Optimal. Leaf size=337 \[ \frac {2 a^{5/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}-\frac {4 a^{5/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}+\frac {4 a \sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2} (3 A b-a B)}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e} \]
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Rubi [A] time = 0.27, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {459, 279, 329, 305, 220, 1196} \[ \frac {2 a^{5/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}-\frac {4 a^{5/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}+\frac {4 a \sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2} (3 A b-a B)}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 305
Rule 329
Rule 459
Rule 1196
Rubi steps
\begin {align*} \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx &=\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}-\frac {\left (2 \left (-\frac {9 A b}{2}+\frac {3 a B}{2}\right )\right ) \int \sqrt {e x} \sqrt {a+b x^2} \, dx}{9 b}\\ &=\frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}+\frac {(2 a (3 A b-a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{15 b}\\ &=\frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}+\frac {(4 a (3 A b-a B)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b e}\\ &=\frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}+\frac {\left (4 a^{3/2} (3 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^{3/2}}-\frac {\left (4 a^{3/2} (3 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} e}}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^{3/2}}\\ &=\frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {4 a (3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}-\frac {4 a^{5/4} (3 A b-a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}+\frac {2 a^{5/4} (3 A b-a B) \sqrt {e} \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 )}{15 b^{7/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 93, normalized size = 0.28 \[ \frac {2 x \sqrt {e x} \sqrt {a+b x^2} \left ((3 A b-a B) \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )+B \sqrt {\frac {b x^2}{a}+1} \left (a+b x^2\right )\right )}{9 b \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}, x\right ) \]
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 \[ \int {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 414, normalized size = 1.23 \[ \frac {2 \sqrt {e x}\, \left (5 B \,b^{3} x^{6}+9 A \,b^{3} x^{4}+7 B a \,b^{2} x^{4}+9 A a \,b^{2} x^{2}+2 B \,a^{2} b \,x^{2}+18 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A \,a^{2} b \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-9 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A \,a^{2} b \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{3} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{3} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right )}{45 \sqrt {b \,x^{2}+a}\, b^{2} x} \]
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 \[ \int {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}\,{d x} \]
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 \[ \int \left (B\,x^2+A\right )\,\sqrt {e\,x}\,\sqrt {b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.75, size = 95, normalized size = 0.28 \[ \frac {A \sqrt {a} \left (e x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} \left (e x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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