\(\int \frac {1}{\sqrt {a x^3+b x^4}} \, dx\) [314]

Optimal result

Integrand size = 15, antiderivative size = 23 \[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=-\frac {2 \sqrt {a x^3+b x^4}}{a x^2} \]

[Out]

-2*(b*x^4+a*x^3)^(1/2)/a/x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2025} \[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=-\frac {2 \sqrt {a x^3+b x^4}}{a x^2} \]

[In]

Int[1/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-2*Sqrt[a*x^3 + b*x^4])/(a*x^2)

Rule 2025

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a x^3+b x^4}}{a x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=-\frac {2 \sqrt {x^3 (a+b x)}}{a x^2} \]

[In]

Integrate[1/Sqrt[a*x^3 + b*x^4],x]

[Out]

(-2*Sqrt[x^3*(a + b*x)])/(a*x^2)

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {2 \sqrt {x^{3} \left (b x +a \right )}}{a \,x^{2}}\)	\(20\)
trager	\(-\frac {2 \sqrt {b \,x^{4}+a \,x^{3}}}{a \,x^{2}}\)	\(22\)
risch	\(-\frac {2 x \left (b x +a \right )}{\sqrt {x^{3} \left (b x +a \right )}\, a}\)	\(23\)
gosper	\(-\frac {2 x \left (b x +a \right )}{a \sqrt {b \,x^{4}+a \,x^{3}}}\)	\(25\)
default	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b \,x^{2}+a x}}{\sqrt {b \,x^{4}+a \,x^{3}}\, a}\)	\(39\)

[In]

int(1/(b*x^4+a*x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*(x^3*(b*x+a))^(1/2)/a/x^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=-\frac {2 \, \sqrt {b x^{4} + a x^{3}}}{a x^{2}} \]

[In]

integrate(1/(b*x^4+a*x^3)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x^4 + a*x^3)/(a*x^2)

Sympy [F]

\[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=\int \frac {1}{\sqrt {a x^{3} + b x^{4}}}\, dx \]

[In]

integrate(1/(b*x**4+a*x**3)**(1/2),x)

[Out]

Integral(1/sqrt(a*x**3 + b*x**4), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a x^{3}}} \,d x } \]

[In]

integrate(1/(b*x^4+a*x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(b*x^4 + a*x^3), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=\frac {2}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/(b*x^4+a*x^3)^(1/2),x, algorithm="giac")

[Out]

2/((sqrt(b)*x - sqrt(b*x^2 + a*x))*sgn(x))

Mupad [B] (verification not implemented)

Time = 9.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx=-\frac {2\,\sqrt {b\,x^4+a\,x^3}}{a\,x^2} \]

[In]

int(1/(a*x^3 + b*x^4)^(1/2),x)

[Out]

-(2*(a*x^3 + b*x^4)^(1/2))/(a*x^2)