Optimal. Leaf size=338 \[ \frac {2 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 a B+A b) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}-\frac {4 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 a B+A b) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}+\frac {4 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (5 a B+A b)}{5 a e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \sqrt {a+b x^2} (5 a B+A b)}{5 a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}} \]
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Rubi [A] time = 0.26, antiderivative size = 338, 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, 277, 329, 305, 220, 1196} \[ \frac {2 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 a B+A b) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}-\frac {4 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 a B+A b) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}-\frac {2 \sqrt {a+b x^2} (5 a B+A b)}{5 a e^3 \sqrt {e x}}+\frac {4 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (5 a B+A b)}{5 a e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 305
Rule 329
Rule 453
Rule 1196
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx &=-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}+\frac {(A b+5 a B) \int \frac {\sqrt {a+b x^2}}{(e x)^{3/2}} \, dx}{5 a e^2}\\ &=-\frac {2 (A b+5 a B) \sqrt {a+b x^2}}{5 a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}+\frac {(2 b (A b+5 a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{5 a e^4}\\ &=-\frac {2 (A b+5 a B) \sqrt {a+b x^2}}{5 a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}+\frac {(4 b (A b+5 a B)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a e^5}\\ &=-\frac {2 (A b+5 a B) \sqrt {a+b x^2}}{5 a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}+\frac {\left (4 \sqrt {b} (A b+5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 \sqrt {a} e^4}-\frac {\left (4 \sqrt {b} (A b+5 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 )}{5 \sqrt {a} e^4}\\ &=-\frac {2 (A b+5 a B) \sqrt {a+b x^2}}{5 a e^3 \sqrt {e x}}+\frac {4 \sqrt {b} (A b+5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}-\frac {4 \sqrt [4]{b} (A b+5 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 )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{b} (A b+5 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 )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 95, normalized size = 0.28 \[ -\frac {2 x \sqrt {a+b x^2} \left (x^2 (5 a B+A b) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};-\frac {b x^2}{a}\right )+A \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1}\right )}{5 a (e x)^{7/2} \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{e^{4} x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{\left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 417, normalized size = 1.23 \[ \frac {-\frac {4 A \,b^{2} x^{4}}{5}-2 B a b \,x^{4}+\frac {4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A a b \,x^{2} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {2 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A a b \,x^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}+4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{2} x^{2} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{2} x^{2} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-\frac {6 A a b \,x^{2}}{5}-2 B \,a^{2} x^{2}-\frac {2 A \,a^{2}}{5}}{\sqrt {b \,x^{2}+a}\, \sqrt {e x}\, a \,e^{3} x^{2}} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{\left (e x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,\sqrt {b\,x^2+a}}{{\left (e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.34, size = 107, normalized size = 0.32 \[ \frac {A \sqrt {a} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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