Optimal. Leaf size=145 \[ \frac {A \sqrt {\frac {c x^2}{a}+1} (e x)^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};-\frac {c x^2}{a}\right )}{a^2 e (m+1) \sqrt {a+c x^2}}+\frac {B \sqrt {\frac {c x^2}{a}+1} (e x)^{m+2} \, _2F_1\left (\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};-\frac {c x^2}{a}\right )}{a^2 e^2 (m+2) \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 145, 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} \[ \frac {A \sqrt {\frac {c x^2}{a}+1} (e x)^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};-\frac {c x^2}{a}\right )}{a^2 e (m+1) \sqrt {a+c x^2}}+\frac {B \sqrt {\frac {c x^2}{a}+1} (e x)^{m+2} \, _2F_1\left (\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};-\frac {c x^2}{a}\right )}{a^2 e^2 (m+2) \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rubi steps
\begin {align*} \int \frac {(e x)^m (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=A \int \frac {(e x)^m}{\left (a+c x^2\right )^{5/2}} \, dx+\frac {B \int \frac {(e x)^{1+m}}{\left (a+c x^2\right )^{5/2}} \, dx}{e}\\ &=\frac {\left (A \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {(e x)^m}{\left (1+\frac {c x^2}{a}\right )^{5/2}} \, dx}{a^2 \sqrt {a+c x^2}}+\frac {\left (B \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {(e x)^{1+m}}{\left (1+\frac {c x^2}{a}\right )^{5/2}} \, dx}{a^2 e \sqrt {a+c x^2}}\\ &=\frac {A (e x)^{1+m} \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};-\frac {c x^2}{a}\right )}{a^2 e (1+m) \sqrt {a+c x^2}}+\frac {B (e x)^{2+m} \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};-\frac {c x^2}{a}\right )}{a^2 e^2 (2+m) \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 111, normalized size = 0.77 \[ \frac {x \sqrt {\frac {c x^2}{a}+1} (e x)^m \left (A (m+2) \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};-\frac {c x^2}{a}\right )+B (m+1) x \, _2F_1\left (\frac {5}{2},\frac {m}{2}+1;\frac {m}{2}+2;-\frac {c x^2}{a}\right )\right )}{a^2 (m+1) (m+2) \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{m}}{c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.67, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (e x \right )^{m}}{\left (c \,x^{2}+a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 159.10, size = 112, normalized size = 0.77 \[ \frac {A e^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {B e^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {m}{2} + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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