Optimal. Leaf size=389 \[ -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b e x \left (c^2 d-3 e\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {2 b c \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}+\frac {2 b c^2 x \left (c^2 d-2 e\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {2 b c^3 x^2 \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.43, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {264, 6302, 12, 474, 583, 531, 418, 492, 411} \[ -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b e x \left (c^2 d-3 e\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {2 b c^3 x^2 \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}-\frac {2 b c \sqrt {-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}+\frac {2 b c^2 x \left (c^2 d-2 e\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 411
Rule 418
Rule 474
Rule 492
Rule 531
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^4} \, dx &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {(b c x) \int -\frac {\left (d+e x^2\right )^{3/2}}{3 d x^4 \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 )}{3 d x^3}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{3 d \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {(b c x) \int \frac {2 d \left (c^2 d-2 e\right )+\left (c^2 d-3 e\right ) e x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {(b c x) \int \frac {d \left (c^2 d-3 e\right ) e+2 c^2 d \left (c^2 d-2 e\right ) e x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {\left (b c \left (c^2 d-3 e\right ) e x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d \sqrt {-c^2 x^2}}-\frac {\left (2 b c^3 \left (c^2 d-2 e\right ) e x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c^3 \left (c^2 d-2 e\right ) x^2 \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {2 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b \left (c^2 d-3 e\right ) e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 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 (2 b c^3 \left (c^2 d-2 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{9 d \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c^3 \left (c^2 d-2 e\right ) x^2 \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {2 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 b c^2 \left (c^2 d-2 e\right ) x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 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 \left (c^2 d-3 e\right ) e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 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 = 0.62, size = 237, normalized size = 0.61 \[ -\frac {\sqrt {d+e x^2} \left (3 a \left (d+e x^2\right )+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (2 c^2 d x^2-d-4 e x^2\right )+3 b \text {csch}^{-1}(c x) \left (d+e x^2\right )\right )}{9 d x^3}-\frac {i b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} \left (2 c^2 d \left (c^2 d-2 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-2 c^4 d^2+5 c^2 d e-3 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{9 \sqrt {c^2} d \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{4}}\, dx \]
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 \[ -\frac {1}{3} \, b {\left (\frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{d x^{3}} + 3 \, \int -\frac {{\left (c^{2} e x^{4} - {\left (3 \, d \log \relax (c) - d\right )} c^{2} x^{2} - 3 \, d \log \relax (c) - 3 \, {\left (c^{2} d x^{2} + d\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d x^{6} + d x^{4}\right )}}\,{d x} + 3 \, \int \frac {{\left (c^{2} e x^{2} + c^{2} d\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d x^{4} + d x^{2} + {\left (c^{2} d x^{4} + d x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, d x^{3}} \]
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 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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