Optimal. Leaf size=256 \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {8 b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e^3 \sqrt {-c^2 x^2}}-\frac {b x \left (9 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 1.13, 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 = {266, 43, 6302, 12, 1615, 157, 63, 217, 203, 93, 204} \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {8 b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e^3 \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {b x \left (9 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 157
Rule 203
Rule 204
Rule 217
Rule 266
Rule 1615
Rule 6302
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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.64, size = 311, normalized size = 1.21 \[ \frac {-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )+b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (d+e x^2\right )-2 b c \text {csch}^{-1}(c x) \left (8 d^2+4 d e x^2-e^2 x^4\right )}{6 c e^3 \sqrt {d+e x^2}}+\frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \left (16 c^5 d^{3/2} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2+1}}{\sqrt {-d-e x^2}}\right )-\sqrt {c^2} \sqrt {e} \sqrt {c^2 d-e} \left (9 c^2 d+e\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2+1}}{\sqrt {c^2} \sqrt {c^2 d-e}}\right )\right )}{6 c^4 e^3 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 1719, normalized size = 6.71 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, 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 \[ \frac {1}{3} \, {\left (\frac {x^{4}}{\sqrt {e x^{2} + d} e} - \frac {4 \, d x^{2}}{\sqrt {e x^{2} + d} e^{2}} - \frac {8 \, d^{2}}{\sqrt {e x^{2} + d} e^{3}}\right )} a + \frac {1}{3} \, b {\left (\frac {{\left (e^{2} x^{4} - 4 \, d e x^{2} - 8 \, d^{2}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x^{2} + d} e^{3}} + 3 \, \int \frac {c^{2} e^{2} x^{5} - 4 \, c^{2} d e x^{3} - 8 \, c^{2} d^{2} x}{3 \, {\left ({\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {e x^{2} + d}\right )}}\,{d x} - 3 \, \int \frac {{\left (3 \, e^{3} \log \relax (c) + e^{3}\right )} c^{2} x^{7} - 12 \, c^{2} d^{2} e x^{3} - 8 \, c^{2} d^{3} x - 3 \, {\left (c^{2} d e^{2} - e^{3} \log \relax (c)\right )} x^{5} + 3 \, {\left (c^{2} e^{3} x^{7} + e^{3} x^{5}\right )} \log \relax (x)}{3 \, {\left (c^{2} e^{4} x^{4} + d e^{3} + {\left (c^{2} d e^{3} + e^{4}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} \]
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 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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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