3.206 \(\int \frac{(a+b x^4)^{7/4}}{(c+d x^4)^2} \, dx\)

Optimal. Leaf size=230 \[ \frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac{b^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.175304, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {413, 530, 240, 212, 206, 203, 377, 208, 205} \[ \frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}+\frac{b^{7/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{(b c-a d)^{3/4} (3 a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} d^2}-\frac{x \left (a+b x^4\right )^{3/4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(7/4)/(c + d*x^4)^2,x]

[Out]

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

Rule 413

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

Rule 530

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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

Mathematica [C]  time = 0.553527, size = 358, normalized size = 1.56 \[ \frac{\frac{15 a^2 \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{\sqrt [4]{b c-a d}}+\frac{16 b^2 c^{3/4} x^5 \sqrt [4]{\frac{b x^4}{a}+1} F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{d \sqrt [4]{a+b x^4}}-\frac{20 c^{3/4} x \left (a+b x^4\right )^{3/4} (b c-a d)}{d \left (c+d x^4\right )}+\frac{5 a b c \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{d \sqrt [4]{b c-a d}}}{80 c^{7/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^4)^(7/4)/(c + d*x^4)^2,x]

[Out]

((-20*c^(3/4)*(b*c - a*d)*x*(a + b*x^4)^(3/4))/(d*(c + d*x^4)) + (16*b^2*c^(3/4)*x^5*(1 + (b*x^4)/a)^(1/4)*App
ellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/(d*(a + b*x^4)^(1/4)) + (15*a^2*(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)]))/(b*c - a*d)^(1/4) + (5*a*b*c*(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)]))/(d*(b*c - a*d)^(1/4)))/(80*c^(7/4))

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Maple [F]  time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(7/4)/(d*x^4+c)^2,x)

[Out]

int((b*x^4+a)^(7/4)/(d*x^4+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{7}{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)^(7/4)/(d*x^4+c)^2,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(7/4)/(d*x^4 + c)^2, x)

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Fricas [B]  time = 6.94258, size = 3656, normalized size = 15.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(7/4)/(d*x^4+c)^2,x, algorithm="fricas")

[Out]

-1/16*(4*(b*x^4 + a)^(3/4)*(b*c - a*d)*x - 4*(c*d^2*x^4 + c^2*d)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3
*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*arct
an(-(c^2*d^2*x*sqrt(((256*b^7*c^10*d^4 - 672*a^2*b^5*c^8*d^6 - 112*a^3*b^4*c^7*d^7 + 609*a^4*b^3*c^6*d^8 + 189
*a^5*b^2*c^5*d^9 - 189*a^6*b*c^4*d^10 - 81*a^7*c^3*d^11)*x^2*sqrt((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3
*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8)) + (4096*b^
10*c^10 + 2048*a*b^9*c^9*d - 14592*a^2*b^8*c^8*d^2 - 9472*a^3*b^7*c^7*d^3 + 18928*a^4*b^6*c^6*d^4 + 15624*a^5*
b^5*c^5*d^5 - 9639*a^6*b^4*c^4*d^6 - 11124*a^7*b^3*c^3*d^7 + 486*a^8*b^2*c^2*d^8 + 2916*a^9*b*c*d^9 + 729*a^10
*d^10)*sqrt(b*x^4 + a))/x^2)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 +
189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4) - (64*b^5*c^7*d^2 + 16*a*b^4*c^6*d^3 - 1
16*a^2*b^3*c^5*d^4 - 45*a^3*b^2*c^4*d^5 + 54*a^4*b*c^3*d^6 + 27*a^5*c^2*d^7)*(b*x^4 + a)^(1/4)*((256*b^7*c^7 -
672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*
a^7*d^7)/(c^7*d^8))^(1/4))/((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 1
89*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)*x)) - 16*(c*d^2*x^4 + c^2*d)*(b^7/d^8)^(1/4)*arctan(-((b*x^
4 + a)^(1/4)*b^5*d^2*(b^7/d^8)^(1/4) - d^2*x*(b^7/d^8)^(1/4)*sqrt((b^7*d^4*x^2*sqrt(b^7/d^8) + sqrt(b*x^4 + a)
*b^10)/x^2))/(b^7*x)) + (c*d^2*x^4 + c^2*d)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^
4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)*log((c^5*d^6*x*((256*b^7*
c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6
- 81*a^7*d^7)/(c^7*d^8))^(3/4) + (64*b^5*c^5 + 16*a*b^4*c^4*d - 116*a^2*b^3*c^3*d^2 - 45*a^3*b^2*c^2*d^3 + 54*
a^4*b*c*d^4 + 27*a^5*d^5)*(b*x^4 + a)^(1/4))/x) - (c*d^2*x^4 + c^2*d)*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 11
2*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(1/4)
*log(-(c^5*d^6*x*((256*b^7*c^7 - 672*a^2*b^5*c^5*d^2 - 112*a^3*b^4*c^4*d^3 + 609*a^4*b^3*c^3*d^4 + 189*a^5*b^2
*c^2*d^5 - 189*a^6*b*c*d^6 - 81*a^7*d^7)/(c^7*d^8))^(3/4) - (64*b^5*c^5 + 16*a*b^4*c^4*d - 116*a^2*b^3*c^3*d^2
- 45*a^3*b^2*c^2*d^3 + 54*a^4*b*c*d^4 + 27*a^5*d^5)*(b*x^4 + a)^(1/4))/x) - 4*(c*d^2*x^4 + c^2*d)*(b^7/d^8)^(
1/4)*log((d^6*x*(b^7/d^8)^(3/4) + (b*x^4 + a)^(1/4)*b^5)/x) + 4*(c*d^2*x^4 + c^2*d)*(b^7/d^8)^(1/4)*log(-(d^6*
x*(b^7/d^8)^(3/4) - (b*x^4 + a)^(1/4)*b^5)/x))/(c*d^2*x^4 + c^2*d)

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Sympy [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)**(7/4)/(d*x**4+c)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{7}{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)^(7/4)/(d*x^4+c)^2,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(7/4)/(d*x^4 + c)^2, x)