Optimal. Leaf size=109 \[ \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b c}{3 e \left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {b \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {264, 4976, 446, 78, 63, 208} \[ \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b c}{3 e \left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {b \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rule 264
Rule 446
Rule 4976
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-(b c) \int \frac {x^3}{\left (3 d+3 c^2 d x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {x}{\left (3 d+3 c^2 d x\right ) (d+e x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {b c}{3 \left (c^2 d-e\right ) e \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{\left (3 d+3 c^2 d x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{2 \left (c^2 d-e\right )}\\ &=\frac {b c}{3 \left (c^2 d-e\right ) e \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{3 d-\frac {3 c^2 d^2}{e}+\frac {3 c^2 d x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{\left (c^2 d-e\right ) e}\\ &=\frac {b c}{3 \left (c^2 d-e\right ) e \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d \left (c^2 d-e\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.11, size = 252, normalized size = 2.31 \[ -\frac {-\frac {2 \left (a x \left (c^2 d-e\right )+b c d\right )}{e \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {2 a d x}{e \left (d+e x^2\right )^{3/2}}+\frac {b \log \left (\frac {12 c d \sqrt {c^2 d-e} \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i)}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac {b \log \left (\frac {12 c d \sqrt {c^2 d-e} \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i)}\right )}{\left (c^2 d-e\right )^{3/2}}-\frac {2 b x^3 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}}}{6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 676, normalized size = 6.20 \[ \left [-\frac {{\left (b e^{3} x^{4} + 2 \, b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} d - e} \log \left (\frac {c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \, {\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt {c^{2} d - e} \sqrt {e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 4 \, {\left (b c^{3} d^{3} - b c d^{2} e + {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \arctan \left (c x\right ) + {\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} + {\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{4} d^{3} e^{3} - 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}, -\frac {{\left (b e^{3} x^{4} + 2 \, b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d}}{2 \, {\left (c^{3} d^{2} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (b c^{3} d^{3} - b c d^{2} e + {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} \arctan \left (c x\right ) + {\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} + {\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{4} d^{3} e^{3} - 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 1.10, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
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 \[ -\frac {1}{3} \, a {\left (\frac {x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} - \frac {x}{\sqrt {e x^{2} + d} d e}\right )} + 2 \, b \int \frac {x^{2} \arctan \left (c x\right )}{2 \, {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \]
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 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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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