Optimal. Leaf size=256 \[ \frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {b \left (9 c^2 d+e\right ) x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}}-\frac {8 b c d^{3/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 e^3 \sqrt {-c^2 x^2}} \]
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Rubi [A]
time = 0.77, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {272, 45,
6437, 12, 1629, 163, 65, 223, 209, 95, 210} \begin {gather*} -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {8 b c d^{3/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e^3 \sqrt {-c^2 x^2}}-\frac {b x \left (9 c^2 d+e\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 163
Rule 209
Rule 210
Rule 223
Rule 272
Rule 1629
Rule 6437
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 e^3 \sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {(b c x) \text {Subst}\left (\int \frac {-8 d^2-4 d e x+e^2 x^2}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^3 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {(b x) \text {Subst}\left (\int \frac {8 c^2 d^2 e+\frac {1}{2} e^2 \left (9 c^2 d+e\right ) x}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c e^4 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (4 b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^3 \sqrt {-c^2 x^2}}+\frac {\left (b \left (9 c^2 d+e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{12 c e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (8 b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1-c^2 x^2}}\right )}{3 e^3 \sqrt {-c^2 x^2}}-\frac {\left (b \left (9 c^2 d+e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {-1-c^2 x^2}\right )}{6 c^3 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {8 b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 e^3 \sqrt {-c^2 x^2}}-\frac {\left (b \left (9 c^2 d+e\right ) x\right ) \text {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {-1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{6 c^3 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {b \left (9 c^2 d+e\right ) x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}}-\frac {8 b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 e^3 \sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.29, size = 229, normalized size = 0.89 \begin {gather*} \frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )-2 b c \left (8 d^2+4 d e x^2-e^2 x^4\right ) \text {csch}^{-1}(c x)}{6 c e^3 \sqrt {d+e x^2}}-\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (-16 c^3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (9 c^2 d+e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{6 c^2 e^3 \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
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 \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. 833 vs.
\(2 (216) = 432\).
time = 0.54, size = 1705, normalized size = 6.66 \begin {gather*} \text {Too large to display} \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 = -0.00 \begin {gather*} \int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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