Optimal. Leaf size=144 \[ -\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \text {ArcTan}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {ArcTan}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {b \left (3 c^2 d-2 e\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {198, 197, 5032,
6820, 12, 585, 79, 65, 214} \begin {gather*} \frac {2 x (a+b \text {ArcTan}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {ArcTan}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b \left (3 c^2 d-2 e\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}}-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 214
Rule 585
Rule 5032
Rule 6820
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {\frac {x}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x}{3 d^2 \sqrt {d+e x^2}}}{1+c^2 x^2} \, dx\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {x \left (3 d+2 e x^2\right )}{3 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x \left (3 d+2 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {3 d+2 e x}{\left (1+c^2 x\right ) (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2}\\ &=-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \left (3 c^2 d-2 e\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d^2 \left (c^2 d-e\right )}\\ &=-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \left (3 c^2 d-2 e\right )\right ) \text {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 )}{3 d^2 \left (c^2 d-e\right ) e}\\ &=-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b \left (3 c^2 d-2 e\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.40, size = 317, normalized size = 2.20 \begin {gather*} \frac {2 \sqrt {c^2 d-e} \left (-b c d \left (d+e x^2\right )+a \left (c^2 d-e\right ) x \left (3 d+2 e x^2\right )\right )+2 b \left (c^2 d-e\right )^{3/2} x \left (3 d+2 e x^2\right ) \text {ArcTan}(c x)+b \left (3 c^2 d-2 e\right ) \left (d+e x^2\right )^{3/2} \log \left (-\frac {12 c d^2 \sqrt {c^2 d-e} \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (3 c^2 d-2 e\right ) (i+c x)}\right )+b \left (3 c^2 d-2 e\right ) \left (d+e x^2\right )^{3/2} \log \left (-\frac {12 c d^2 \sqrt {c^2 d-e} \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (3 c^2 d-2 e\right ) (-i+c x)}\right )}{6 d^2 \left (c^2 d-e\right )^{3/2} \left (d+e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.89, size = 0, normalized size = 0.00 \[\int \frac {a +b \arctan \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 429 vs.
\(2 (132) = 264\).
time = 2.41, size = 897, normalized size = 6.23 \begin {gather*} \left [\frac {{\left (3 \, b c^{2} d^{3} - 2 \, b x^{4} e^{3} + {\left (3 \, b c^{2} d x^{4} - 4 \, b d x^{2}\right )} e^{2} + 2 \, {\left (3 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} e\right )} \sqrt {c^{2} d - e} \log \left (\frac {8 \, c^{4} d^{2} + 4 \, {\left (2 \, c^{3} d + {\left (c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} d - e} \sqrt {x^{2} e + d} + {\left (c^{4} x^{4} - 6 \, c^{2} x^{2} + 1\right )} e^{2} + 8 \, {\left (c^{4} d x^{2} - c^{2} d\right )} e}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 4 \, {\left (3 \, a c^{4} d^{3} x - b c^{3} d^{3} + 2 \, a x^{3} e^{3} + {\left (3 \, b c^{4} d^{3} x + 2 \, b x^{3} e^{3} - {\left (4 \, b c^{2} d x^{3} - 3 \, b d x\right )} e^{2} + 2 \, {\left (b c^{4} d^{2} x^{3} - 3 \, b c^{2} d^{2} x\right )} e\right )} \arctan \left (c x\right ) - {\left (4 \, a c^{2} d x^{3} - b c d x^{2} - 3 \, a d x\right )} e^{2} + {\left (2 \, a c^{4} d^{2} x^{3} - b c^{3} d^{2} x^{2} - 6 \, a c^{2} d^{2} x + b c d^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{12 \, {\left (c^{4} d^{6} + d^{2} x^{4} e^{4} - 2 \, {\left (c^{2} d^{3} x^{4} - d^{3} x^{2}\right )} e^{3} + {\left (c^{4} d^{4} x^{4} - 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{2} + 2 \, {\left (c^{4} d^{5} x^{2} - c^{2} d^{5}\right )} e\right )}}, \frac {{\left (3 \, b c^{2} d^{3} - 2 \, b x^{4} e^{3} + {\left (3 \, b c^{2} d x^{4} - 4 \, b d x^{2}\right )} e^{2} + 2 \, {\left (3 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} e\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (2 \, c^{2} d + {\left (c^{2} x^{2} - 1\right )} e\right )} \sqrt {-c^{2} d + e} \sqrt {x^{2} e + d}}{2 \, {\left (c^{3} d^{2} - c x^{2} e^{2} + {\left (c^{3} d x^{2} - c d\right )} e\right )}}\right ) + 2 \, {\left (3 \, a c^{4} d^{3} x - b c^{3} d^{3} + 2 \, a x^{3} e^{3} + {\left (3 \, b c^{4} d^{3} x + 2 \, b x^{3} e^{3} - {\left (4 \, b c^{2} d x^{3} - 3 \, b d x\right )} e^{2} + 2 \, {\left (b c^{4} d^{2} x^{3} - 3 \, b c^{2} d^{2} x\right )} e\right )} \arctan \left (c x\right ) - {\left (4 \, a c^{2} d x^{3} - b c d x^{2} - 3 \, a d x\right )} e^{2} + {\left (2 \, a c^{4} d^{2} x^{3} - b c^{3} d^{2} x^{2} - 6 \, a c^{2} d^{2} x + b c d^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{6 \, {\left (c^{4} d^{6} + d^{2} x^{4} e^{4} - 2 \, {\left (c^{2} d^{3} x^{4} - d^{3} x^{2}\right )} e^{3} + {\left (c^{4} d^{4} x^{4} - 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{2} + 2 \, {\left (c^{4} d^{5} x^{2} - c^{2} d^{5}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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