Optimal. Leaf size=492 \[ -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}-\frac {4 b c \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{75 x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}+\frac {b e x \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{75 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c^2 x \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{75 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^3 x^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.58, antiderivative size = 492, 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 = {264, 6302, 12, 474, 580, 583, 531, 418, 492, 411} \[ -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {b c^3 x^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}+\frac {b c \sqrt {-c^2 x^2-1} \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}+\frac {b e x \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{75 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c^2 x \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{75 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {4 b c \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{75 x^2 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 411
Rule 418
Rule 474
Rule 492
Rule 531
Rule 580
Rule 583
Rule 6302
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}-\frac {(b c x) \int -\frac {\left (d+e x^2\right )^{5/2}}{5 d x^6 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6 \sqrt {-1-c^2 x^2}} \, dx}{5 d \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (4 d \left (c^2 d-2 e\right )+\left (c^2 d-5 e\right ) e x^2\right )}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{25 d \sqrt {-c^2 x^2}}\\ &=-\frac {4 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 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 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {d \left (8 c^4 d^2-23 c^2 d e+23 e^2\right )+e \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d \sqrt {-c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}-\frac {4 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 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 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {(b c x) \int \frac {d e \left (4 c^4 d^2-11 c^2 d e+15 e^2\right )+c^2 d e \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}-\frac {4 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 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 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {\left (b c e \left (4 c^4 d^2-11 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}{75 d \sqrt {-c^2 x^2}}+\frac {\left (b c^3 e \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x^2 \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}-\frac {4 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 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 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}+\frac {b e \left (4 c^4 d^2-11 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 )}{75 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 {\left (b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{75 d \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x^2 \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}-\frac {4 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 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 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}-\frac {b c^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{75 d \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 {b e \left (4 c^4 d^2-11 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 )}{75 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 )}}}\\ \end {align*}
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Mathematica [C] time = 0.70, size = 291, normalized size = 0.59 \[ \frac {\sqrt {d+e x^2} \left (-15 a \left (d+e x^2\right )^2+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (d e x^2 \left (11-23 c^2 x^2\right )+d^2 \left (8 c^4 x^4-4 c^2 x^2+3\right )+23 e^2 x^4\right )-15 b \text {csch}^{-1}(c x) \left (d+e x^2\right )^2\right )}{75 d x^5}+\frac {i b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-8 c^6 d^3+27 c^4 d^2 e-34 c^2 d e^2+15 e^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{75 \sqrt {c^2} d \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsch}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{6}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \]
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 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{6}}\, dx \]
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 \[ -\frac {1}{5} \, b {\left (\frac {{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{d x^{5}} + 5 \, \int -\frac {{\left (c^{2} e^{2} x^{6} - {\left (5 \, d e \log \relax (c) - 2 \, d e\right )} c^{2} x^{4} - {\left ({\left (5 \, d^{2} \log \relax (c) - d^{2}\right )} c^{2} + 5 \, d e \log \relax (c)\right )} x^{2} - 5 \, d^{2} \log \relax (c) - 5 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{5 \, {\left (c^{2} d x^{8} + d x^{6}\right )}}\,{d x} + 5 \, \int \frac {{\left (c^{2} e^{2} x^{4} + 2 \, c^{2} d e x^{2} + c^{2} d^{2}\right )} \sqrt {e x^{2} + d}}{5 \, {\left (c^{2} d x^{6} + d x^{4} + {\left (c^{2} d x^{6} + d x^{4}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}} a}{5 \, d x^{5}} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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