Optimal. Leaf size=333 \[ \frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt {a+b x^2}}+\frac {\sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 333, 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 = {453, 290, 329, 305, 220, 1196} \[ \frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt {a+b x^2}}+\frac {\sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 305
Rule 329
Rule 453
Rule 1196
Rubi steps
\begin {align*} \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/2}} \, dx}{a e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{2 a^2 e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(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 )}{a^2 e^3}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a^{3/2} \sqrt {b} e^2}-\frac {(3 A b-a B) \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 )}{a^{3/2} \sqrt {b} e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {(3 A b-a B) \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 )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(3 A b-a B) \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 )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 77, normalized size = 0.23 \[ \frac {x \left (2 x^2 \sqrt {\frac {b x^2}{a}+1} (a B-3 A b) \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {b x^2}{a}\right )-6 a A\right )}{3 a^2 (e x)^{3/2} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 386, normalized size = 1.16 \[ \frac {-6 A \,b^{2} x^{2}+2 B a b \,x^{2}+6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A a b \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A a b \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{2} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-4 A a b}{2 \sqrt {b \,x^{2}+a}\, \sqrt {e x}\, a^{2} b e} \]
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 \[ \int \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {B\,x^2+A}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 24.79, size = 97, normalized size = 0.29 \[ \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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