Optimal. Leaf size=407 \[ \frac {x^3 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{a c^2 (n+3) \sqrt {c-a^2 c x^2}}-\frac {3 (2-n) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 c^2 \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {3 \left (-n^2+2 n+1\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 c^2 (3-n) (n+1) (n+3) \sqrt {c-a^2 c x^2}}-\frac {3 \left (-n^2+2 n+1\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}}}{a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}-\frac {3 x \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^3 c^2 (n+3) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.49, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6153, 6150, 94, 90, 79, 45, 37} \[ -\frac {3 (2-n) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 c^2 \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {3 \left (-n^2+2 n+1\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^4 c^2 (3-n) (n+1) (n+3) \sqrt {c-a^2 c x^2}}-\frac {3 \left (-n^2+2 n+1\right ) \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}}}{a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}}+\frac {x^3 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{a c^2 (n+3) \sqrt {c-a^2 c x^2}}-\frac {3 x \sqrt {1-a^2 x^2} (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{a^3 c^2 (n+3) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 79
Rule 90
Rule 94
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int x^3 (1-a x)^{-\frac {5}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x^3 (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {1-a^2 x^2}\right ) \int x^2 (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}} \, dx}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {x^3 (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 x (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {1-a^2 x^2}\right ) \int (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{-\frac {5}{2}+\frac {n}{2}} (-1+a (1-n) x) \, dx}{a^3 c^2 (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {x^3 (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 (2-n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 x (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {\left (3 \left (1+2 n-n^2\right ) \sqrt {1-a^2 x^2}\right ) \int (1-a x)^{-\frac {3}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \, dx}{a^3 c^2 (3-n) (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {x^3 (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 (2-n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 x (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {3 \left (1+2 n-n^2\right ) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 (3-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}+\frac {\left (3 \left (1+2 n-n^2\right ) \sqrt {1-a^2 x^2}\right ) \int (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \, dx}{a^3 c^2 (3-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}\\ &=\frac {x^3 (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 (2-n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {3 x (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{a^3 c^2 (3+n) \sqrt {c-a^2 c x^2}}+\frac {3 \left (1+2 n-n^2\right ) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 (3-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {3 \left (1+2 n-n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{a^4 c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 112, normalized size = 0.28 \[ -\frac {\sqrt {1-a^2 x^2} (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}} \left (-a^3 n \left (n^2-7\right ) x^3+3 a^2 \left (n^2-3\right ) x^2-6 a n x+6\right )}{a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 175, normalized size = 0.43 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x - 3 \, {\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} - 6\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} + {\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \, {\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} c^{3}\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 93, normalized size = 0.23 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a^{3} n^{3} x^{3}-7 x^{3} a^{3} n -3 a^{2} n^{2} x^{2}+9 a^{2} x^{2}+6 n a x -6\right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{a^{4} \left (n^{4}-10 n^{2}+9\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 162, normalized size = 0.40 \[ -\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {6}{a^6\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {6\,n\,x}{a^5\,c^2\,\left (n^4-10\,n^2+9\right )}+\frac {x^2\,\left (3\,n^2-9\right )}{a^4\,c^2\,\left (n^4-10\,n^2+9\right )}-\frac {n\,x^3\,\left (n^2-7\right )}{a^3\,c^2\,\left (n^4-10\,n^2+9\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}-x^2\,\sqrt {c-a^2\,c\,x^2}\right )} \]
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 \[ \int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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