Optimal. Leaf size=187 \[ a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )-\frac {3 a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};1-a^2 x^2\right )}{2 p+1}+\frac {4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.33, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6153, 6148, 1807, 1652, 446, 79, 65, 12, 245} \[ a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )-\frac {3 a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};1-a^2 x^2\right )}{2 p+1}+\frac {4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 245
Rule 446
Rule 1652
Rule 1807
Rule 6148
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {(1+a x)^3 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p}}{x^2} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}-\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {\left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \left (-3 a-a^2 (5-2 p) x-a^3 x^2\right )}{x} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int a^2 (5-2 p) \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 \frac {\left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \left (-3 a-a^3 x^2\right )}{x} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{x \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 \frac {\left (1-a^2 x\right )^{-\frac {3}{2}+p} \left (-3 a-a^3 x\right )}{x} \, dx,x,x^2\right )+\left (a^2 (5-2 p) \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \, dx\\ &=\frac {4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )+\frac {1}{2} \left (3 a \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1-a^2 x\right )^{-\frac {1}{2}+p}}{x} \, dx,x,x^2\right )\\ &=\frac {4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )-\frac {3 a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,\frac {1}{2}+p;\frac {3}{2}+p;1-a^2 x^2\right )}{1+2 p}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 133, normalized size = 0.71 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a \left (\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}} \left (3 \, _2F_1\left (1,p-\frac {1}{2};p+\frac {1}{2};1-a^2 x^2\right )+1\right )}{1-2 p}+3 a x \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )\right )-\frac {\, _2F_1\left (-\frac {1}{2},\frac {3}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{4} - 2 \, a x^{3} + x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{3} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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