Optimal. Leaf size=224 \[ \frac {a (6 p+17) x^5 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )}{10 (p+2)}-\frac {a x^5 \left (c-a^2 c x^2\right )^p}{2 (p+2) \sqrt {1-a^2 x^2}}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^4 (2 p+3)}+\frac {7 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^4 (2 p+1)}+\frac {4 \left (c-a^2 c x^2\right )^p}{a^4 (1-2 p) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.36, antiderivative size = 224, 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, 446, 77, 459, 364} \[ \frac {a (6 p+17) x^5 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )}{10 (p+2)}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^4 (2 p+3)}+\frac {7 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^4 (2 p+1)}-\frac {a x^5 \left (c-a^2 c x^2\right )^p}{2 (p+2) \sqrt {1-a^2 x^2}}+\frac {4 \left (c-a^2 c x^2\right )^p}{a^4 (1-2 p) \sqrt {1-a^2 x^2}} \]
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^3 \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^3 \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^3 (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^3 \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^4 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \left (3 a+a^3 x^2\right ) \, dx\\ &=-\frac {a x^5 \left (c-a^2 c x^2\right )^p}{2 (2+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 (1+3 a^2 x\right ) \, dx,x,x^2\right )+\frac {\left (a (17+6 p) \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^4 \left (1-a^2 x^2\right )^{-\frac {3}{2}+p} \, dx}{2 (2+p)}\\ &=-\frac {a x^5 \left (c-a^2 c x^2\right )^p}{2 (2+p) \sqrt {1-a^2 x^2}}+\frac {a (17+6 p) x^5 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )}{10 (2+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^2}-\frac {7 \left (1-a^2 x\right )^{-\frac {1}{2}+p}}{a^2}+\frac {3 \left (1-a^2 x\right )^{\frac {1}{2}+p}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {4 \left (c-a^2 c x^2\right )^p}{a^4 (1-2 p) \sqrt {1-a^2 x^2}}-\frac {a x^5 \left (c-a^2 c x^2\right )^p}{2 (2+p) \sqrt {1-a^2 x^2}}+\frac {7 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^4 (1+2 p)}-\frac {3 \left (1-a^2 x^2\right )^{3/2} \left (c-a^2 c x^2\right )^p}{a^4 (3+2 p)}+\frac {a (17+6 p) x^5 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )}{10 (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 176, normalized size = 0.79 \[ \frac {\left (c-a^2 c x^2\right )^p \left (-\frac {105 \left (1-a^2 x^2\right )^{3/2}}{2 p+3}+\frac {245 \sqrt {1-a^2 x^2}}{2 p+1}+\frac {140}{(1-2 p) \sqrt {1-a^2 x^2}}+5 a^7 x^7 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac {7}{2},\frac {3}{2}-p;\frac {9}{2};a^2 x^2\right )+21 a^5 x^5 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac {5}{2},\frac {3}{2}-p;\frac {7}{2};a^2 x^2\right )\right )}{35 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x^{4} + x^{3}\right )} \sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{3} x^{3} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ -\frac {{\left (a^{2} c^{p} {\left (2 \, p - 1\right )} x^{2} + 2 \, c^{p}\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} {\left (4 \, p^{2} - 1\right )} a^{4}} - \int \frac {{\left (a^{3} c^{p} x^{6} + 3 \, a^{2} c^{p} x^{5} + 3 \, a c^{p} x^{4}\right )} e^{\left (p \log \left (a x + 1\right ) + p \log \left (-a x + 1\right )\right )}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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