3.197 \(\int \frac{1}{(a+b x^4)^{9/4} (c+d x^4)} \, dx\)

Optimal. Leaf size=180 \[ \frac{b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]

[Out]

(b*x)/(5*a*(b*c - a*d)*(a + b*x^4)^(5/4)) + (b*(4*b*c - 9*a*d)*x)/(5*a^2*(b*c - a*d)^2*(a + b*x^4)^(1/4)) + (d
^2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(9/4)) + (d^2*ArcTanh[((b
*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(9/4))

________________________________________________________________________________________

Rubi [A]  time = 0.202568, antiderivative size = 180, 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} \[ \frac{b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]

[Out]

(b*x)/(5*a*(b*c - a*d)*(a + b*x^4)^(5/4)) + (b*(4*b*c - 9*a*d)*x)/(5*a^2*(b*c - a*d)^2*(a + b*x^4)^(1/4)) + (d
^2*ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(b*c - a*d)^(9/4)) + (d^2*ArcTanh[((b
*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*(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 )^{9/4} \left (c+d x^4\right )} \, dx &=\frac{b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}-\frac{\int \frac{-4 b c+5 a d-4 b d x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{5 a (b c-a d)}\\ &=\frac{b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac{b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac{\int \frac{5 a^2 d^2}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{5 a^2 (b c-a d)^2}\\ &=\frac{b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac{b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac{d^2 \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{(b c-a d)^2}\\ &=\frac{b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac{b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{(b c-a d)^2}\\ &=\frac{b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac{b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac{d^2 \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 )}{2 \sqrt{c} (b c-a d)^2}+\frac{d^2 \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 )}{2 \sqrt{c} (b c-a d)^2}\\ &=\frac{b x}{5 a (b c-a d) \left (a+b x^4\right )^{5/4}}+\frac{b (4 b c-9 a d) x}{5 a^2 (b c-a d)^2 \sqrt [4]{a+b x^4}}+\frac{d^2 \tan ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}\\ \end{align*}

Mathematica [C]  time = 1.66171, size = 621, normalized size = 3.45 \[ \frac{80 c^2 x^{12} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{13}{4}\right \},\left \{1,\frac{17}{4}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )+80 d^2 x^{20} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{13}{4}\right \},\left \{1,\frac{17}{4}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )+160 c d x^{16} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{13}{4}\right \},\left \{1,\frac{17}{4}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )+2080 c^3 d^2 x^8 \left (a+b x^4\right )^3 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-416 c^2 d^2 x^{12} \left (a+b x^4\right )^2 (b c-a d)-2080 c^3 d^2 x^8 \left (a+b x^4\right )^3+280 c^2 x^{12} (b c-a d)^3 \, _2F_1\left (2,\frac{13}{4};\frac{17}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+4680 c^4 d x^4 \left (a+b x^4\right )^3 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+2925 c^5 \left (a+b x^4\right )^3 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )-936 c^3 d x^8 \left (a+b x^4\right )^2 (b c-a d)-4680 c^4 d x^4 \left (a+b x^4\right )^3-585 c^4 x^4 \left (a+b x^4\right )^2 (b c-a d)-2925 c^5 \left (a+b x^4\right )^3+240 d^2 x^{20} (b c-a d)^3 \, _2F_1\left (2,\frac{13}{4};\frac{17}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )+520 c d x^{16} (b c-a d)^3 \, _2F_1\left (2,\frac{13}{4};\frac{17}{4};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{325 c^4 x^7 \left (a+b x^4\right )^{13/4} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]

[Out]

(-585*c^4*(b*c - a*d)*x^4*(a + b*x^4)^2 - 936*c^3*d*(b*c - a*d)*x^8*(a + b*x^4)^2 - 416*c^2*d^2*(b*c - a*d)*x^
12*(a + b*x^4)^2 - 2925*c^5*(a + b*x^4)^3 - 4680*c^4*d*x^4*(a + b*x^4)^3 - 2080*c^3*d^2*x^8*(a + b*x^4)^3 + 29
25*c^5*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 4680*c^4*d*x^4*(a + b
*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 2080*c^3*d^2*x^8*(a + b*x^4)^3*Hyp
ergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 280*c^2*(b*c - a*d)^3*x^12*Hypergeometric2F1[
2, 13/4, 17/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 520*c*d*(b*c - a*d)^3*x^16*Hypergeometric2F1[2, 13/4, 17/4
, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 240*d^2*(b*c - a*d)^3*x^20*Hypergeometric2F1[2, 13/4, 17/4, ((b*c - a*d
)*x^4)/(c*(a + b*x^4))] + 80*c^2*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 13/4}, {1, 17/4}, ((b*c - a*d)*x^
4)/(c*(a + b*x^4))] + 160*c*d*(b*c - a*d)^3*x^16*HypergeometricPFQ[{2, 2, 13/4}, {1, 17/4}, ((b*c - a*d)*x^4)/
(c*(a + b*x^4))] + 80*d^2*(b*c - a*d)^3*x^20*HypergeometricPFQ[{2, 2, 13/4}, {1, 17/4}, ((b*c - a*d)*x^4)/(c*(
a + b*x^4))])/(325*c^4*(b*c - a*d)^2*x^7*(a + b*x^4)^(13/4))

________________________________________________________________________________________

Maple [F]  time = 0.41, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{9}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(9/4)/(d*x^4+c),x)

[Out]

int(1/(b*x^4+a)^(9/4)/(d*x^4+c),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)), 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(1/(b*x^4+a)^(9/4)/(d*x^4+c),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{4}\right )^{\frac{9}{4}} \left (c + d x^{4}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(9/4)/(d*x**4+c),x)

[Out]

Integral(1/((a + b*x**4)**(9/4)*(c + d*x**4)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)), x)