Optimal. Leaf size=333 \[ -\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 333, 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 = {464, 296, 335,
311, 226, 1210} \begin {gather*} \frac {\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 )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {\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) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt {a+b x^2}}+\frac {\sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 296
Rule 311
Rule 335
Rule 464
Rule 1210
Rubi steps
\begin {align*} \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/2}} \, dx}{a e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{2 a^2 e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a^2 e^3}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(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 )}{a^{3/2} \sqrt {b} e^2}-\frac {(3 A b-a B) \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 )}{a^{3/2} \sqrt {b} e^2}\\ &=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {(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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(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 )}{2 a^{7/4} b^{3/4} e^{3/2} \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 = 10.03, size = 77, normalized size = 0.23 \begin {gather*} \frac {x \left (-6 a A+2 (-3 A b+a B) x^2 \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 )}{3 a^2 (e x)^{3/2} \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 386, normalized size = 1.16
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {x^{2} \left (A b -B a \right )}{e \,a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {2 \left (b e \,x^{2}+a e \right ) A}{a^{2} e^{2} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {\left (\frac {A b -B a}{2 a^{2} e}+\frac {b A}{a^{2} e}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(281\) |
default | \(\frac {6 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b -3 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b -2 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 A \,b^{2} x^{2}+2 B a b \,x^{2}-4 a b A}{2 \sqrt {b \,x^{2}+a}\, b e \sqrt {e x}\, a^{2}}\) | \(386\) |
risch | \(-\frac {2 A \sqrt {b \,x^{2}+a}}{a^{2} e \sqrt {e x}}+\frac {\left (\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b e \,x^{3}+a e x}}-a \left (A b -B a \right ) \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{a^{2} e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(413\) |
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 = 0.40, size = 109, normalized size = 0.33 \begin {gather*} \frac {{\left ({\left ({\left (B a b - 3 \, A b^{2}\right )} x^{3} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (2 \, A a b - {\left (B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{a^{2} b^{2} x^{3} + a^{3} b x} \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 = 9.40, size = 97, normalized size = 0.29 \begin {gather*} \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{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 )}^{3/2}\,{\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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