Optimal. Leaf size=72 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.083009, antiderivative size = 72, 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 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(3/2)*(c + d*x)^(1/5)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.9482, size = 68, normalized size = 0.94 \[ \frac{5 d \sqrt{a + b x} \left (c + d x\right )^{\frac{4}{5}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{4}{5} \\ \frac{9}{5} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{4 \sqrt{\frac{d \left (a + b x\right )}{a d - b c}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/5),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.139928, size = 84, normalized size = 1.17 \[ \frac{(c+d x)^{4/5} \left (3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{4}{5};\frac{9}{5};\frac{b (c+d x)}{b c-a d}\right )-8\right )}{4 \sqrt{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(1/5)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.082, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [5]{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(3/2)/(d*x+c)^(1/5),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt [5]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/5),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)),x, algorithm="giac")
[Out]