Optimal. Leaf size=527 \[ \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}-\frac {b c \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c^2 x \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {2 b e x \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^3 x^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.64, antiderivative size = 527, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {271, 264, 6302, 12, 580, 583, 531, 418, 492, 411} \[ \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {b c^3 x^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}+\frac {b c \sqrt {-c^2 x^2-1} \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}+\frac {2 b e x \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c^2 x \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {b c \sqrt {-c^2 x^2-1} \left (12 c^2 d+e\right ) \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 411
Rule 418
Rule 492
Rule 531
Rule 580
Rule 583
Rule 6302
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{15 d^2 x^6 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-3 d+2 e x^2\right )}{x^6 \sqrt {-1-c^2 x^2}} \, dx}{15 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}+\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d \left (12 c^2 d+e\right )-e \left (3 c^2 d+10 e\right ) x^2\right )}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{75 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c x) \int \frac {-d \left (24 c^4 d^2-19 c^2 d e-31 e^2\right )-2 e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {(b c x) \int \frac {-2 d e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right )-c^2 d e \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^3 \sqrt {-c^2 x^2}}\\ &=\frac {b c \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}+\frac {\left (b c^3 e \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {-c^2 x^2}}+\frac {\left (2 b c e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{225 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x^2 \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}+\frac {2 b e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {\left (b c^3 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{225 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x^2 \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {b c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {2 b e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{225 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end {align*}
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Mathematica [C] time = 0.70, size = 314, normalized size = 0.60 \[ \frac {\sqrt {d+e x^2} \left (-15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (d e x^2 \left (8-19 c^2 x^2\right )+3 d^2 \left (8 c^4 x^4-4 c^2 x^2+3\right )-31 e^2 x^4\right )-15 b \text {csch}^{-1}(c x) \left (3 d^2+d e x^2-2 e^2 x^4\right )\right )}{225 d^2 x^5}+\frac {i b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-24 c^6 d^3+31 c^4 d^2 e+23 c^2 d e^2-30 e^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{225 \sqrt {c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{6}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{6}}\, dx \]
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 \[ \frac {1}{15} \, a {\left (\frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d^{2} x^{3}} - \frac {3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} + \frac {1}{15} \, b {\left (\frac {{\left (2 \, e^{2} x^{4} - d e x^{2} - 3 \, d^{2}\right )} \sqrt {e x^{2} + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{d^{2} x^{5}} - 15 \, \int \frac {{\left (2 \, c^{2} e^{2} x^{6} - c^{2} d e x^{4} + 3 \, {\left (5 \, d^{2} \log \relax (c) - d^{2}\right )} c^{2} x^{2} + 15 \, d^{2} \log \relax (c) + 15 \, {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{15 \, {\left (c^{2} d^{2} x^{8} + d^{2} x^{6}\right )}}\,{d x} + 15 \, \int \frac {{\left (2 \, c^{2} e^{2} x^{4} - c^{2} d e x^{2} - 3 \, c^{2} d^{2}\right )} \sqrt {e x^{2} + d}}{15 \, {\left (c^{2} d^{2} x^{6} + d^{2} x^{4} + {\left (c^{2} d^{2} x^{6} + d^{2} x^{4}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} \]
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 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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