} \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx &=((b c-a d) e) \operatorname {Subst}\left (\int \frac {x^4 \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 )^{5/2}}{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) \operatorname {Subst}\left (\int \frac {x^4 \left (-6 b^2 c^2 e^2+5 (b c e-a d e)^2+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)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {a c (b c-a d) (b c+11 a d) e^3+4 c^2 (b c-a d) (b c+11 a d) e^2 x^2-24 a c^3 d^2 e x^4}{\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^5}\\ &=\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {3 a c \left (b^2 c^2+10 a b c d-19 a^2 d^2\right ) e^3-48 a^2 c^2 d^2 e^2 x^2}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{48 a^2 c^5 e}\\ &=\frac {d^2 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^2\right ) \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 )}{16 a c^4}\\ &=\frac {d^2 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^{3/2} \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^{3/2} c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 245, normalized size = 0.67 \[ \frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (3 x^6 \sqrt {c+d x^2} \left (35 a^3 d^3-45 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )-\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )+2 a b c x^2 \left (7 c^2-19 c d x^2-50 d^2 x^4\right )+3 b^2 c^2 x^4 \left (c+d x^2\right )\right )\right )}{48 a^{3/2} c^{9/2} x^6 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 14.42, size = 573, normalized size = 1.57 \[ \left [\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e + {\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, a c^{4} x^{6}}, -\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e x^{6} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left ({\left (3 \, b^{2} c^{2} d - 100 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} e x^{6} + 8 \, a^{2} c^{3} e + {\left (3 \, b^{2} c^{3} - 38 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} e x^{4} + 14 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, a c^{4} x^{6}}\right ] \]
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 = 0.10, size = 1498, normalized size = 4.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 453, normalized size = 1.24 \[ \frac {1}{96} \, e {\left (\frac {2 \, {\left (3 \, {\left (b^{3} c^{5} - 23 \, a b^{2} c^{4} d + 51 \, a^{2} b c^{3} d^{2} - 29 \, a^{3} c^{2} d^{3}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {5}{2}} e + 8 \, {\left (a b^{3} c^{4} + 9 \, a^{2} b^{2} c^{3} d - 27 \, a^{3} b c^{2} d^{2} + 17 \, a^{4} c d^{3}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} e^{2} - 3 \, {\left (a^{2} b^{3} c^{3} + 9 \, a^{3} b^{2} c^{2} d - 29 \, a^{4} b c d^{2} + 19 \, a^{5} d^{3}\right )} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} e^{3}\right )}}{a^{4} c^{4} e^{3} - \frac {3 \, {\left (b x^{2} + a\right )} a^{3} c^{5} e^{3}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a^{2} c^{6} e^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} a c^{7} e^{3}}{{\left (d x^{2} + c\right )}^{3}}} + \frac {96 \, {\left (b c d^{2} - a d^{3}\right )} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{c^{4}} - \frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} e \log \left (\frac {c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {a c e}}{c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {a c e}}\right )}{\sqrt {a c e} a c^{4}}\right )} \]
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 \[ \int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^7} \,d x \]
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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