Optimal. Leaf size=194 \[ \frac {a^2 (9-2 p) \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p-\frac {1}{2};p+\frac {1}{2};1-a^2 x^2\right )}{2 (1-2 p) \sqrt {1-a^2 x^2}}-\frac {3 a \left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \sqrt {1-a^2 x^2}}+a^3 (7-6 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right ) \]
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Rubi [A] time = 0.37, antiderivative size = 194, 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, 1807, 764, 266, 65, 245} \[ a^3 (7-6 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )+\frac {a^2 (9-2 p) \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p-\frac {1}{2};p+\frac {1}{2};1-a^2 x^2\right )}{2 (1-2 p) \sqrt {1-a^2 x^2}}-\frac {3 a \left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 245
Rule 266
Rule 764
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^3} \, 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^3} \, 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^3} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \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 ) \int \frac {\left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \left (-6 a-a^2 (9-2 p) x-2 a^3 x^2\right )}{x^2} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {3 a \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 ) \int \frac {\left (a^2 (9-2 p)+2 a^3 (7-6 p) x\right ) \left (1-a^2 x^2\right )^{-\frac {3}{2}+p}}{x} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {3 a \left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+\left (a^3 (7-6 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 {1}{2} \left (a^2 (9-2 p) \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}}{x} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {3 a \left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+a^3 (7-6 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}{4} \left (a^2 (9-2 p) \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}}{x} \, dx,x,x^2\right )\\ &=-\frac {\left (c-a^2 c x^2\right )^p}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {3 a \left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+a^3 (7-6 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 {a^2 (9-2 p) \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,-\frac {1}{2}+p;\frac {1}{2}+p;1-a^2 x^2\right )}{2 (1-2 p) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 154, normalized size = 0.79 \[ a \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a \left (\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}} \left (3 \, _2F_1\left (1,p-\frac {1}{2};p+\frac {1}{2};1-a^2 x^2\right )+\, _2F_1\left (2,p-\frac {1}{2};p+\frac {1}{2};1-a^2 x^2\right )\right )}{1-2 p}+a x \, _2F_1\left (\frac {1}{2},\frac {3}{2}-p;\frac {3}{2};a^2 x^2\right )\right )-\frac {3 \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{5} - 2 \, a x^{4} + x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {\left (a x +1\right )^{3} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x^{3}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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