\(\int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [2131]

Optimal result

Integrand size = 48, antiderivative size = 154 \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=-\frac {4 \sqrt {-b x+a^2 x^2} \left (3003 b^3-4032 a^2 b^2 x-5120 a^4 b x^2-8192 a^6 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{45045 b^5 x^5}+\frac {4 \left (6699 a b^3+448 a^3 b^2 x+1024 a^5 b x^2+8192 a^7 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{45045 b^5 x^4} \]

[Out]

-4/45045*(a^2*x^2-b*x)^(1/2)*(-8192*a^6*x^3-5120*a^4*b*x^2-4032*a^2*b^2*x+3003*b^3)*(x*(a*x+(a^2*x^2-b*x)^(1/2
)))^(1/2)/b^5/x^5+4/45045*(8192*a^7*x^3+1024*a^5*b*x^2+448*a^3*b^2*x+6699*a*b^3)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))
^(1/2)/b^5/x^4

Rubi [F]

\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]

[In]

Int[1/(x^2*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][1/(x^4*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 {1}{x^{5/2} \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 {1}{x^4 \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 = 3.54 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (3003 b^4+1024 a^5 b x^2 \left (-3 a x+\sqrt {x \left (-b+a^2 x\right )}\right )+8192 a^7 x^3 \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )+64 a^3 b^2 x \left (-17 a x+7 \sqrt {x \left (-b+a^2 x\right )}\right )+21 a b^3 \left (-335 a x+319 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{45045 b^5 x^4 \sqrt {x \left (-b+a^2 x\right )}} \]

[In]

Integrate[1/(x^2*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)])]*(3003*b^4 + 1024*a^5*b*x^2*(-3*a*x + Sqrt[x*(-b + a^2*x)]) + 8192*a^7*
x^3*(a*x + Sqrt[x*(-b + a^2*x)]) + 64*a^3*b^2*x*(-17*a*x + 7*Sqrt[x*(-b + a^2*x)]) + 21*a*b^3*(-335*a*x + 319*
Sqrt[x*(-b + a^2*x)])))/(45045*b^5*x^4*Sqrt[x*(-b + a^2*x)])

Maple [F]

\[\int \frac {1}{x^{2} \sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]

[In]

int(1/x^2/(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(1/x^2/(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 = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \, {\left (8192 \, a^{7} x^{4} + 1024 \, a^{5} b x^{3} + 448 \, a^{3} b^{2} x^{2} + 6699 \, a b^{3} x + {\left (8192 \, a^{6} x^{3} + 5120 \, a^{4} b x^{2} + 4032 \, a^{2} b^{2} x - 3003 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{45045 \, b^{5} x^{5}} \]

[In]

integrate(1/x^2/(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]

4/45045*(8192*a^7*x^4 + 1024*a^5*b*x^3 + 448*a^3*b^2*x^2 + 6699*a*b^3*x + (8192*a^6*x^3 + 5120*a^4*b*x^2 + 403
2*a^2*b^2*x - 3003*b^3)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)/(b^5*x^5)

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \sqrt {x \left (a^{2} x - b\right )}}\, dx \]

[In]

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

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

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

Giac [F]

\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^2\,\sqrt {a^2\,x^2-b\,x}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]

[In]

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