3.875 \(\int \frac{\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=169 \[ -\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac{2 \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \]

[Out]

(-2*a^(1/4)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/c^(1/4)
+ (2*b^(1/4)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/d^(1/
4) - (2*a^(1/4)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/c^
(1/4) + (2*b^(1/4)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])
/d^(1/4)

_______________________________________________________________________________________

Rubi [A]  time = 0.25389, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac{2 \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/4)/(x*(c + d*x)^(1/4)),x]

[Out]

(-2*a^(1/4)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/c^(1/4)
+ (2*b^(1/4)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/d^(1/
4) - (2*a^(1/4)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/c^
(1/4) + (2*b^(1/4)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])
/d^(1/4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.8903, size = 162, normalized size = 0.96 \[ - \frac{2 \sqrt [4]{a} \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{\sqrt [4]{c}} - \frac{2 \sqrt [4]{a} \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{\sqrt [4]{c}} - \frac{2 \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{\sqrt [4]{d}} + \frac{2 \sqrt [4]{b} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{\sqrt [4]{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/4)/x/(d*x+c)**(1/4),x)

[Out]

-2*a**(1/4)*atan(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/c**(1/4)
- 2*a**(1/4)*atanh(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/c**(1
/4) - 2*b**(1/4)*atan(b**(1/4)*(c + d*x)**(1/4)/(d**(1/4)*(a + b*x)**(1/4)))/d**
(1/4) + 2*b**(1/4)*atanh(b**(1/4)*(c + d*x)**(1/4)/(d**(1/4)*(a + b*x)**(1/4)))/
d**(1/4)

_______________________________________________________________________________________

Mathematica [C]  time = 0.521569, size = 216, normalized size = 1.28 \[ \frac{36 a (a+b x)^{5/4} (b c-a d) F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )}{5 b x \sqrt [4]{c+d x} \left (9 a (b c-a d) F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )-(a+b x) \left ((4 a d-4 b c) F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )+a d F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^(1/4)/(x*(c + d*x)^(1/4)),x]

[Out]

(36*a*(b*c - a*d)*(a + b*x)^(5/4)*AppellF1[5/4, 1/4, 1, 9/4, (d*(a + b*x))/(-(b*
c) + a*d), 1 + (b*x)/a])/(5*b*x*(c + d*x)^(1/4)*(9*a*(b*c - a*d)*AppellF1[5/4, 1
/4, 1, 9/4, (d*(a + b*x))/(-(b*c) + a*d), 1 + (b*x)/a] - (a + b*x)*((-4*b*c + 4*
a*d)*AppellF1[9/4, 1/4, 2, 13/4, (d*(a + b*x))/(-(b*c) + a*d), 1 + (b*x)/a] + a*
d*AppellF1[9/4, 5/4, 1, 13/4, (d*(a + b*x))/(-(b*c) + a*d), 1 + (b*x)/a])))

_______________________________________________________________________________________

Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/4)/x/(d*x+c)^(1/4),x)

[Out]

int((b*x+a)^(1/4)/x/(d*x+c)^(1/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.315641, size = 490, normalized size = 2.9 \[ 4 \, \left (\frac{a}{c}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (d x + c\right )} \left (\frac{a}{c}\right )^{\frac{1}{4}}}{{\left (d x + c\right )} \sqrt{\frac{{\left (d x + c\right )} \sqrt{\frac{a}{c}} + \sqrt{b x + a} \sqrt{d x + c}}{d x + c}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\right ) - 4 \, \left (\frac{b}{d}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (d x + c\right )} \left (\frac{b}{d}\right )^{\frac{1}{4}}}{{\left (d x + c\right )} \sqrt{\frac{{\left (d x + c\right )} \sqrt{\frac{b}{d}} + \sqrt{b x + a} \sqrt{d x + c}}{d x + c}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\right ) - \left (\frac{a}{c}\right )^{\frac{1}{4}} \log \left (\frac{{\left (d x + c\right )} \left (\frac{a}{c}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) + \left (\frac{a}{c}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (d x + c\right )} \left (\frac{a}{c}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) + \left (\frac{b}{d}\right )^{\frac{1}{4}} \log \left (\frac{{\left (d x + c\right )} \left (\frac{b}{d}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) - \left (\frac{b}{d}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (d x + c\right )} \left (\frac{b}{d}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x),x, algorithm="fricas")

[Out]

4*(a/c)^(1/4)*arctan((d*x + c)*(a/c)^(1/4)/((d*x + c)*sqrt(((d*x + c)*sqrt(a/c)
+ sqrt(b*x + a)*sqrt(d*x + c))/(d*x + c)) + (b*x + a)^(1/4)*(d*x + c)^(3/4))) -
4*(b/d)^(1/4)*arctan((d*x + c)*(b/d)^(1/4)/((d*x + c)*sqrt(((d*x + c)*sqrt(b/d)
+ sqrt(b*x + a)*sqrt(d*x + c))/(d*x + c)) + (b*x + a)^(1/4)*(d*x + c)^(3/4))) -
(a/c)^(1/4)*log(((d*x + c)*(a/c)^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x +
c)) + (a/c)^(1/4)*log(-((d*x + c)*(a/c)^(1/4) - (b*x + a)^(1/4)*(d*x + c)^(3/4)
)/(d*x + c)) + (b/d)^(1/4)*log(((d*x + c)*(b/d)^(1/4) + (b*x + a)^(1/4)*(d*x + c
)^(3/4))/(d*x + c)) - (b/d)^(1/4)*log(-((d*x + c)*(b/d)^(1/4) - (b*x + a)^(1/4)*
(d*x + c)^(3/4))/(d*x + c))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{x \sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/4)/x/(d*x+c)**(1/4),x)

[Out]

Integral((a + b*x)**(1/4)/(x*(c + d*x)**(1/4)), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x),x, algorithm="giac")

[Out]

Timed out