Optimal. Leaf size=137 \[ \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}+\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}-\frac {b \left (2 c^2 d-e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{c e^2 \sqrt {c^2 d-e}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c e^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {266, 43, 4976, 12, 523, 217, 206, 377, 203} \[ \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}+\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}-\frac {b \left (2 c^2 d-e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{c e^2 \sqrt {c^2 d-e}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c e^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 203
Rule 206
Rule 217
Rule 266
Rule 377
Rule 523
Rule 4976
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}-(b c) \int \frac {2 d+e x^2}{e^2 \left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx\\ &=\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac {(b c) \int \frac {2 d+e x^2}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{e^2}\\ &=\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac {b \int \frac {1}{\sqrt {d+e x^2}} \, dx}{c e}-\frac {\left (b c \left (2 d-\frac {e}{c^2}\right )\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{e^2}\\ &=\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c e}-\frac {\left (b c \left (2 d-\frac {e}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{e^2}\\ &=\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac {b c \left (2 d-\frac {e}{c^2}\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c^2 d-e} e^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c e^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.66, size = 321, normalized size = 2.34 \[ \frac {\frac {2 a \left (2 d+e x^2\right )}{\sqrt {d+e x^2}}-\frac {i b \left (2 c^2 d-e\right ) \log \left (\frac {4 c^2 e^2 \left (-i \sqrt {c^2 d-e} \sqrt {d+e x^2}-i c d+e x\right )}{b (c x-i) \sqrt {c^2 d-e} \left (2 c^2 d-e\right )}\right )}{c \sqrt {c^2 d-e}}+\frac {i b \left (2 c^2 d-e\right ) \log \left (\frac {4 c^2 e^2 \left (i \sqrt {c^2 d-e} \sqrt {d+e x^2}+i c d+e x\right )}{b (c x+i) \sqrt {c^2 d-e} \left (2 c^2 d-e\right )}\right )}{c \sqrt {c^2 d-e}}-\frac {2 b \sqrt {e} \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{c}+\frac {2 b \tan ^{-1}(c x) \left (2 d+e x^2\right )}{\sqrt {d+e x^2}}}{2 e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 1291, normalized size = 9.42 \[ \text {result too large to display} \]
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 = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {x^{2}}{\sqrt {e x^{2} + d} e} + \frac {2 \, d}{\sqrt {e x^{2} + d} e^{2}}\right )} + 2 \, b \int \frac {x^{3} \arctan \left (c x\right )}{2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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