\(\int \frac {x}{\sqrt {-b x+a^2 x^2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [1250]

Optimal result

Integrand size = 46, antiderivative size = 90 \[ \int \frac {x}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {20 a \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{3 b^2 x}-\frac {4 \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{3 b^2 x^2} \]

[Out]

20/3*a*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b^2/x-4/3*(a^2*x^2-b*x)^(1/2)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b
^2/x^2

Rubi [F]

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

[In]

Int[x/(Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]

[Out]

(2*Sqrt[x]*Sqrt[-b + a^2*x]*Defer[Subst][Defer[Int][x^2/(Sqrt[-b + a^2*x^2]*(a*x^4 + x^2*Sqrt[-(b*x^2) + a^2*x
^4])^(3/2)), x], x, Sqrt[x]])/Sqrt[-(b*x) + a^2*x^2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-b+a^2 x}\right ) \int \frac {\sqrt {x}}{\sqrt {-b+a^2 x} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx}{\sqrt {-b x+a^2 x^2}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-b+a^2 x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-b+a^2 x^2} \left (a x^4+x^2 \sqrt {-b x^2+a^2 x^4}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.00 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (b+a \left (-a x+5 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{3 b^2 x \sqrt {x \left (-b+a^2 x\right )}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x}{\sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.59 \[ \int \frac {x}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \, \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} {\left (5 \, a x - \sqrt {a^{2} x^{2} - b x}\right )}}{3 \, b^{2} x^{2}} \]

[In]

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

[Out]

4/3*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*(5*a*x - sqrt(a^2*x^2 - b*x))/(b^2*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {x}{\sqrt {a^2\,x^2-b\,x}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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