Optimal. Leaf size=178 \[ \frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac {\left (d+e x^2\right )^{5/2} (a+b \text {ArcTan}(c x))}{5 d x^5}-\frac {b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 \sqrt {d}}+\frac {b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{5 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {270, 5096, 12,
457, 100, 154, 162, 65, 214} \begin {gather*} -\frac {\left (d+e x^2\right )^{5/2} (a+b \text {ArcTan}(c x))}{5 d x^5}+\frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}+\frac {b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{5 d}-\frac {b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 \sqrt {d}}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 100
Rule 154
Rule 162
Rule 214
Rule 270
Rule 457
Rule 5096
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-(b c) \int \frac {\left (d+e x^2\right )^{5/2}}{5 x^5 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac {1}{5} (b c) \int \frac {\left (d+e x^2\right )^{5/2}}{x^5 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac {1}{10} (b c) \text {Subst}\left (\int \frac {(d+e x)^{5/2}}{x^3 \left (-d-c^2 d x\right )} \, dx,x,x^2\right )\\ &=-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac {(b c) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (-\frac {1}{2} d^2 \left (4 c^2 d-7 e\right )-\frac {1}{2} d \left (c^2 d-4 e\right ) e x\right )}{x^2 \left (-d-c^2 d x\right )} \, dx,x,x^2\right )}{20 d}\\ &=\frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {(b c) \text {Subst}\left (\int \frac {-\frac {1}{4} d^3 \left (8 c^4 d^2-20 c^2 d e+15 e^2\right )-\frac {1}{4} d^2 e \left (4 c^4 d^2-9 c^2 d e+8 e^2\right ) x}{x \left (-d-c^2 d x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{20 d^2}\\ &=\frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {1}{10} \left (b c \left (c^2 d-e\right )^3\right ) \text {Subst}\left (\int \frac {1}{\left (-d-c^2 d x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )+\frac {1}{80} \left (b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=\frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}+\frac {\left (b c \left (c^2 d-e\right )^3\right ) \text {Subst}\left (\int \frac {1}{-d+\frac {c^2 d^2}{e}-\frac {c^2 d x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{5 e}+\frac {\left (b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{40 e}\\ &=\frac {b c \left (4 c^2 d-7 e\right ) \sqrt {d+e x^2}}{40 x^2}-\frac {b c \left (d+e x^2\right )^{3/2}}{20 x^4}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{5 d x^5}-\frac {b c \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 \sqrt {d}}+\frac {b \left (c^2 d-e\right )^{5/2} \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{5 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.34, size = 334, normalized size = 1.88 \begin {gather*} \frac {-\sqrt {d+e x^2} \left (8 a \left (d+e x^2\right )^2+b c d x \left (9 e x^2+d \left (2-4 c^2 x^2\right )\right )\right )-8 b \left (d+e x^2\right )^{5/2} \text {ArcTan}(c x)+b c \sqrt {d} \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) x^5 \log (x)-b c \sqrt {d} \left (8 c^4 d^2-20 c^2 d e+15 e^2\right ) x^5 \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+4 b \left (c^2 d-e\right )^{5/2} x^5 \log \left (-\frac {20 c d \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} (i+c x)}\right )+4 b \left (c^2 d-e\right )^{5/2} x^5 \log \left (-\frac {20 c d \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} (-i+c x)}\right )}{40 d x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{6}}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 1227, normalized size = 6.89 \begin {gather*} \left [\frac {4 \, {\left (b c^{4} d^{2} x^{5} - 2 \, b c^{2} d x^{5} e + b x^{5} e^{2}\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 ) + {\left (8 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e + 15 \, b c x^{5} e^{2}\right )} \sqrt {d} \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (4 \, b c^{3} d^{2} x^{3} - 8 \, a x^{4} e^{2} - 2 \, b c d^{2} x - 8 \, a d^{2} - 8 \, {\left (b x^{4} e^{2} + 2 \, b d x^{2} e + b d^{2}\right )} \arctan \left (c x\right ) - {\left (9 \, b c d x^{3} + 16 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{80 \, d x^{5}}, \frac {8 \, {\left (b c^{4} d^{2} x^{5} - 2 \, b c^{2} d x^{5} e + b x^{5} e^{2}\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 ) + {\left (8 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e + 15 \, b c x^{5} e^{2}\right )} \sqrt {d} \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (4 \, b c^{3} d^{2} x^{3} - 8 \, a x^{4} e^{2} - 2 \, b c d^{2} x - 8 \, a d^{2} - 8 \, {\left (b x^{4} e^{2} + 2 \, b d x^{2} e + b d^{2}\right )} \arctan \left (c x\right ) - {\left (9 \, b c d x^{3} + 16 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{80 \, d x^{5}}, \frac {{\left (8 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e + 15 \, b c x^{5} e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + 2 \, {\left (b c^{4} d^{2} x^{5} - 2 \, b c^{2} d x^{5} e + b x^{5} e^{2}\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 ) + {\left (4 \, b c^{3} d^{2} x^{3} - 8 \, a x^{4} e^{2} - 2 \, b c d^{2} x - 8 \, a d^{2} - 8 \, {\left (b x^{4} e^{2} + 2 \, b d x^{2} e + b d^{2}\right )} \arctan \left (c x\right ) - {\left (9 \, b c d x^{3} + 16 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{40 \, d x^{5}}, \frac {4 \, {\left (b c^{4} d^{2} x^{5} - 2 \, b c^{2} d x^{5} e + b x^{5} e^{2}\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 ) + {\left (8 \, b c^{5} d^{2} x^{5} - 20 \, b c^{3} d x^{5} e + 15 \, b c x^{5} e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + {\left (4 \, b c^{3} d^{2} x^{3} - 8 \, a x^{4} e^{2} - 2 \, b c d^{2} x - 8 \, a d^{2} - 8 \, {\left (b x^{4} e^{2} + 2 \, b d x^{2} e + b d^{2}\right )} \arctan \left (c x\right ) - {\left (9 \, b c d x^{3} + 16 \, a d x^{2}\right )} e\right )} \sqrt {x^{2} e + d}}{40 \, d x^{5}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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