Optimal. Leaf size=423 \[ \frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{7/2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}+\frac {b c e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}+\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c e}{2 d^3 \sqrt {d+e x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {d+e x^2}} \]
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Rubi [A] time = 1.10, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {271, 192, 191, 4976, 12, 6725, 266, 51, 63, 208, 261, 444} \[ \frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}-\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{7/2}}+\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c \sqrt {d+e x^2}}{2 d^3 x^2}+\frac {b c}{3 d^2 x^2 \sqrt {d+e x^2}}+\frac {b c e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 191
Rule 192
Rule 208
Rule 261
Rule 266
Rule 271
Rule 444
Rule 4976
Rule 6725
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{5/2}} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}-(b c) \int \frac {-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{3 d^4 x^3 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {-d^3+6 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6}{x^3 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{3 d^4}\\ &=-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {(b c) \int \left (-\frac {d^3}{x^3 \left (d+e x^2\right )^{3/2}}+\frac {d^2 \left (c^2 d+6 e\right )}{x \left (d+e x^2\right )^{3/2}}+\frac {16 e^3 x}{c^2 \left (d+e x^2\right )^{3/2}}+\frac {\left (c^2 d-2 e\right ) \left (-c^4 d^2-8 c^2 d e+8 e^2\right ) x}{c^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}}\right ) \, dx}{3 d^4}\\ &=-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {(b c) \int \frac {1}{x^3 \left (d+e x^2\right )^{3/2}} \, dx}{3 d}-\frac {\left (16 b e^3\right ) \int \frac {x}{\left (d+e x^2\right )^{3/2}} \, dx}{3 c d^4}-\frac {\left (b c \left (c^2 d+6 e\right )\right ) \int \frac {1}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}+\frac {\left (b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \int \frac {x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{3 c d^4}\\ &=\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x^2 (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d}-\frac {\left (b c \left (c^2 d+6 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2}+\frac {\left (b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 c d^4}\\ &=\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {b c}{3 d^2 x^2 \sqrt {d+e x^2}}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {d+e x}} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (b c \left (c^2 d+6 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d^3}+\frac {\left (b c \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d^4 \left (c^2 d-e\right )}\\ &=\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {b c}{3 d^2 x^2 \sqrt {d+e x^2}}-\frac {b c \sqrt {d+e x^2}}{2 d^3 x^2}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}-\frac {(b c e) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{4 d^3}-\frac {\left (b c \left (c^2 d+6 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 d^3 e}+\frac {\left (b c \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )\right ) \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 )}{3 d^4 \left (c^2 d-e\right ) e}\\ &=\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {b c}{3 d^2 x^2 \sqrt {d+e x^2}}-\frac {b c \sqrt {d+e x^2}}{2 d^3 x^2}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{7/2}}-\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{2 d^3}\\ &=\frac {16 b e^2}{3 c d^4 \sqrt {d+e x^2}}-\frac {b c \left (c^2 d+6 e\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right )}{3 c d^4 \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {b c}{3 d^2 x^2 \sqrt {d+e x^2}}-\frac {b c \sqrt {d+e x^2}}{2 d^3 x^2}-\frac {a+b \tan ^{-1}(c x)}{3 d x^3 \left (d+e x^2\right )^{3/2}}+\frac {2 e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x \left (d+e x^2\right )^{3/2}}+\frac {8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \left (d+e x^2\right )^{3/2}}+\frac {16 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^4 \sqrt {d+e x^2}}+\frac {b c e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{7/2}}+\frac {b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{7/2}}-\frac {b \left (c^2 d-2 e\right ) \left (c^4 d^2+8 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^4 \left (c^2 d-e\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.96, size = 510, normalized size = 1.21 \[ -\frac {\frac {2 a \left (d^3-6 d^2 e x^2-24 d e^2 x^4-16 e^3 x^6\right )}{x^3 \left (d+e x^2\right )^{3/2}}+\frac {b c d \left (c^2 d \left (d+e x^2\right )+e \left (e x^2-d\right )\right )}{x^2 \left (c^2 d-e\right ) \sqrt {d+e x^2}}-b c \sqrt {d} \left (2 c^2 d+15 e\right ) \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )+b c \sqrt {d} \log (x) \left (2 c^2 d+15 e\right )+\frac {b \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right ) \log \left (\frac {12 c d^4 \sqrt {c^2 d-e} \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right )}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac {b \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right ) \log \left (\frac {12 c d^4 \sqrt {c^2 d-e} \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right )}\right )}{\left (c^2 d-e\right )^{3/2}}+\frac {2 b \tan ^{-1}(c x) \left (d^3-6 d^2 e x^2-24 d e^2 x^4-16 e^3 x^6\right )}{x^3 \left (d+e x^2\right )^{3/2}}}{6 d^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.21, size = 3460, normalized size = 8.18 \[ \text {result too large to display} \]
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 = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctan \left (c x \right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
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 \[ \frac {1}{3} \, a {\left (\frac {16 \, e^{2} x}{\sqrt {e x^{2} + d} d^{4}} + \frac {8 \, e^{2} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3}} + \frac {6 \, e}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} x} - \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e^{2} x^{8} + 2 \, d e x^{6} + d^{2} x^{4}\right )} \sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^4\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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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