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

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))

________________________________________________________________________________________

Rubi [A]  time = 0.0762624, 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, Rules used = {378, 377, 212, 208, 205} \[ \frac{3 a \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} \sqrt [4]{b c-a d}}+\frac{3 a \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} \sqrt [4]{b c-a d}}+\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )} \]

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))

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && 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{\left (a+b x^4\right )^{3/4}}{\left (c+d x^4\right )^2} \, dx &=\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )}+\frac{(3 a) \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c}\\ &=\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )}+\frac{(3 a) \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}\\ &=\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )}+\frac{(3 a) \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}}+\frac{(3 a) \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}}\\ &=\frac{x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right )}+\frac{3 a \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} \sqrt [4]{b c-a d}}+\frac{3 a \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} \sqrt [4]{b c-a d}}\\ \end{align*}

Mathematica [C]  time = 0.0329985, size = 78, normalized size = 0.58 \[ \frac{x \left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{(a d-b c) x^4}{a \left (d x^4+c\right )}\right )}{c^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \sqrt [4]{\frac{d x^4}{c}+1}} \]

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

________________________________________________________________________________________

Maple [F]  time = 0.41, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

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]

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

________________________________________________________________________________________

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

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)