Optimal. Leaf size=337 \[ -\frac {(5 A b-7 a B) e (e x)^{3/2}}{5 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}}+\frac {3 (5 A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{5 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {3 \sqrt [4]{a} (5 A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 b^{11/4} \sqrt {a+b x^2}}+\frac {3 \sqrt [4]{a} (5 A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 337, 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, 294, 335,
311, 226, 1210} \begin {gather*} \frac {3 \sqrt [4]{a} e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 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 )}{10 b^{11/4} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{a} e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 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 )}{5 b^{11/4} \sqrt {a+b x^2}}+\frac {3 e^2 \sqrt {e x} \sqrt {a+b x^2} (5 A b-7 a B)}{5 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {e (e x)^{3/2} (5 A b-7 a B)}{5 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 294
Rule 311
Rule 335
Rule 470
Rule 1210
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}}-\frac {\left (2 \left (-\frac {5 A b}{2}+\frac {7 a B}{2}\right )\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{5 b}\\ &=-\frac {(5 A b-7 a B) e (e x)^{3/2}}{5 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}}+\frac {\left (3 (5 A b-7 a B) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{10 b^2}\\ &=-\frac {(5 A b-7 a B) e (e x)^{3/2}}{5 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}}+\frac {(3 (5 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 )}{5 b^2}\\ &=-\frac {(5 A b-7 a B) e (e x)^{3/2}}{5 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}}+\frac {\left (3 \sqrt {a} (5 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 )}{5 b^{5/2}}-\frac {\left (3 \sqrt {a} (5 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 )}{5 b^{5/2}}\\ &=-\frac {(5 A b-7 a B) e (e x)^{3/2}}{5 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{7/2}}{5 b e \sqrt {a+b x^2}}+\frac {3 (5 A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{5 b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {3 \sqrt [4]{a} (5 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 )}{5 b^{11/4} \sqrt {a+b x^2}}+\frac {3 \sqrt [4]{a} (5 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 )}{10 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 = 10.11, size = 84, normalized size = 0.25 \begin {gather*} \frac {2 e (e x)^{3/2} \left (5 A b-7 a B+b B x^2+(-5 A b+7 a B) \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{5 b^2 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 391, normalized size = 1.16
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {e^{3} x^{2} \left (A b -B a \right )}{b^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {2 B \,e^{2} x \sqrt {b e \,x^{3}+a e x}}{5 b^{2}}+\frac {\left (\frac {3 \left (A b -B a \right ) e^{3}}{2 b^{2}}-\frac {3 B \,e^{3} a}{5 b^{2}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(278\) |
default | \(\frac {e^{2} \sqrt {e x}\, \left (30 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b -15 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b -42 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+21 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+4 b^{2} B \,x^{4}-10 A \,b^{2} x^{2}+14 B a b \,x^{2}\right )}{10 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(391\) |
risch | \(\frac {2 B \,x^{2} \sqrt {b \,x^{2}+a}\, e^{3}}{5 b^{2} \sqrt {e x}}+\frac {\left (\frac {\left (5 A b -8 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}-5 a \left (A b -B a \right ) \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) e^{3} \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 b^{2} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(428\) |
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 = 0.30, size = 109, normalized size = 0.32 \begin {gather*} \frac {3 \, {\left (7 \, B a^{2} - 5 \, A a b + {\left (7 \, B a b - 5 \, A b^{2}\right )} x^{2}\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 (2 \, B b^{2} x^{3} + {\left (7 \, B a b - 5 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {x} e^{\frac {5}{2}}}{5 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \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 = 52.69, 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 {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \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 {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \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}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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