3.1719 \(\int \frac{\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=108 \[ \frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.110975, antiderivative size = 108, 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 \[ \frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}+\frac{2 \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}-\frac{4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(1/4)/(c + d*x)^(5/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.3012, size = 100, normalized size = 0.93 \[ \frac{2 \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{d^{\frac{5}{4}}} + \frac{2 \sqrt [4]{b} \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{d^{\frac{5}{4}}} - \frac{4 \sqrt [4]{a + b x}}{d \sqrt [4]{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/4)/(d*x+c)**(5/4),x)

[Out]

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

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Mathematica [C]  time = 0.164692, size = 89, normalized size = 0.82 \[ \frac{4 \left (b (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )-3 d (a+b x)\right )}{3 d^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(1/4)/(c + d*x)^(5/4),x]

[Out]

(4*(-3*d*(a + b*x) + b*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*(c + d*x)*Hypergeome
tric2F1[3/4, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(3*d^2*(a + b*x)^(3/4)*(c +
d*x)^(1/4))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{bx+a} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/4)/(d*x+c)^(5/4),x)

[Out]

int((b*x+a)^(1/4)/(d*x+c)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/(d*x + c)^(5/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/(d*x + c)^(5/4), x)

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Fricas [A]  time = 0.231707, size = 346, normalized size = 3.2 \[ -\frac{4 \,{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}}}{{\left (d x + c\right )} \sqrt{\frac{{\left (d^{3} x + c d^{2}\right )} \sqrt{\frac{b}{d^{5}}} + \sqrt{b x + a} \sqrt{d x + c}}{d x + c}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\right ) -{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} +{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) +{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (d^{2} x + c d\right )} \left (\frac{b}{d^{5}}\right )^{\frac{1}{4}} -{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d x + c}\right ) + 4 \,{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{d^{2} x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/4)/(d*x + c)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{a + b x}}{\left (c + d x\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/4)/(d*x+c)**(5/4),x)

[Out]

Integral((a + b*x)**(1/4)/(c + d*x)**(5/4), 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)^(5/4),x, algorithm="giac")

[Out]

Timed out