Optimal. Leaf size=139 \[ -\frac {(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}}+\frac {(5 c d-2 b e) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}}+\frac {x \sqrt {d+e x^2}}{2 c} \]
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Rubi [A] time = 0.28, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1149, 416, 523, 217, 206, 377, 208} \[ -\frac {(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}}+\frac {(5 c d-2 b e) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}}+\frac {x \sqrt {d+e x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 217
Rule 377
Rule 416
Rule 523
Rule 1149
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac {\left (d+e x^2\right )^{3/2}}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\frac {x \sqrt {d+e x^2}}{2 c}+\frac {\int \frac {d e (3 c d-b e)+e^2 (5 c d-2 b e) x^2}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{2 c e}\\ &=\frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{2 c^2}+\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{c^2}\\ &=\frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{2 c^2}+\frac {(2 c d-b e)^2 \operatorname {Subst}\left (\int \frac {1}{\frac {-c d^2+b d e}{d}-\left (-c d e+\frac {e \left (-c d^2+b d e\right )}{d}\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c^2}\\ &=\frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}}-\frac {(2 c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 134, normalized size = 0.96 \[ -\frac {\frac {(2 b e-5 c d) \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{\sqrt {e}}-\frac {2 (b e-2 c d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {b e-2 c d}}{\sqrt {d+e x^2} \sqrt {b e-c d}}\right )}{\sqrt {e} \sqrt {b e-c d}}-c x \sqrt {d+e x^2}}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.97, size = 1079, normalized size = 7.76 \[ \left [\frac {2 \, \sqrt {e x^{2} + d} c e x - {\left (5 \, c d - 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, c d e - b e^{2}\right )} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, {\left ({\left (3 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, c^{2} e}, \frac {2 \, \sqrt {e x^{2} + d} c e x - 2 \, {\left (5 \, c d - 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (2 \, c d e - b e^{2}\right )} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, {\left ({\left (3 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, c^{2} e}, \frac {2 \, \sqrt {e x^{2} + d} c e x + 2 \, {\left (2 \, c d e - b e^{2}\right )} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}} \arctan \left (\frac {{\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}}}{2 \, {\left ({\left (2 \, c d e - b e^{2}\right )} x^{3} + {\left (2 \, c d^{2} - b d e\right )} x\right )}}\right ) - {\left (5 \, c d - 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right )}{4 \, c^{2} e}, \frac {\sqrt {e x^{2} + d} c e x - {\left (5 \, c d - 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, c d e - b e^{2}\right )} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}} \arctan \left (\frac {{\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}}}{2 \, {\left ({\left (2 \, c d e - b e^{2}\right )} x^{3} + {\left (2 \, c d^{2} - b d e\right )} x\right )}}\right )}{2 \, c^{2} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.39, size = 54, normalized size = 0.39 \[ -\frac {{\left (5 \, c d - 2 \, b e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2}\right )}{4 \, c^{2}} + \frac {\sqrt {x^{2} e + d} x}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 7043, normalized size = 50.67 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x^2+d\right )}^{5/2}}{-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x^{2}\right )^{\frac {3}{2}}}{b e - c d + c e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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