Optimal. Leaf size=213 \[ -\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {5 (3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{12 a^{13/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 213, 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 = {464, 296, 335,
226} \begin {gather*} -\frac {5 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-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 )}{12 a^{13/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {e x} (3 A b-a B)}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {\sqrt {e x} (3 A b-a B)}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 296
Rule 335
Rule 464
Rubi steps
\begin {align*} \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{5/2}} \, dx}{a e^2}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {(5 (3 A b-a B)) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/2}} \, dx}{6 a^2 e^2}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {(5 (3 A b-a B)) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{12 a^3 e^2}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {(5 (3 A b-a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a^3 e^3}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {5 (3 A b-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 )}{12 a^{13/4} \sqrt [4]{b} e^{5/2} \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 = 10.08, size = 120, normalized size = 0.56 \begin {gather*} \frac {x \left (-15 A b^2 x^4+a^2 \left (-4 A+7 B x^2\right )+a \left (-21 A b x^2+5 b B x^4\right )+5 (-3 A b+a B) x^2 \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{6 a^3 (e x)^{5/2} \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs.
\(2(214)=428\).
time = 0.14, size = 446, normalized size = 2.09
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {\left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 a^{2} e^{3} b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {x \left (11 A b -5 B a \right )}{6 e^{2} a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{3 a^{3} e^{3} x^{2}}+\frac {\left (-\frac {11 A b -5 B a}{12 a^{3} e^{2}}-\frac {b A}{3 a^{3} e^{2}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(267\) |
default | \(-\frac {15 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, b^{2} x^{3}-5 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b \,x^{3}+15 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b x -5 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2} x +30 A \,b^{3} x^{4}-10 B a \,b^{2} x^{4}+42 A a \,b^{2} x^{2}-14 B \,a^{2} b \,x^{2}+8 A \,a^{2} b}{12 x \,e^{2} \sqrt {e x}\, a^{3} b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(446\) |
risch | \(-\frac {2 A \sqrt {b \,x^{2}+a}}{3 a^{3} x \,e^{2} \sqrt {e x}}-\frac {\left (\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}+3 a^{2} \left (A b -B a \right ) \left (\frac {\sqrt {b e \,x^{3}+a e x}}{3 a e \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 x}{6 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} b \sqrt {b e \,x^{3}+a e x}}\right )+3 a b A \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 a^{3} e^{2} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(492\) |
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 = 0.41, size = 163, normalized size = 0.77 \begin {gather*} \frac {{\left (5 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (5 \, {\left (B a b^{2} - 3 \, A b^{3}\right )} x^{4} - 4 \, A a^{2} b + 7 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{6 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\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 = 148.76, size = 97, normalized size = 0.46 \begin {gather*} \frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {5}{2}} \Gamma \left (\frac {5}{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 {B\,x^2+A}{{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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