Optimal. Leaf size=527 \[ \frac {b c^3 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x^2 \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {-c^2 x^2}}-\frac {b c \left (12 c^2 d+e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{15 d^2 x^3}-\frac {b c^2 \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) x \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {2 b e \left (6 c^4 d^2-4 c^2 d e-15 e^2\right ) x \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{225 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
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Rubi [A]
time = 0.43, 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 = {277, 270,
6437, 12, 594, 597, 545, 429, 506, 422} \begin {gather*} \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^2 x \left (24 c^4 d^2-19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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 (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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 \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^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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 277
Rule 422
Rule 429
Rule 506
Rule 545
Rule 594
Rule 597
Rule 6437
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] Result contains complex when optimal does not.
time = 10.07, size = 314, normalized size = 0.60 \begin {gather*} \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 \sqrt {1+\frac {1}{c^2 x^2}} x \left (-31 e^2 x^4+d e x^2 \left (8-19 c^2 x^2\right )+3 d^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d^2+d e x^2-2 e^2 x^4\right ) \text {csch}^{-1}(c x)\right )}{225 d^2 x^5}+\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \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 {1+c^2 x^2} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \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 {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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