Optimal. Leaf size=90 \[ \frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{3 b^3 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{10 \sqrt{2} c^2 \sqrt [4]{b x+c x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0779072, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{(b+2 c x) \left (b x+c x^2\right )^{3/4}}{5 c}-\frac{3 b^3 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{10 \sqrt{2} c^2 \sqrt [4]{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(3/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.028, size = 80, normalized size = 0.89 \[ - \frac{3 \sqrt{2} b^{3} \sqrt [4]{\frac{c \left (- b x - c x^{2}\right )}{b^{2}}} E\left (\frac{\operatorname{asin}{\left (1 + \frac{2 c x}{b} \right )}}{2}\middle | 2\right )}{20 c^{2} \sqrt [4]{b x + c x^{2}}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{4}}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(3/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0639374, size = 70, normalized size = 0.78 \[ \frac{x \left (b^2 \left (-\sqrt [4]{\frac{c x}{b}+1}\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{c x}{b}\right )+b^2+3 b c x+2 c^2 x^2\right )}{5 c \sqrt [4]{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(3/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.056, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(3/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac{3}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x + c x^{2}\right )^{\frac{3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(3/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/4),x, algorithm="giac")
[Out]