Optimal. Leaf size=210 \[ \frac {4 a^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 a B+7 A b) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}}+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2} (3 a B+7 A b)}{21 a e^3}+\frac {4 \sqrt {e x} \sqrt {a+b x^2} (3 a B+7 A b)}{21 e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {453, 279, 329, 220} \[ \frac {4 a^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 a B+7 A b) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}}+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2} (3 a B+7 A b)}{21 a e^3}+\frac {4 \sqrt {e x} \sqrt {a+b x^2} (3 a B+7 A b)}{21 e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 329
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx &=-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(7 A b+3 a B) \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{3 a e^2}\\ &=\frac {2 (7 A b+3 a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{21 a e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(2 (7 A b+3 a B)) \int \frac {\sqrt {a+b x^2}}{\sqrt {e x}} \, dx}{7 e^2}\\ &=\frac {4 (7 A b+3 a B) \sqrt {e x} \sqrt {a+b x^2}}{21 e^3}+\frac {2 (7 A b+3 a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{21 a e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(4 a (7 A b+3 a B)) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{21 e^2}\\ &=\frac {4 (7 A b+3 a B) \sqrt {e x} \sqrt {a+b x^2}}{21 e^3}+\frac {2 (7 A b+3 a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{21 a e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {(8 a (7 A b+3 a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 e^3}\\ &=\frac {4 (7 A b+3 a B) \sqrt {e x} \sqrt {a+b x^2}}{21 e^3}+\frac {2 (7 A b+3 a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{21 a e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {4 a^{3/4} (7 A b+3 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 )}{21 \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 85, normalized size = 0.40 \[ \frac {2 x \sqrt {a+b x^2} \left (\frac {x^2 (3 a B+7 A b) \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )}{\sqrt {\frac {b x^2}{a}+1}}-\frac {A \left (a+b x^2\right )^2}{a}\right )}{3 (e x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B b x^{4} + {\left (B a + A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{e^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 255, normalized size = 1.21 \[ \frac {\frac {2 B \,b^{3} x^{6}}{7}+\frac {2 A \,b^{3} x^{4}}{3}+\frac {8 B a \,b^{2} x^{4}}{7}+\frac {6 B \,a^{2} b \,x^{2}}{7}+\frac {4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, A a b x \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}\, B \,a^{2} x \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7}-\frac {2 A \,a^{2} b}{3}}{\sqrt {b \,x^{2}+a}\, \sqrt {e x}\, b \,e^{2} x} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 14.67, size = 202, normalized size = 0.96 \[ \frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {A \sqrt {a} b \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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