Optimal. Leaf size=266 \[ \frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.363655, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {107, 106, 490, 1218} \[ \frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 107
Rule 106
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt [4]{e+f x}} \, dx &=\frac{\sqrt{-\frac{f (c+d x)}{d e-c f}} \int \frac{1}{(a+b x) \sqrt [4]{e+f x} \sqrt{-\frac{c f}{d e-c f}-\frac{d f x}{d e-c f}}} \, dx}{\sqrt{c+d x}}\\ &=-\frac{\left (4 \sqrt{-\frac{f (c+d x)}{d e-c f}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (b e-a f-b x^4\right ) \sqrt{\frac{d e}{d e-c f}-\frac{c f}{d e-c f}-\frac{d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt{c+d x}}\\ &=-\frac{\left (2 \sqrt{-\frac{f (c+d x)}{d e-c f}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{b e-a f}-\sqrt{b} x^2\right ) \sqrt{\frac{d e}{d e-c f}-\frac{c f}{d e-c f}-\frac{d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt{b} \sqrt{c+d x}}+\frac{\left (2 \sqrt{-\frac{f (c+d x)}{d e-c f}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{b e-a f}+\sqrt{b} x^2\right ) \sqrt{\frac{d e}{d e-c f}-\frac{c f}{d e-c f}-\frac{d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt{b} \sqrt{c+d x}}\\ &=\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{b e-a f} \sqrt{c+d x}}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{b e-a f} \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 0.113706, size = 118, normalized size = 0.44 \[ -\frac{4 \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt [4]{\frac{b (e+f x)}{f (a+b x)}} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{4};\frac{7}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{3 b \sqrt{c+d x} \sqrt [4]{e+f x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt [4]{fx+e}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )} \sqrt{d x + c}{\left (f x + e\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right ) \sqrt{c + d x} \sqrt [4]{e + f x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )} \sqrt{d x + c}{\left (f x + e\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]