Optimal. Leaf size=141 \[ \frac {a A \sqrt {a+c x^2} (e x)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};-\frac {c x^2}{a}\right )}{e (m+1) \sqrt {\frac {c x^2}{a}+1}}+\frac {a B \sqrt {a+c x^2} (e x)^{m+2} \, _2F_1\left (-\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};-\frac {c x^2}{a}\right )}{e^2 (m+2) \sqrt {\frac {c x^2}{a}+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {808, 365, 364} \[ \frac {a A \sqrt {a+c x^2} (e x)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};-\frac {c x^2}{a}\right )}{e (m+1) \sqrt {\frac {c x^2}{a}+1}}+\frac {a B \sqrt {a+c x^2} (e x)^{m+2} \, _2F_1\left (-\frac {3}{2},\frac {m+2}{2};\frac {m+4}{2};-\frac {c x^2}{a}\right )}{e^2 (m+2) \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rubi steps
\begin {align*} \int (e x)^m (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=A \int (e x)^m \left (a+c x^2\right )^{3/2} \, dx+\frac {B \int (e x)^{1+m} \left (a+c x^2\right )^{3/2} \, dx}{e}\\ &=\frac {\left (a A \sqrt {a+c x^2}\right ) \int (e x)^m \left (1+\frac {c x^2}{a}\right )^{3/2} \, dx}{\sqrt {1+\frac {c x^2}{a}}}+\frac {\left (a B \sqrt {a+c x^2}\right ) \int (e x)^{1+m} \left (1+\frac {c x^2}{a}\right )^{3/2} \, dx}{e \sqrt {1+\frac {c x^2}{a}}}\\ &=\frac {a A (e x)^{1+m} \sqrt {a+c x^2} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};-\frac {c x^2}{a}\right )}{e (1+m) \sqrt {1+\frac {c x^2}{a}}}+\frac {a B (e x)^{2+m} \sqrt {a+c x^2} \, _2F_1\left (-\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};-\frac {c x^2}{a}\right )}{e^2 (2+m) \sqrt {1+\frac {c x^2}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 109, normalized size = 0.77 \[ \frac {a x \sqrt {a+c x^2} (e x)^m \left (A (m+2) \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};-\frac {c x^2}{a}\right )+B (m+1) x \, _2F_1\left (-\frac {3}{2},\frac {m}{2}+1;\frac {m}{2}+2;-\frac {c x^2}{a}\right )\right )}{(m+1) (m+2) \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a} \left (e x\right )^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.15, size = 0, normalized size = 0.00 \[ \int \left (B x +A \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \left (e x \right )^{m}\, dx \]
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 \[ \int {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \]
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 \[ \int {\left (e\,x\right )}^m\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.55, size = 238, normalized size = 1.69 \[ \frac {A a^{\frac {3}{2}} e^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A \sqrt {a} c e^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a^{\frac {3}{2}} e^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {B \sqrt {a} c e^{m} x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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