Optimal. Leaf size=111 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.167426, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d}-\frac{8 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} d^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/(c + d*x)^(3/4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.1155, size = 168, normalized size = 1.51 \[ \frac{4 \sqrt{a + b x} \sqrt [4]{c + d x}}{3 d} + \frac{4 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{5}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{b} d^{2} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/(d*x+c)**(3/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.161622, size = 77, normalized size = 0.69 \[ \frac{4 \sqrt{a+b x} \sqrt [4]{c+d x} \left (\frac{2 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt{\frac{d (a+b x)}{a d-b c}}}+1\right )}{3 d} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/(c + d*x)^(3/4),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)/(d*x+c)^(3/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/(d*x + c)^(3/4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/(d*x + c)^(3/4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x}}{\left (c + d x\right )^{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/2)/(d*x+c)**(3/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/(d*x + c)^(3/4),x, algorithm="giac")
[Out]