Optimal. Leaf size=643 \[ \frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {b c \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {b c \sqrt {-c^2 x^2-1} \left (30 c^2 d-11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b c \sqrt {-c^2 x^2-1} \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2}}-\frac {b e x \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^2 x \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c^3 x^2 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.87, antiderivative size = 643, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}-\frac {b c^3 x^2 \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}-\frac {b c \sqrt {-c^2 x^2-1} \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b e x \left (-249 c^4 d^2 e+120 c^6 d^3+71 c^2 d e^2+210 e^3\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^2 x \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {b c \sqrt {-c^2 x^2-1} \left (30 c^2 d-11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}} \]
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 {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{35 d^2 x^8 \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 )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{x^8 \sqrt {-1-c^2 x^2}} \, dx}{35 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-d \left (30 c^2 d-11 e\right )-e \left (5 c^2 d+14 e\right ) x^2\right )}{x^6 \sqrt {-1-c^2 x^2}} \, dx}{245 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (30 c^2 d-11 e\right ) \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d \left (120 c^4 d^2-159 c^2 d e-37 e^2\right )-2 e \left (15 c^4 d^2-18 c^2 d e-35 e^2\right ) x^2\right )}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{1225 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-11 e\right ) \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {(b c x) \int \frac {-d \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right )-e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2}}+\frac {b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-11 e\right ) \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {(b c x) \int \frac {-d e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right )-c^2 d e \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^3 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2}}+\frac {b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-11 e\right ) \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (b c e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^2 \sqrt {-c^2 x^2}}-\frac {\left (b c^3 e \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3675 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c^3 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x^2 \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2}}+\frac {b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-11 e\right ) \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {b e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 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 (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{3675 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c^3 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x^2 \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 \sqrt {-c^2 x^2}}+\frac {b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-11 e\right ) \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {b c^2 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 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 {b e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{3675 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 = 0.83, size = 372, normalized size = 0.58 \[ -\frac {\sqrt {d+e x^2} \left (105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (d e^2 x^4 \left (193 c^2 x^2-71\right )-3 d^2 e x^2 \left (176 c^4 x^4-83 c^2 x^2+61\right )+15 d^3 \left (16 c^6 x^6-8 c^4 x^4+6 c^2 x^2-5\right )+247 e^3 x^6\right )+105 b \text {csch}^{-1}(c x) \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2\right )}{3675 d^2 x^7}-\frac {i b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )-2 \left (120 c^8 d^4-324 c^6 d^3 e+221 c^4 d^2 e^2+88 c^2 d e^3-105 e^4\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{3675 \sqrt {c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsch}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{8}}\, dx \]
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 \[ \frac {1}{35} \, a {\left (\frac {2 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} e}{d^{2} x^{5}} - \frac {5 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} + \frac {1}{35} \, b {\left (\frac {{\left (2 \, e^{3} x^{6} - d e^{2} x^{4} - 8 \, d^{2} e x^{2} - 5 \, d^{3}\right )} \sqrt {e x^{2} + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{d^{2} x^{7}} - 35 \, \int \frac {{\left (2 \, c^{2} e^{3} x^{8} - c^{2} d e^{2} x^{6} + {\left (35 \, d^{2} e \log \relax (c) - 8 \, d^{2} e\right )} c^{2} x^{4} + 35 \, d^{3} \log \relax (c) + 5 \, {\left (7 \, d^{2} e \log \relax (c) + {\left (7 \, d^{3} \log \relax (c) - d^{3}\right )} c^{2}\right )} x^{2} + 35 \, {\left (c^{2} d^{2} e x^{4} + d^{3} + {\left (c^{2} d^{3} + d^{2} e\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{35 \, {\left (c^{2} d^{2} x^{10} + d^{2} x^{8}\right )}}\,{d x} + 35 \, \int \frac {{\left (2 \, c^{2} e^{3} x^{6} - c^{2} d e^{2} x^{4} - 8 \, c^{2} d^{2} e x^{2} - 5 \, c^{2} d^{3}\right )} \sqrt {e x^{2} + d}}{35 \, {\left (c^{2} d^{2} x^{8} + d^{2} x^{6} + {\left (c^{2} d^{2} x^{8} + d^{2} x^{6}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]
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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