3.1731 \(\int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\)

Optimal. Leaf size=193 \[ -\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{a} \]

[Out]

(Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/a - (Sqrt[2]*Arc
Tan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/a - Log[1 + Sqrt[1 - a*x]/Sq
rt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) + Log[1 + S
qrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]
*a)

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Rubi [A]  time = 0.220637, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - a*x)^(1/4)*(1 + a*x)^(3/4)),x]

[Out]

(Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/a - (Sqrt[2]*Arc
Tan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/a - Log[1 + Sqrt[1 - a*x]/Sq
rt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) + Log[1 + S
qrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]
*a)

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Rubi in Sympy [A]  time = 22.1406, size = 165, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{- a x + 1}}{\sqrt [4]{a x + 1}} + \frac{\sqrt{- a x + 1}}{\sqrt{a x + 1}} + 1 \right )}}{2 a} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{- a x + 1}}{\sqrt [4]{a x + 1}} + \frac{\sqrt{- a x + 1}}{\sqrt{a x + 1}} + 1 \right )}}{2 a} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- a x + 1}}{\sqrt [4]{a x + 1}} - 1 \right )}}{a} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- a x + 1}}{\sqrt [4]{a x + 1}} + 1 \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-a*x+1)**(1/4)/(a*x+1)**(3/4),x)

[Out]

-sqrt(2)*log(-sqrt(2)*(-a*x + 1)**(1/4)/(a*x + 1)**(1/4) + sqrt(-a*x + 1)/sqrt(a
*x + 1) + 1)/(2*a) + sqrt(2)*log(sqrt(2)*(-a*x + 1)**(1/4)/(a*x + 1)**(1/4) + sq
rt(-a*x + 1)/sqrt(a*x + 1) + 1)/(2*a) - sqrt(2)*atan(sqrt(2)*(-a*x + 1)**(1/4)/(
a*x + 1)**(1/4) - 1)/a - sqrt(2)*atan(sqrt(2)*(-a*x + 1)**(1/4)/(a*x + 1)**(1/4)
+ 1)/a

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Mathematica [C]  time = 0.0257151, size = 38, normalized size = 0.2 \[ \frac{2\ 2^{3/4} \sqrt [4]{a x+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (a x+1)\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - a*x)^(1/4)*(1 + a*x)^(3/4)),x]

[Out]

(2*2^(3/4)*(1 + a*x)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (1 + a*x)/2])/a

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [4]{-ax+1}}} \left ( ax+1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x)

[Out]

int(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + 1\right )}^{\frac{3}{4}}{\left (-a x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)),x, algorithm="maxima")

[Out]

integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)), x)

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Fricas [A]  time = 0.25399, size = 608, normalized size = 3.15 \[ 2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{2} x - a\right )} \frac{1}{a^{4}}^{\frac{1}{4}}}{\sqrt{2}{\left (a^{2} x - a\right )} \frac{1}{a^{4}}^{\frac{1}{4}} + 2 \,{\left (a x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}} + 2 \,{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}}}\right ) + 2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (a^{2} x - a\right )} \frac{1}{a^{4}}^{\frac{1}{4}}}{\sqrt{2}{\left (a^{2} x - a\right )} \frac{1}{a^{4}}^{\frac{1}{4}} - 2 \,{\left (a x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}} - 2 \,{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}}}\right ) - \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}\right ) + \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)),x, algorithm="fricas")

[Out]

2*sqrt(2)*(a^(-4))^(1/4)*arctan(sqrt(2)*(a^2*x - a)*(a^(-4))^(1/4)/(sqrt(2)*(a^2
*x - a)*(a^(-4))^(1/4) + 2*(a*x - 1)*sqrt((sqrt(2)*(a*x + 1)^(1/4)*(-a*x + 1)^(3
/4)*a*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(a*x + 1)*sqrt(-a*x + 1)
)/(a*x - 1)) + 2*(a*x + 1)^(1/4)*(-a*x + 1)^(3/4))) + 2*sqrt(2)*(a^(-4))^(1/4)*a
rctan(-sqrt(2)*(a^2*x - a)*(a^(-4))^(1/4)/(sqrt(2)*(a^2*x - a)*(a^(-4))^(1/4) -
2*(a*x - 1)*sqrt(-(sqrt(2)*(a*x + 1)^(1/4)*(-a*x + 1)^(3/4)*a*(a^(-4))^(1/4) - (
a^3*x - a^2)*sqrt(a^(-4)) + sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*x - 1)) - 2*(a*x +
1)^(1/4)*(-a*x + 1)^(3/4))) - 1/2*sqrt(2)*(a^(-4))^(1/4)*log((sqrt(2)*(a*x + 1)^
(1/4)*(-a*x + 1)^(3/4)*a*(a^(-4))^(1/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(a*x
+ 1)*sqrt(-a*x + 1))/(a*x - 1)) + 1/2*sqrt(2)*(a^(-4))^(1/4)*log(-(sqrt(2)*(a*x
+ 1)^(1/4)*(-a*x + 1)^(3/4)*a*(a^(-4))^(1/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt
(a*x + 1)*sqrt(-a*x + 1))/(a*x - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{- a x + 1} \left (a x + 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-a*x+1)**(1/4)/(a*x+1)**(3/4),x)

[Out]

Integral(1/((-a*x + 1)**(1/4)*(a*x + 1)**(3/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + 1\right )}^{\frac{3}{4}}{\left (-a x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)),x, algorithm="giac")

[Out]

integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)), x)