Optimal. Leaf size=70 \[ \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d \sqrt {c^2 d-e}} \]
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Rubi [A] time = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {191, 4912, 12, 444, 63, 208} \[ \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d \sqrt {c^2 d-e}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 191
Rule 208
Rule 444
Rule 4912
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-(b c) \int \frac {x}{d \left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{d}\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{1-\frac {c^2 d}{e}+\frac {c^2 x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{d e}\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{d \sqrt {c^2 d-e}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 202, normalized size = 2.89 \[ \frac {\frac {2 a x}{\sqrt {d+e x^2}}+\frac {b \log \left (-\frac {4 c d \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \sqrt {c^2 d-e}}\right )}{\sqrt {c^2 d-e}}+\frac {b \log \left (-\frac {4 c d \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \sqrt {c^2 d-e}}\right )}{\sqrt {c^2 d-e}}+\frac {2 b x \tan ^{-1}(c x)}{\sqrt {d+e x^2}}}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 388, normalized size = 5.54 \[ \left [\frac {{\left (b e x^{2} + b d\right )} \sqrt {c^{2} d - e} \log \left (\frac {c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \, {\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt {c^{2} d - e} \sqrt {e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 4 \, \sqrt {e x^{2} + d} {\left ({\left (b c^{2} d - b e\right )} x \arctan \left (c x\right ) + {\left (a c^{2} d - a e\right )} x\right )}}{4 \, {\left (c^{2} d^{3} - d^{2} e + {\left (c^{2} d^{2} e - d e^{2}\right )} x^{2}\right )}}, \frac {{\left (b e x^{2} + b d\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d}}{2 \, {\left (c^{3} d^{2} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {e x^{2} + d} {\left ({\left (b c^{2} d - b e\right )} x \arctan \left (c x\right ) + {\left (a c^{2} d - a e\right )} x\right )}}{2 \, {\left (c^{2} d^{3} - d^{2} e + {\left (c^{2} d^{2} e - d e^{2}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.14, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctan \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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