Optimal. Leaf size=70 \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2+1}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e}} \]
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Rubi [A] time = 0.10, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {191, 5704, 12, 444, 63, 217, 206} \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2+1}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 191
Rule 206
Rule 217
Rule 444
Rule 5704
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-(b c) \int \frac {x}{d \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \, dx\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{c d}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{c d}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 75, normalized size = 1.07 \[ \frac {x \left (2 \left (a+b \sinh ^{-1}(c x)\right )-b c x \sqrt {\frac {e x^2}{d}+1} F_1\left (1;\frac {1}{2},\frac {1}{2};2;-c^2 x^2,-\frac {e x^2}{d}\right )\right )}{2 d \sqrt {d+e x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.54, size = 326, normalized size = 4.66 \[ \left [\frac {4 \, \sqrt {e x^{2} + d} b e x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 4 \, \sqrt {e x^{2} + d} a e x + {\left (b e x^{2} + b d\right )} \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} + 6 \, c^{2} d e + 8 \, {\left (c^{4} d e + c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d + c e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right )}{4 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {2 \, \sqrt {e x^{2} + d} b e x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {e x^{2} + d} a e x + {\left (b e x^{2} + b d\right )} \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d + e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} + c d e + {\left (c^{3} d e + c e^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arcsinh \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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