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

Optimal. Leaf size=127 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{3/4} d^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{3/4} d^{5/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{d} \]

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4))/d - ((b*c - a*d)*ArcTan[(d^(1/4)*(a + b*x)^(1/
4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(3/4)*d^(5/4)) - ((b*c - a*d)*ArcTanh[(d^(1
/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(3/4)*d^(5/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.134333, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{3/4} d^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{3/4} d^{5/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{d} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4))/d - ((b*c - a*d)*ArcTan[(d^(1/4)*(a + b*x)^(1/
4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(3/4)*d^(5/4)) - ((b*c - a*d)*ArcTanh[(d^(1
/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(3/4)*d^(5/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.8803, size = 112, normalized size = 0.88 \[ \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{d} - \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{2 b^{\frac{3}{4}} d^{\frac{5}{4}}} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{2 b^{\frac{3}{4}} d^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(1/4)*(c + d*x)**(3/4)/d - (a*d - b*c)*atan(b**(1/4)*(c + d*x)**(1/4)
/(d**(1/4)*(a + b*x)**(1/4)))/(2*b**(3/4)*d**(5/4)) + (a*d - b*c)*atanh(b**(1/4)
*(c + d*x)**(1/4)/(d**(1/4)*(a + b*x)**(1/4)))/(2*b**(3/4)*d**(5/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.17498, size = 76, normalized size = 0.6 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac{\, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [4]{\frac{d (a+b x)}{a d-b c}}}+3\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4)*(3 + Hypergeometric2F1[3/4, 3/4, 7/4, (b*(c + d
*x))/(b*c - a*d)]/((d*(a + b*x))/(-(b*c) + a*d))^(1/4)))/(3*d)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.252218, size = 865, normalized size = 6.81 \[ -\frac{4 \, d \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b d^{2} x + b c d\right )} \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}\right )^{\frac{1}{4}}}{{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} -{\left (d x + c\right )} \sqrt{\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (b^{2} d^{3} x + b^{2} c d^{2}\right )} \sqrt{\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}}}{d x + c}}}\right ) + d \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} +{\left (b d^{2} x + b c d\right )} \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) - d \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} -{\left (b d^{2} x + b c d\right )} \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{b^{3} d^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) - 4 \,{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^
4)/(b^3*d^5))^(1/4)*arctan(-(b*d^2*x + b*c*d)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*
b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b^3*d^5))^(1/4)/((b*c - a*d)*(b*x + a)^(
1/4)*(d*x + c)^(3/4) - (d*x + c)*sqrt(((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(b*x
+ a)*sqrt(d*x + c) + (b^2*d^3*x + b^2*c*d^2)*sqrt((b^4*c^4 - 4*a*b^3*c^3*d + 6*a
^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b^3*d^5)))/(d*x + c)))) + d*((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b^3*d^5))^(1/4)
*log(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b*d^2*x + b*c*d)*((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b^3*d^5))^(1/4)
)/(d*x + c)) - d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)/(b^3*d^5))^(1/4)*log(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (
b*d^2*x + b*c*d)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)/(b^3*d^5))^(1/4))/(d*x + c)) - 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/d

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(1/4)/(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, algorithm="giac")

[Out]

Timed out