Optimal. Leaf size=275 \[ -\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \left (4+n+n^2+a (1-n) n x\right ) \sqrt {1-a^2 x^2}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {1}{2} (-1+n)} n \left (5+n^2\right ) (1-a x)^{\frac {3-n}{2}} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (1-n) (3-n) \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6288, 6285,
102, 151, 71} \begin {gather*} -\frac {x^2 \sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {n-1}{2}} n \left (n^2+5\right ) \sqrt {1-a^2 x^2} (1-a x)^{\frac {3-n}{2}} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (1-n) (3-n) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} (a x+1)^{\frac {n+1}{2}} \left (a (1-n) n x+n^2+n+4\right ) (1-a x)^{\frac {1-n}{2}}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 151
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \tanh ^{-1}(a x)} x^3}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int x^3 (1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {1}{2}+\frac {n}{2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \int x (1-a x)^{-\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {1}{2}+\frac {n}{2}} (-2-a n x) \, dx}{3 a^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \left (4+n+n^2+a (1-n) n x\right ) \sqrt {1-a^2 x^2}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}}+\frac {\left (n \left (5+n^2\right ) \sqrt {1-a^2 x^2}\right ) \int (1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{-\frac {1}{2}+\frac {n}{2}} \, dx}{6 a^3 (1-n) \sqrt {c-a^2 c x^2}}\\ &=-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{3 a^2 \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \left (4+n+n^2+a (1-n) n x\right ) \sqrt {1-a^2 x^2}}{6 a^4 (1-n) \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {1}{2} (-1+n)} n \left (5+n^2\right ) (1-a x)^{\frac {3-n}{2}} \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1-n}{2},\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-a x)\right )}{3 a^4 (1-n) (3-n) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 187, normalized size = 0.68 \begin {gather*} \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-a^2 x^2} \left (-\sqrt {2} (-3+n) (1+a x)^{\frac {1+n}{2}} \left (n^2 (-1+a x)-2 \left (2+a^2 x^2\right )+n \left (-1-a x+2 a^2 x^2\right )\right )+2^{1+\frac {n}{2}} n \left (5+n^2\right ) (-1+a x) \, _2F_1\left (\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {5}{2}-\frac {n}{2};\frac {1}{2}-\frac {a x}{2}\right )\right )}{6 \sqrt {2} a^4 (-3+n) (-1+n) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{3}}{\sqrt {-a^{2} c \,x^{2}+c}}\, dx\]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {x^3\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-a^2\,c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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