\(\int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx\) [300]

Optimal result

Integrand size = 21, antiderivative size = 36 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \]

[Out]

2/3*arctanh(x^(5/2)*b^(1/2)/(b*x^5+a*x^2)^(1/2))/b^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2054, 212} \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \]

[In]

Int[x^(3/2)/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a*x^2 + b*x^5]])/(3*Sqrt[b])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{5/2}}{\sqrt {a x^2+b x^5}}\right ) \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 x \sqrt {a+b x^3} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{3 \sqrt {b} \sqrt {x^2 \left (a+b x^3\right )}} \]

[In]

Integrate[x^(3/2)/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*x*Sqrt[a + b*x^3]*Log[Sqrt[b]*x^(3/2) + Sqrt[a + b*x^3]])/(3*Sqrt[b]*Sqrt[x^2*(a + b*x^3)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(26)=52\).

Time = 1.78 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {2 x^{\frac {3}{2}} \left (b \,x^{3}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {b \,x^{5}+a \,x^{2}}\, \sqrt {x \left (b \,x^{3}+a \right )}\, \sqrt {b}}\)	\(59\)

[In]

int(x^(3/2)/(b*x^5+a*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3/(b*x^5+a*x^2)^(1/2)*x^(3/2)*(b*x^3+a)/(x*(b*x^3+a))^(1/2)/b^(1/2)*arctanh(1/x^2*(x*(b*x^3+a))^(1/2)/b^(1/2
))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.81 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} + a\right )} \sqrt {b} \sqrt {x} - a^{2}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {-b} \sqrt {x}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \]

[In]

integrate(x^(3/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/6*log(-8*b^2*x^6 - 8*a*b*x^3 - 4*sqrt(b*x^5 + a*x^2)*(2*b*x^3 + a)*sqrt(b)*sqrt(x) - a^2)/sqrt(b), -1/3*sqr
t(-b)*arctan(2*sqrt(b*x^5 + a*x^2)*sqrt(-b)*sqrt(x)/(2*b*x^3 + a))/b]

Sympy [F]

\[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{\frac {3}{2}}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]

[In]

integrate(x**(3/2)/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(x**(3/2)/sqrt(x**2*(a + b*x**3)), x)

Maxima [F]

\[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {x^{\frac {3}{2}}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \]

[In]

integrate(x^(3/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(3/2)/sqrt(b*x^5 + a*x^2), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{3 \, \sqrt {b}} - \frac {2 \, \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} + \sqrt {b x^{3} + a} \right |}\right )}{3 \, \sqrt {b} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^(3/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

1/3*log(abs(a))*sgn(x)/sqrt(b) - 2/3*log(abs(-sqrt(b)*x^(3/2) + sqrt(b*x^3 + a)))/(sqrt(b)*sgn(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{3/2}}{\sqrt {b\,x^5+a\,x^2}} \,d x \]

[In]

int(x^(3/2)/(a*x^2 + b*x^5)^(1/2),x)

[Out]

int(x^(3/2)/(a*x^2 + b*x^5)^(1/2), x)