Optimal. Leaf size=338 \[ -\frac {a^{5/4} e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (9 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^2}}+\frac {2 a^{5/4} e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (9 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^2}}-\frac {2 a e^2 \sqrt {e x} \sqrt {a+b x^2} (9 A b-7 a B)}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 e (e x)^{3/2} \sqrt {a+b x^2} (9 A b-7 a B)}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e} \]
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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 = {459, 321, 329, 305, 220, 1196} \[ -\frac {a^{5/4} e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (9 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^2}}+\frac {2 a^{5/4} e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (9 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^2}}-\frac {2 a e^2 \sqrt {e x} \sqrt {a+b x^2} (9 A b-7 a B)}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 e (e x)^{3/2} \sqrt {a+b x^2} (9 A b-7 a B)}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 321
Rule 329
Rule 459
Rule 1196
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx &=\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e}-\frac {\left (2 \left (-\frac {9 A b}{2}+\frac {7 a B}{2}\right )\right ) \int \frac {(e x)^{5/2}}{\sqrt {a+b x^2}} \, dx}{9 b}\\ &=\frac {2 (9 A b-7 a B) e (e x)^{3/2} \sqrt {a+b x^2}}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e}-\frac {\left (a (9 A b-7 a B) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{15 b^2}\\ &=\frac {2 (9 A b-7 a B) e (e x)^{3/2} \sqrt {a+b x^2}}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e}-\frac {(2 a (9 A b-7 a B) e) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^2}\\ &=\frac {2 (9 A b-7 a B) e (e x)^{3/2} \sqrt {a+b x^2}}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e}-\frac {\left (2 a^{3/2} (9 A b-7 a B) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^{5/2}}+\frac {\left (2 a^{3/2} (9 A b-7 a B) e^2\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^{5/2}}\\ &=\frac {2 (9 A b-7 a B) e (e x)^{3/2} \sqrt {a+b x^2}}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e}-\frac {2 a (9 A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 a^{5/4} (9 A b-7 a B) e^{5/2} \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^{11/4} \sqrt {a+b x^2}}-\frac {a^{5/4} (9 A b-7 a B) e^{5/2} \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 )}{15 b^{11/4} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 96, normalized size = 0.28 \[ \frac {2 e (e x)^{3/2} \left (a \sqrt {\frac {b x^2}{a}+1} (7 a B-9 A b) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )-\left (a+b x^2\right ) \left (7 a B-9 A b-5 b B x^2\right )\right )}{45 b^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{2} x^{4} + A e^{2} x^{2}\right )} \sqrt {e x}}{\sqrt {b x^{2} + a}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {5}{2}}}{\sqrt {b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 417, normalized size = 1.23 \[ -\frac {\sqrt {e x}\, \left (-10 B \,b^{3} x^{6}-18 A \,b^{3} x^{4}+4 B a \,b^{2} x^{4}-18 A a \,b^{2} x^{2}+14 B \,a^{2} b \,x^{2}+54 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A \,a^{2} b \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-27 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, A \,a^{2} b \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-42 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{3} \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, B \,a^{3} \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{2}}{45 \sqrt {b \,x^{2}+a}\, b^{3} x} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {5}{2}}}{\sqrt {b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{5/2}}{\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 25.89, size = 94, normalized size = 0.28 \[ \frac {A e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {B e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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