3.876 \(\int \frac{\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=131 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]

[Out]

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

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Rubi [A]  time = 0.184273, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.7706, size = 114, normalized size = 0.87 \[ - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{c x} + \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{2 a^{\frac{3}{4}} c^{\frac{5}{4}}} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{2 a^{\frac{3}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/4)/x**2/(d*x+c)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.411209, size = 176, normalized size = 1.34 \[ \frac{\frac{2 b d x^2 (b c-a d) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}-(a+b x) (c+d x)}{c x (a+b x)^{3/4} \sqrt [4]{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x]

[Out]

(-((a + b*x)*(c + d*x)) + (2*b*d*(b*c - a*d)*x^2*AppellF1[1, 3/4, 1/4, 2, -(a/(b
*x)), -(c/(d*x))])/(-8*b*d*x*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))] +
b*c*AppellF1[2, 3/4, 5/4, 3, -(a/(b*x)), -(c/(d*x))] + 3*a*d*AppellF1[2, 7/4, 1/
4, 3, -(a/(b*x)), -(c/(d*x))]))/(c*x*(a + b*x)^(3/4)*(c + d*x)^(1/4))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)

[Out]

int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2), x)

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Fricas [A]  time = 0.2557, size = 873, normalized size = 6.66 \[ -\frac{4 \, c x \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (a c d x + a c^{2}\right )} \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac{1}{4}}}{{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} -{\left (d x + c\right )} \sqrt{\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (a^{2} c^{2} d x + a^{2} c^{3}\right )} \sqrt{\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}}}{d x + c}}}\right ) + c x \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} +{\left (a c d x + a c^{2}\right )} \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) - c x \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} -{\left (a c d x + a c^{2}\right )} \left (\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) + 4 \,{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{4 \, c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{x^{2} \sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/4)/x**2/(d*x+c)**(1/4),x)

[Out]

Integral((a + b*x)**(1/4)/(x**2*(c + d*x)**(1/4)), x)

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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^2),x, algorithm="giac")

[Out]

Timed out