\(\int \frac {1}{(-b x+a^2 x^2)^{5/2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [2835]

Optimal result

Integrand size = 45, antiderivative size = 288 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {-b x+a^2 x^2} \left (-4368 b^5+2688 a^2 b^4 x+5248 a^4 b^3 x^2+12416 a^6 b^2 x^3-148243 a^8 b x^4+121339 a^{10} x^5\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{16380 b^7 x^5 \left (b-a^2 x\right )^2}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (-\frac {-4872 a b^4-9352 a^3 b^3 x-24840 a^5 b^2 x^2-229768 a^7 b x^3+283847 a^9 x^4}{8190 b^7 x^4 \left (b-a^2 x\right )}+\frac {109 a^{15/2} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{4 b^{15/2} x}\right ) \]

[Out]

1/16380*(a^2*x^2-b*x)^(1/2)*(121339*a^10*x^5-148243*a^8*b*x^4+12416*a^6*b^2*x^3+5248*a^4*b^3*x^2+2688*a^2*b^4*
x-4368*b^5)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b^7/x^5/(-a^2*x+b)^2+(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)*(-1/8
190*(283847*a^9*x^4-229768*a^7*b*x^3-24840*a^5*b^2*x^2-9352*a^3*b^3*x-4872*a*b^4)/b^7/x^4/(-a^2*x+b)+109/4*a^(
15/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)*arctan(a^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)/b^(1/2))/b^(15/2)/x)

Rubi [F]

\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]

[In]

Int[1/((-(b*x) + a^2*x^2)^(5/2)*(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*(-b + a^2*x^2)^(5/2)*(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} \left (-b+a^2 x\right )^{5/2} \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 \left (-b+a^2 x^2\right )^{5/2} \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 = 5.37 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (\sqrt {b} \left (-4368 b^5+16 a^5 b^2 x^2 \left (776 a x-3105 \sqrt {x \left (-b+a^2 x\right )}\right )+16 a^3 b^3 x \left (328 a x-1169 \sqrt {x \left (-b+a^2 x\right )}\right )+336 a b^4 \left (8 a x-29 \sqrt {x \left (-b+a^2 x\right )}\right )-a^7 b x^3 \left (148243 a x+459536 \sqrt {x \left (-b+a^2 x\right )}\right )+a^9 x^4 \left (121339 a x+567694 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+446355 a^{15/2} x^2 \left (x \left (-b+a^2 x\right )\right )^{3/2} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{16380 b^{15/2} x^3 \left (x \left (-b+a^2 x\right )\right )^{3/2}} \]

[In]

Integrate[1/((-(b*x) + a^2*x^2)^(5/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]

[Out]

(Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(Sqrt[b]*(-4368*b^5 + 16*a^5*b^2*x^2*(776*a*x - 3105*Sqrt[x*(-b + a^2*x)
]) + 16*a^3*b^3*x*(328*a*x - 1169*Sqrt[x*(-b + a^2*x)]) + 336*a*b^4*(8*a*x - 29*Sqrt[x*(-b + a^2*x)]) - a^7*b*
x^3*(148243*a*x + 459536*Sqrt[x*(-b + a^2*x)]) + a^9*x^4*(121339*a*x + 567694*Sqrt[x*(-b + a^2*x)])) + 446355*
a^(15/2)*x^2*(x*(-b + a^2*x))^(3/2)*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]*ArcTan[(Sqrt[a]*Sqrt[-(a*x) + Sqrt[x*(
-b + a^2*x)]])/Sqrt[b]]))/(16380*b^(15/2)*x^3*(x*(-b + a^2*x))^(3/2))

Maple [F]

\[\int \frac {1}{\left (a^{2} x^{2}-b x \right )^{\frac {5}{2}} \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]

[In]

int(1/(a^2*x^2-b*x)^(5/2)/(a*x^2+x*(a^2*x^2-b*x)^(1/2))^(3/2),x)

[Out]

int(1/(a^2*x^2-b*x)^(5/2)/(a*x^2+x*(a^2*x^2-b*x)^(1/2))^(3/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {446355 \, {\left (a^{11} x^{7} - 2 \, a^{9} b x^{6} + a^{7} b^{2} x^{5}\right )} \sqrt {a} \log \left (\frac {a^{2} x^{2} + 2 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, \sqrt {a^{2} x^{2} - b x} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {a}}{a^{2} x^{2} - b x}\right ) + 2 \, {\left (567694 \, a^{11} x^{6} - 1027230 \, a^{9} b x^{5} + 409856 \, a^{7} b^{2} x^{4} + 30976 \, a^{5} b^{3} x^{3} + 8960 \, a^{3} b^{4} x^{2} + 9744 \, a b^{5} x + {\left (121339 \, a^{10} x^{5} - 148243 \, a^{8} b x^{4} + 12416 \, a^{6} b^{2} x^{3} + 5248 \, a^{4} b^{3} x^{2} + 2688 \, a^{2} b^{4} x - 4368 \, b^{5}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{32760 \, {\left (a^{4} b^{7} x^{7} - 2 \, a^{2} b^{8} x^{6} + b^{9} x^{5}\right )}}, \frac {446355 \, {\left (a^{11} x^{7} - 2 \, a^{9} b x^{6} + a^{7} b^{2} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{a x}\right ) + {\left (567694 \, a^{11} x^{6} - 1027230 \, a^{9} b x^{5} + 409856 \, a^{7} b^{2} x^{4} + 30976 \, a^{5} b^{3} x^{3} + 8960 \, a^{3} b^{4} x^{2} + 9744 \, a b^{5} x + {\left (121339 \, a^{10} x^{5} - 148243 \, a^{8} b x^{4} + 12416 \, a^{6} b^{2} x^{3} + 5248 \, a^{4} b^{3} x^{2} + 2688 \, a^{2} b^{4} x - 4368 \, b^{5}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{16380 \, {\left (a^{4} b^{7} x^{7} - 2 \, a^{2} b^{8} x^{6} + b^{9} x^{5}\right )}}\right ] \]

[In]

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

[Out]

[1/32760*(446355*(a^11*x^7 - 2*a^9*b*x^6 + a^7*b^2*x^5)*sqrt(a)*log((a^2*x^2 + 2*sqrt(a^2*x^2 - b*x)*a*x - b*x
- 2*sqrt(a^2*x^2 - b*x)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(a))/(a^2*x^2 - b*x)) + 2*(567694*a^11*x^6 -
1027230*a^9*b*x^5 + 409856*a^7*b^2*x^4 + 30976*a^5*b^3*x^3 + 8960*a^3*b^4*x^2 + 9744*a*b^5*x + (121339*a^10*x^
5 - 148243*a^8*b*x^4 + 12416*a^6*b^2*x^3 + 5248*a^4*b^3*x^2 + 2688*a^2*b^4*x - 4368*b^5)*sqrt(a^2*x^2 - b*x))*
sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^4*b^7*x^7 - 2*a^2*b^8*x^6 + b^9*x^5), 1/16380*(446355*(a^11*x^7 - 2*a^
9*b*x^6 + a^7*b^2*x^5)*sqrt(-a)*arctan(sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a*x)) + (567694*a^11*x^6
- 1027230*a^9*b*x^5 + 409856*a^7*b^2*x^4 + 30976*a^5*b^3*x^3 + 8960*a^3*b^4*x^2 + 9744*a*b^5*x + (121339*a^10*
x^5 - 148243*a^8*b*x^4 + 12416*a^6*b^2*x^3 + 5248*a^4*b^3*x^2 + 2688*a^2*b^4*x - 4368*b^5)*sqrt(a^2*x^2 - b*x)
)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^4*b^7*x^7 - 2*a^2*b^8*x^6 + b^9*x^5)]

Sympy [F]

\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \left (x \left (a^{2} x - b\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral(1/((x*(a*x + sqrt(a**2*x**2 - b*x)))**(3/2)*(x*(a**2*x - b))**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - b x\right )}^{\frac {5}{2}} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - b x\right )}^{\frac {5}{2}} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{{\left (a^2\,x^2-b\,x\right )}^{5/2}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]

[In]

int(1/((a^2*x^2 - b*x)^(5/2)*(a*x^2 + x*(a^2*x^2 - b*x)^(1/2))^(3/2)),x)

[Out]

int(1/((a^2*x^2 - b*x)^(5/2)*(a*x^2 + x*(a^2*x^2 - b*x)^(1/2))^(3/2)), x)