Optimal. Leaf size=59 \[ \frac{4 \sqrt [4]{c x} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{8},\frac{7}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{a c \left (a+b x^2\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0668793, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{4 \sqrt [4]{c x} \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{8},\frac{7}{4};\frac{9}{8};-\frac{b x^2}{a}\right )}{a c \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(3/4)*(a + b*x^2)^(7/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.32128, size = 49, normalized size = 0.83 \[ \frac{4 \sqrt [4]{c x} \sqrt [4]{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{4}, \frac{1}{8} \\ \frac{9}{8} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a^{2} c \sqrt [4]{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(3/4)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.045259, size = 63, normalized size = 1.07 \[ \frac{2 \left (5 x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{8},\frac{3}{4};\frac{9}{8};-\frac{b x^2}{a}\right )+x\right )}{3 a (c x)^{3/4} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(3/4)*(a + b*x^2)^(7/4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{3}{4}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(3/4)/(b*x^2+a)^(7/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (c x\right )^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (c x\right )^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(3/4)/(b*x**2+a)**(7/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (c x\right )^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(7/4)*(c*x)^(3/4)),x, algorithm="giac")
[Out]