3.209 \(\int \frac{1}{(a+b x^4)^{5/4} (c+d x^4)^2} \, dx\)

Optimal. Leaf size=205 \[ -\frac{d (8 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac{d (8 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}+\frac{b x (a d+4 b c)}{4 a c \sqrt [4]{a+b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]

[Out]

(b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c - a*d)*(a + b*x^4)^(1/4)*(c + d*
x^4)) - (d*(8*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)^(
9/4)) - (d*(8*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)^
(9/4))

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Rubi [A]  time = 0.184153, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {414, 527, 12, 377, 212, 208, 205} \[ -\frac{d (8 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}-\frac{d (8 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{9/4}}+\frac{b x (a d+4 b c)}{4 a c \sqrt [4]{a+b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt [4]{a+b x^4} \left (c+d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c - a*d)*(a + b*x^4)^(1/4)*(c + d*
x^4)) - (d*(8*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)^(
9/4)) - (d*(8*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)^
(9/4))

Rule 414

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[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )^2} \, dx &=-\frac{d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}+\frac{\int \frac{4 b c-3 a d-4 b d x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac{b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac{\int \frac{a d (8 b c-3 a d)}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 a c (b c-a d)^2}\\ &=\frac{b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac{(d (8 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)^2}\\ &=\frac{b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac{(d (8 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)^2}\\ &=\frac{b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac{(d (8 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)^2}-\frac{(d (8 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)^2}\\ &=\frac{b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^4}}-\frac{d x}{4 c (b c-a d) \sqrt [4]{a+b x^4} \left (c+d x^4\right )}-\frac{d (8 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)^{9/4}}-\frac{d (8 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)^{9/4}}\\ \end{align*}

Mathematica [C]  time = 1.37312, size = 625, normalized size = 3.05 \[ \frac{c \left (a+b x^4\right )^{3/4} \left (\frac{320 d^2 x^{20} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{9}{4}\right \},\left \{1,\frac{17}{4}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )}{c^5 \left (a+b x^4\right )^3}+\frac{640 d x^{16} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{9}{4}\right \},\left \{1,\frac{17}{4}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )}{c^4 \left (a+b x^4\right )^3}+\frac{320 x^{12} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{9}{4}\right \},\left \{1,\frac{17}{4}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )}{c^3 \left (a+b x^4\right )^3}+\frac{16380 d^2 x^{12} (a d-b c) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^3 \left (a+b x^4\right )}+\frac{44460 d^2 x^8 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^2}+\frac{7488 d^2 x^{12} (b c-a d)}{c^3 \left (a+b x^4\right )}+\frac{33930 d x^8 (a d-b c) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c^2 \left (a+b x^4\right )}+\frac{14976 d x^8 (b c-a d)}{c^2 \left (a+b x^4\right )}+\frac{94770 d x^4 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c}-\frac{14625 x^4 (b c-a d) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{c \left (a+b x^4\right )}+47385 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+\frac{5148 x^4 (b c-a d)}{c \left (a+b x^4\right )}-\frac{44460 d^2 x^8}{c^2}-\frac{94770 d x^4}{c}-47385\right )}{2340 x^7 \left (c+d x^4\right ) (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(a + b*x^4)^(3/4)*(-47385 - (94770*d*x^4)/c - (44460*d^2*x^8)/c^2 + (5148*(b*c - a*d)*x^4)/(c*(a + b*x^4))
+ (14976*d*(b*c - a*d)*x^8)/(c^2*(a + b*x^4)) + (7488*d^2*(b*c - a*d)*x^12)/(c^3*(a + b*x^4)) + 47385*Hypergeo
metric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + (94770*d*x^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c
- a*d)*x^4)/(c*(a + b*x^4))])/c + (44460*d^2*x^8*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^
4))])/c^2 - (14625*(b*c - a*d)*x^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c*(a +
b*x^4)) + (33930*d*(-(b*c) + a*d)*x^8*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^2*
(a + b*x^4)) + (16380*d^2*(-(b*c) + a*d)*x^12*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))
])/(c^3*(a + b*x^4)) + (320*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 9/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*
(a + b*x^4))])/(c^3*(a + b*x^4)^3) + (640*d*(b*c - a*d)^3*x^16*HypergeometricPFQ[{2, 2, 9/4}, {1, 17/4}, ((b*c
- a*d)*x^4)/(c*(a + b*x^4))])/(c^4*(a + b*x^4)^3) + (320*d^2*(b*c - a*d)^3*x^20*HypergeometricPFQ[{2, 2, 9/4}
, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(c^5*(a + b*x^4)^3)))/(2340*(b*c - a*d)^2*x^7*(c + d*x^4))

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Maple [F]  time = 0.409, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(5/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(5/4)/(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{5}{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)^(5/4)/(d*x^4+c)^2,x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*(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)^(5/4)/(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)**(5/4)/(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{5}{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)^(5/4)/(d*x^4+c)^2,x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*(d*x^4 + c)^2), x)