Optimal. Leaf size=220 \[ \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (32 a^4 x^2-88 a^2 b x+115 b^2\right )}{40 b^2 x}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {9 \sqrt {b} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a} x}+\frac {-32 a^5 x^2+104 a^3 b x-145 a b^2}{40 b^2}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 4.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {-b x+a^2 x^2} \int \frac {x^{3/2} \left (-b+a^2 x\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx}{\sqrt {x} \sqrt {-b+a^2 x}}\\ &=\frac {\left (2 \sqrt {-b x+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-b+a^2 x^2\right )^{3/2}}{\left (a x^4+x^2 \sqrt {-b x^2+a^2 x^4}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a^2 x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 7.44, size = 220, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (32 a^4 x^2-88 a^2 b x+115 b^2\right )}{40 b^2 x}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {9 \sqrt {b} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a} x}+\frac {-32 a^5 x^2+104 a^3 b x-145 a b^2}{40 b^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 367, normalized size = 1.67 \begin {gather*} \left [\frac {45 \, \sqrt {2} \sqrt {a} b^{3} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (32 \, a^{6} x^{3} - 104 \, a^{4} b x^{2} + 145 \, a^{2} b^{2} x - {\left (32 \, a^{5} x^{2} - 88 \, a^{3} b x + 115 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{160 \, a b^{2} x}, \frac {45 \, \sqrt {2} \sqrt {-a} b^{3} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (32 \, a^{6} x^{3} - 104 \, a^{4} b x^{2} + 145 \, a^{2} b^{2} x - {\left (32 \, a^{5} x^{2} - 88 \, a^{3} b x + 115 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{80 \, a b^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}}}{{\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^{2} x^{2}-b x \right )^{\frac {3}{2}}}{\left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}}}{{\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a^2\,x^2-b\,x\right )}^{3/2}}{{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (a^{2} x - b\right )\right )^{\frac {3}{2}}}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________