3.108 \(\int \frac{\left (a+b x^4\right )^{3/4}}{\left (c+d x^4\right )^2} \, dx\)

Optimal. Leaf size=135 \[ \frac{3 a \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} \sqrt [4]{b c-a d}}+\frac{3 a \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} \sqrt [4]{b c-a d}}+\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )} \]

[Out]

(x*(a + b*x^4)^(3/4))/(4*c*(c + d*x^4)) + (3*a*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(
1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(1/4)) + (3*a*ArcTanh[((b*c - a
*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(1/4))

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Rubi [A]  time = 0.203699, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{3 a \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} \sqrt [4]{b c-a d}}+\frac{3 a \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} \sqrt [4]{b c-a d}}+\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/(c + d*x^4)^2,x]

[Out]

(x*(a + b*x^4)^(3/4))/(4*c*(c + d*x^4)) + (3*a*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(
1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(1/4)) + (3*a*ArcTanh[((b*c - a
*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(8*c^(7/4)*(b*c - a*d)^(1/4))

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Rubi in Sympy [A]  time = 31.2364, size = 119, normalized size = 0.88 \[ \frac{3 a \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \sqrt [4]{- a d + b c}} + \frac{3 a \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{8 c^{\frac{7}{4}} \sqrt [4]{- a d + b c}} + \frac{x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 c \left (c + d x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/(d*x**4+c)**2,x)

[Out]

3*a*atan(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(8*c**(7/4)*(-a*d
+ b*c)**(1/4)) + 3*a*atanh(x*(-a*d + b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4))
)/(8*c**(7/4)*(-a*d + b*c)**(1/4)) + x*(a + b*x**4)**(3/4)/(4*c*(c + d*x**4))

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Mathematica [A]  time = 0.350861, size = 152, normalized size = 1.13 \[ \frac{3 a \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{16 c^{7/4} \sqrt [4]{b c-a d}}+\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^4)^(3/4)/(c + d*x^4)^2,x]

[Out]

(x*(a + b*x^4)^(3/4))/(4*c*(c + d*x^4)) + (3*a*(2*ArcTan[((b*c - a*d)^(1/4)*x)/(
c^(1/4)*(b + a*x^4)^(1/4))] - Log[c^(1/4) - ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1
/4)] + Log[c^(1/4) + ((b*c - a*d)^(1/4)*x)/(b + a*x^4)^(1/4)]))/(16*c^(7/4)*(b*c
- a*d)^(1/4))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/(d*x^4+c)^2,x)

[Out]

int((b*x^4+a)^(3/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/(d*x^4 + c)^2,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/(d*x^4 + c)^2, x)

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Fricas [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^4 + a)^(3/4)/(d*x^4 + c)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [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**4+a)**(3/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{{\left (d x^{4} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/(d*x^4 + c)^2,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/(d*x^4 + c)^2, x)