3.208 \(\int \frac{1}{\sqrt [4]{a+b x^4} (c+d x^4)^2} \, dx\)

Optimal. Leaf size=162 \[ \frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

[Out]

-(d*x*(a + b*x^4)^(3/4))/(4*c*(b*c - a*d)*(c + d*x^4)) + ((4*b*c - 3*a*d)*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)^(5/4)) + ((4*b*c - 3*a*d)*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)^(5/4))

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Rubi [A]  time = 0.108515, antiderivative size = 162, 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, Rules used = {382, 377, 212, 208, 205} \[ \frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*x*(a + b*x^4)^(3/4))/(4*c*(b*c - a*d)*(c + d*x^4)) + ((4*b*c - 3*a*d)*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)^(5/4)) + ((4*b*c - 3*a*d)*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)^(5/4))

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{4 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c}-\sqrt{b c-a d} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)}+\frac{(4 b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c}+\sqrt{b c-a d} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)}\\ &=-\frac{d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac{(4 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac{(4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}\\ \end{align*}

Mathematica [C]  time = 0.105946, size = 99, normalized size = 0.61 \[ \frac{x \left (\left (c+d x^4\right ) (4 b c-3 a d) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-c d \left (a+b x^4\right )\right )}{4 c^2 \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.41, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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