\(\int \frac {a g+e x+f x^3-b g x^4}{(a+b x^4)^{3/2}} \, dx\) [218]

Optimal result

Integrand size = 31, antiderivative size = 38 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {a f-2 a b g x-b e x^2}{2 a b \sqrt {a+b x^4}} \]

[Out]

1/2*(2*a*b*g*x+b*e*x^2-a*f)/a/b/(b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1870} \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {-2 a b g x+a f-b e x^2}{2 a b \sqrt {a+b x^4}} \]

[In]

Int[(a*g + e*x + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x]

[Out]

-1/2*(a*f - 2*a*b*g*x - b*e*x^2)/(a*b*Sqrt[a + b*x^4])

Rule 1870

Int[(P4_)/((a_) + (b_.)*(x_)^4)^(3/2), x_Symbol] :> With[{d = Coeff[P4, x, 0], e = Coeff[P4, x, 1], f = Coeff[
P4, x, 3], g = Coeff[P4, x, 4]}, Simp[-(a*f + 2*a*g*x - b*e*x^2)/(2*a*b*Sqrt[a + b*x^4]), x] /; EqQ[b*d + a*g,
0]] /; FreeQ[{a, b}, x] && PolyQ[P4, x, 4] && EqQ[Coeff[P4, x, 2], 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {a f-2 a b g x-b e x^2}{2 a b \sqrt {a+b x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {-a f+2 a b g x+b e x^2}{2 a b \sqrt {a+b x^4}} \]

[In]

Integrate[(a*g + e*x + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x]

[Out]

(-(a*f) + 2*a*b*g*x + b*e*x^2)/(2*a*b*Sqrt[a + b*x^4])

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92

method	result	size
gosper	\(\frac {2 a b g x +b e \,x^{2}-a f}{2 a b \sqrt {b \,x^{4}+a}}\)	\(35\)
trager	\(\frac {2 a b g x +b e \,x^{2}-a f}{2 a b \sqrt {b \,x^{4}+a}}\)	\(35\)
elliptic	\(-\frac {-b e \,x^{2}+a f}{2 \sqrt {b \,x^{4}+a}\, a b}+\frac {g x}{\sqrt {b \,x^{4}+a}}\)	\(42\)
default	\(-\frac {f}{2 b \sqrt {b \,x^{4}+a}}+a g \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {e \,x^{2}}{2 a \sqrt {b \,x^{4}+a}}-g b \left (-\frac {x}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(228\)

[In]

int((-b*g*x^4+f*x^3+a*g+e*x)/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(2*a*b*g*x+b*e*x^2-a*f)/a/b/(b*x^4+a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {\sqrt {b x^{4} + a} {\left (2 \, a b g x + b e x^{2} - a f\right )}}{2 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} \]

[In]

integrate((-b*g*x^4+f*x^3+a*g+e*x)/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

1/2*sqrt(b*x^4 + a)*(2*a*b*g*x + b*e*x^2 - a*f)/(a*b^2*x^4 + a^2*b)

Sympy [A] (verification not implemented)

Time = 5.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.50 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=f \left (\begin {cases} - \frac {1}{2 b \sqrt {a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {g x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} - \frac {b g x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {e x^{2}}{2 a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{4}}{a}}} \]

[In]

integrate((-b*g*x**4+f*x**3+a*g+e*x)/(b*x**4+a)**(3/2),x)

[Out]

f*Piecewise((-1/(2*b*sqrt(a + b*x**4)), Ne(b, 0)), (x**4/(4*a**(3/2)), True)) + g*x*gamma(1/4)*hyper((1/4, 3/2
), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(5/4)) - b*g*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), b*
x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(9/4)) + e*x**2/(2*a**(3/2)*sqrt(1 + b*x**4/a))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {\sqrt {b x^{4} + a} {\left (2 \, a b g x + b e x^{2} - a f\right )}}{2 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} \]

[In]

integrate((-b*g*x^4+f*x^3+a*g+e*x)/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

1/2*sqrt(b*x^4 + a)*(2*a*b*g*x + b*e*x^2 - a*f)/(a*b^2*x^4 + a^2*b)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {{\left (2 \, g + \frac {e x}{a}\right )} x - \frac {f}{b}}{2 \, \sqrt {b x^{4} + a}} \]

[In]

integrate((-b*g*x^4+f*x^3+a*g+e*x)/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

1/2*((2*g + e*x/a)*x - f/b)/sqrt(b*x^4 + a)

Mupad [B] (verification not implemented)

Time = 9.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {a g+e x+f x^3-b g x^4}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {g\,x-\frac {f}{2\,b}+\frac {e\,x^2}{2\,a}}{\sqrt {b\,x^4+a}} \]

[In]

int((a*g + e*x + f*x^3 - b*g*x^4)/(a + b*x^4)^(3/2),x)

[Out]

(g*x - f/(2*b) + (e*x^2)/(2*a))/(a + b*x^4)^(1/2)