Optimal. Leaf size=222 \[ \frac {(2 p+11) x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {3}{2}-p;\frac {5}{2};a^2 x^2\right )}{6 (p+1)}-\frac {3 x^3 \left (c-a^2 c x^2\right )^p}{2 (p+1) \sqrt {1-a^2 x^2}}-\frac {\left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^3 (2 p+3)}+\frac {5 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^3 (2 p+1)}+\frac {4 \left (c-a^2 c x^2\right )^p}{a^3 (1-2 p) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.34, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6153, 6148, 1652, 459, 364, 446, 77} \[ \frac {(2 p+11) x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {3}{2}-p;\frac {5}{2};a^2 x^2\right )}{6 (p+1)}-\frac {\left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^3 (2 p+3)}+\frac {5 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^3 (2 p+1)}-\frac {3 x^3 \left (c-a^2 c x^2\right )^p}{2 (p+1) \sqrt {1-a^2 x^2}}+\frac {4 \left (c-a^2 c x^2\right )^p}{a^3 (1-2 p) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 364
Rule 446
Rule 459
Rule 1652
Rule 6148
Rule 6153
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{3 \tanh ^{-1}(a x)} x^2 \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^2 (1+a x)^3 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^2 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \left (1+3 a^2 x^2\right ) \, dx+\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^3 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \left (3 a+a^3 x^2\right ) \, dx\\ &=-\frac {3 x^3 \left (c-a^2 c x^2\right )^p}{2 (1+p) \sqrt {1-a^2 x^2}}+\frac {1}{2} \left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \operatorname {Subst}\left (\int x \left (1-a^2 x\right )^{-\frac {3}{2}+p} \left (3 a+a^3 x\right ) \, dx,x,x^2\right )+\frac {\left ((11+2 p) \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^2 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \, dx}{2 (1+p)}\\ &=-\frac {3 x^3 \left (c-a^2 c x^2\right )^p}{2 (1+p) \sqrt {1-a^2 x^2}}+\frac {(11+2 p) x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {3}{2}-p;\frac {5}{2};a^2 x^2\right )}{6 (1+p)}+\frac {1}{2} \left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \operatorname {Subst}\left (\int \left (\frac {4 \left (1-a^2 x\right )^{-\frac {3}{2}+p}}{a}-\frac {5 \left (1-a^2 x\right )^{-\frac {1}{2}+p}}{a}+\frac {\left (1-a^2 x\right )^{\frac {1}{2}+p}}{a}\right ) \, dx,x,x^2\right )\\ &=\frac {4 \left (c-a^2 c x^2\right )^p}{a^3 (1-2 p) \sqrt {1-a^2 x^2}}-\frac {3 x^3 \left (c-a^2 c x^2\right )^p}{2 (1+p) \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^3 (1+2 p)}-\frac {\left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^3 (3+2 p)}+\frac {(11+2 p) x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {3}{2}-p;\frac {5}{2};a^2 x^2\right )}{6 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 179, normalized size = 0.81 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {3}{5} a^2 x^5 \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )+\frac {1}{3} x^3 \, _2F_1\left (\frac {3}{2},\frac {3}{2}-p;\frac {5}{2};a^2 x^2\right )+\frac {\left (a^4 x^4-4 a^2 p^2 x^2 \left (a^2 x^2+3\right )-4 p \left (5 a^2 x^2+3\right )+13 a^2 x^2-26\right ) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{a^3 (2 p-1) (2 p+1) (2 p+3)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + x^{2}\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{3} x^{2} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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