Optimal. Leaf size=279 \[ -\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a x+b x^3}}+\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a x+b x^3}}-\frac{21 a x \left (a+b x^2\right )}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{7 x \sqrt{a x+b x^3}}{5 b^2}-\frac{x^4}{b \sqrt{a x+b x^3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.533806, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a x+b x^3}}+\frac{21 a^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a x+b x^3}}-\frac{21 a x \left (a+b x^2\right )}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{7 x \sqrt{a x+b x^3}}{5 b^2}-\frac{x^4}{b \sqrt{a x+b x^3}} \]
Antiderivative was successfully verified.
[In] Int[x^6/(a*x + b*x^3)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 51.415, size = 260, normalized size = 0.93 \[ \frac{21 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{11}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{21 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{11}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{21 a \sqrt{a x + b x^{3}}}{5 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{x^{4}}{b \sqrt{a x + b x^{3}}} + \frac{7 x \sqrt{a x + b x^{3}}}{5 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(b*x**3+a*x)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.178064, size = 173, normalized size = 0.62 \[ \frac{x \left (21 a^{3/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-21 a^{3/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (7 a+2 b x^2\right )\right )}{5 b^{5/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(a*x + b*x^3)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.028, size = 200, normalized size = 0.7 \[{\frac{a{x}^{2}}{{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{2}+{\frac{a}{b}} \right ) xb}}}}+{\frac{2\,x}{5\,{b}^{2}}\sqrt{b{x}^{3}+ax}}-{\frac{21\,a}{10\,{b}^{3}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(b*x^3+a*x)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^3 + a*x)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^3 + a*x)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(b*x**3+a*x)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^3 + a*x)^(3/2),x, algorithm="giac")
[Out]