Optimal. Leaf size=229 \[ \frac {(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {32 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {8 (a x+1)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6192, 6193, 43} \[ \frac {(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {8 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {32 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {8 (a x+1)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac {\left (c-a^2 c x^2\right )^{9/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {\left (c-a^2 c x^2\right )^{9/2} \int (-1+a x)^4 (1+a x)^5 \, dx}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {\left (c-a^2 c x^2\right )^{9/2} \int \left (16 (1+a x)^5-32 (1+a x)^6+24 (1+a x)^7-8 (1+a x)^8+(1+a x)^9\right ) \, dx}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {8 (1+a x)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {32 (1+a x)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {3 (1+a x)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}-\frac {8 (1+a x)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}+\frac {(1+a x)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} x^9}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 79, normalized size = 0.34 \[ \frac {c^4 (a x+1)^6 \left (63 a^4 x^4-308 a^3 x^3+588 a^2 x^2-528 a x+193\right ) \sqrt {c-a^2 c x^2}}{630 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 117, normalized size = 0.51 \[ \frac {{\left (63 \, a^{9} c^{4} x^{10} + 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} - 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} + 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} - 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} + 630 \, c^{4} x\right )} \sqrt {-a^{2} c}}{630 \, a} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 116, normalized size = 0.51 \[ \frac {x \left (63 a^{9} x^{9}+70 x^{8} a^{8}-315 a^{7} x^{7}-360 x^{6} a^{6}+630 x^{5} a^{5}+756 x^{4} a^{4}-630 x^{3} a^{3}-840 a^{2} x^{2}+315 a x +630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{630 \left (a x -1\right )^{4} \left (a x +1\right )^{5} \sqrt {\frac {a x -1}{a x +1}}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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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