Optimal. Leaf size=492 \[ \frac {b c^3 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x^2 \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 d \sqrt {-c^2 x^2}}-\frac {4 b c \left (c^2 d-2 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{75 x^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^4 \sqrt {-c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 d x^5}-\frac {b c^2 \left (8 c^4 d^2-23 c^2 d e+23 e^2\right ) x \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{75 d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {b e \left (4 c^4 d^2-11 c^2 d e+15 e^2\right ) x \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{75 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
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Rubi [A]
time = 0.39, 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 = {270, 6437, 12,
485, 594, 597, 545, 429, 506, 422} \begin {gather*} -\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 x \left (4 c^4 d^2-11 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 )}{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 (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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 {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 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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 422
Rule 429
Rule 485
Rule 506
Rule 545
Rule 594
Rule 597
Rule 6437
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] Result contains complex when optimal does not.
time = 10.19, size = 291, normalized size = 0.59 \begin {gather*} \frac {\sqrt {d+e x^2} \left (-15 a \left (d+e x^2\right )^2+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (23 e^2 x^4+d e x^2 \left (11-23 c^2 x^2\right )+d^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (d+e x^2\right )^2 \text {csch}^{-1}(c x)\right )}{75 d 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 (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 {1+c^2 x^2} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, 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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (e\,x^2+d\right )}^{3/2}\,\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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