Optimal. Leaf size=174 \[ -\frac {a^{3/4} e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}}+\frac {2 e \sqrt {e x} \sqrt {a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e} \]
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Rubi [A] time = 0.11, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {459, 321, 329, 220} \[ -\frac {a^{3/4} e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}}+\frac {2 e \sqrt {e x} \sqrt {a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e} \]
Antiderivative was successfully verified.
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Rule 220
Rule 321
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx &=\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {5 a B}{2}\right )\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{7 b}\\ &=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {\left (a (7 A b-5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{21 b^2}\\ &=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {(2 a (7 A b-5 a B) e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^2}\\ &=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {a^{3/4} (7 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 96, normalized size = 0.55 \[ \frac {2 e \sqrt {e x} \left (a \sqrt {\frac {b x^2}{a}+1} (5 a B-7 A b) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )-\left (a+b x^2\right ) \left (5 a B-7 A b-3 b B x^2\right )\right )}{21 b^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e x^{3} + A e x\right )} \sqrt {e x}}{\sqrt {b x^{2} + a}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{\sqrt {b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 250, normalized size = 1.44 \[ -\frac {\sqrt {e x}\, \left (-6 B \,b^{3} x^{5}-14 A \,b^{3} x^{3}+4 B a \,b^{2} x^{3}-14 A a \,b^{2} x +10 B \,a^{2} b x +7 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, A a b \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-5 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) e}{21 \sqrt {b \,x^{2}+a}\, b^{3} x} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{\sqrt {b x^{2} + a}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}}{\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.27, size = 94, normalized size = 0.54 \[ \frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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