Optimal. Leaf size=187 \[ \frac {(A b-a B) \sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(5 A b+a B) \sqrt {e x}}{6 a^2 b e \sqrt {a+b x^2}}+\frac {(5 A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{12 a^{9/4} b^{5/4} \sqrt {e} \sqrt {a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {468, 296, 335,
226} \begin {gather*} \frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+5 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^{9/4} b^{5/4} \sqrt {e} \sqrt {a+b x^2}}+\frac {\sqrt {e x} (a B+5 A b)}{6 a^2 b e \sqrt {a+b x^2}}+\frac {\sqrt {e x} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 226
Rule 296
Rule 335
Rule 468
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/2}} \, dx &=\frac {(A b-a B) \sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {\left (\frac {5 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/2}} \, dx}{3 a b}\\ &=\frac {(A b-a B) \sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(5 A b+a B) \sqrt {e x}}{6 a^2 b e \sqrt {a+b x^2}}+\frac {(5 A b+a B) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{12 a^2 b}\\ &=\frac {(A b-a B) \sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(5 A b+a B) \sqrt {e x}}{6 a^2 b e \sqrt {a+b x^2}}+\frac {(5 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^2 b e}\\ &=\frac {(A b-a B) \sqrt {e x}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(5 A b+a B) \sqrt {e x}}{6 a^2 b e \sqrt {a+b x^2}}+\frac {(5 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^{9/4} b^{5/4} \sqrt {e} \sqrt {a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.07, size = 108, normalized size = 0.58 \begin {gather*} \frac {-a^2 B x+5 A b^2 x^3+a b x \left (7 A+B x^2\right )+(5 A b+a B) x \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 )}{6 a^2 b \sqrt {e x} \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs.
\(2(194)=388\).
time = 0.12, size = 425, normalized size = 2.27
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {\left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 a e \,b^{3} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {x \left (5 A b +B a \right )}{6 b \,a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (5 A b +B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 b^{2} a^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(225\) |
default | \(\frac {5 A \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, b^{2} x^{2}+B \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, a b \,x^{2}+5 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b +B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}+10 A \,b^{3} x^{3}+2 B a \,b^{2} x^{3}+14 A a \,b^{2} x -2 B \,a^{2} b x}{12 \sqrt {e x}\, a^{2} b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(425\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.18, size = 144, normalized size = 0.77 \begin {gather*} \frac {{\left ({\left ({\left (B a b^{2} + 5 \, A b^{3}\right )} x^{4} + B a^{3} + 5 \, A a^{2} b + 2 \, {\left (B a^{2} b + 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (B a^{2} b - 7 \, A a b^{2} - {\left (B a b^{2} + 5 \, A b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {1}{2}\right )}}{6 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 42.55, size = 94, normalized size = 0.50 \begin {gather*} \frac {A \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}} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {B x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {B\,x^2+A}{\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________