Optimal. Leaf size=338 \[ \frac {2 (9 A b-7 a B) e (e x)^{3/2} \sqrt {a+b x^2}}{45 b^2}+\frac {2 B (e x)^{7/2} \sqrt {a+b x^2}}{9 b e}-\frac {2 a (9 A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{15 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 a^{5/4} (9 A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^2}}-\frac {a^{5/4} (9 A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.18, 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 = {470, 327, 335,
311, 226, 1210} \begin {gather*} -\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 \text {ArcTan}\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 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 327
Rule 335
Rule 470
Rule 1210
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) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.10, size = 96, normalized size = 0.28 \begin {gather*} \frac {2 e (e x)^{3/2} \left (-\left (\left (a+b x^2\right ) \left (-9 A b+7 a B-5 b B x^2\right )\right )+a (-9 A b+7 a B) \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{45 b^2 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 417, normalized size = 1.23
method | result | size |
risch | \(\frac {2 x^{2} \left (5 b B \,x^{2}+9 A b -7 B a \right ) \sqrt {b \,x^{2}+a}\, e^{3}}{45 b^{2} \sqrt {e x}}-\frac {a \left (9 A b -7 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) e^{3} \sqrt {\left (b \,x^{2}+a \right ) e x}}{15 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(242\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,e^{2} x^{3} \sqrt {b e \,x^{3}+a e x}}{9 b}+\frac {2 \left (A \,e^{3}-\frac {7 B \,e^{3} a}{9 b}\right ) x \sqrt {b e \,x^{3}+a e x}}{5 b e}-\frac {3 \left (A \,e^{3}-\frac {7 B \,e^{3} a}{9 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(275\) |
default | \(-\frac {e^{2} \sqrt {e x}\, \left (-10 B \,b^{3} x^{6}+54 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -27 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -42 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+21 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-18 A \,b^{3} x^{4}+4 B a \,b^{2} x^{4}-18 A a \,b^{2} x^{2}+14 B \,a^{2} b \,x^{2}\right )}{45 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(417\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.39, size = 83, normalized size = 0.25 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (7 \, B a^{2} - 9 \, A a b\right )} \sqrt {b} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, B b^{2} x^{3} - {\left (7 \, B a b - 9 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {x} e^{\frac {5}{2}}\right )}}{45 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 14.89, size = 94, normalized size = 0.28 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{5/2}}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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