3.3.10 \(\int \frac {1}{(a+b x^4)^{9/4} (c+d x^4)^2} \, dx\) [210]

Optimal. Leaf size=266 \[ \frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {3 d^2 (4 b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac {3 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}} \]

[Out]

1/20*b*(5*a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^4+a)^(5/4)+1/20*b*(-5*a^2*d^2-56*a*b*c*d+16*b^2*c^2)*x/a^2/c/(-a*
d+b*c)^3/(b*x^4+a)^(1/4)-1/4*d*x/c/(-a*d+b*c)/(b*x^4+a)^(5/4)/(d*x^4+c)+3/8*d^2*(-a*d+4*b*c)*arctan((-a*d+b*c)
^(1/4)*x/c^(1/4)/(b*x^4+a)^(1/4))/c^(7/4)/(-a*d+b*c)^(13/4)+3/8*d^2*(-a*d+4*b*c)*arctanh((-a*d+b*c)^(1/4)*x/c^
(1/4)/(b*x^4+a)^(1/4))/c^(7/4)/(-a*d+b*c)^(13/4)

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Rubi [A]
time = 0.22, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {425, 541, 12, 385, 218, 214, 211} \begin {gather*} \frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{20 a^2 c \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac {3 d^2 (4 b c-a d) \text {ArcTan}\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)^{13/4}}+\frac {3 d^2 (4 b c-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)^{13/4}}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}+\frac {b x (5 a d+4 b c)}{20 a c \left (a+b x^4\right )^{5/4} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]

[Out]

(b*(4*b*c + 5*a*d)*x)/(20*a*c*(b*c - a*d)^2*(a + b*x^4)^(5/4)) + (b*(16*b^2*c^2 - 56*a*b*c*d - 5*a^2*d^2)*x)/(
20*a^2*c*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c - a*d)*(a + b*x^4)^(5/4)*(c + d*x^4)) + (3*d^2*(4*
b*c - 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)^(13/4)) + (3*d^2*
(4*b*c - 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)^(13/4))

Rule 12

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

Rule 211

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

Rule 214

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 385

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 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

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]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx &=-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\int \frac {4 b c-3 a d-8 b d x^4}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}-\frac {\int \frac {-16 b^2 c^2+40 a b c d-15 a^2 d^2-4 b d (4 b c+5 a d) x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{20 a c (b c-a d)^2}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\int \frac {15 a^2 d^2 (4 b c-a d)}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{20 a^2 c (b c-a d)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\left (3 d^2 (4 b c-a d)\right ) \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\left (3 d^2 (4 b c-a d)\right ) \text {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)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {\left (3 d^2 (4 b c-a d)\right ) \text {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)^3}+\frac {\left (3 d^2 (4 b c-a d)\right ) \text {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)^3}\\ &=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {3 d^2 (4 b c-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)^{13/4}}+\frac {3 d^2 (4 b c-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)^{13/4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.68, size = 357, normalized size = 1.34 \begin {gather*} \frac {\left (\frac {1}{80}+\frac {i}{80}\right ) \left (\frac {(2-2 i) c^{3/4} x \left (5 a^4 d^3+10 a^3 b d^3 x^4-16 b^4 c^2 x^4 \left (c+d x^4\right )+5 a^2 b^2 d \left (12 c^2+12 c d x^4+d^2 x^8\right )+4 a b^3 c \left (-5 c^2+9 c d x^4+14 d^2 x^8\right )\right )}{a^2 (-b c+a d)^3 \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {15 d^2 (4 b c-a d) \tan ^{-1}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}-\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{(b c-a d)^{13/4}}+\frac {15 d^2 (4 b c-a d) \tanh ^{-1}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}+\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{(b c-a d)^{13/4}}\right )}{c^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]

[Out]

((1/80 + I/80)*(((2 - 2*I)*c^(3/4)*x*(5*a^4*d^3 + 10*a^3*b*d^3*x^4 - 16*b^4*c^2*x^4*(c + d*x^4) + 5*a^2*b^2*d*
(12*c^2 + 12*c*d*x^4 + d^2*x^8) + 4*a*b^3*c*(-5*c^2 + 9*c*d*x^4 + 14*d^2*x^8)))/(a^2*(-(b*c) + a*d)^3*(a + b*x
^4)^(5/4)*(c + d*x^4)) + (15*d^2*(4*b*c - a*d)*ArcTan[(((1 - I)*(b*c - a*d)^(1/4)*x^2)/(c^(1/4)*(a + b*x^4)^(1
/4)) - ((1 + I)*c^(1/4)*(a + b*x^4)^(1/4))/(b*c - a*d)^(1/4))/(2*x)])/(b*c - a*d)^(13/4) + (15*d^2*(4*b*c - a*
d)*ArcTanh[(((1 - I)*(b*c - a*d)^(1/4)*x^2)/(c^(1/4)*(a + b*x^4)^(1/4)) + ((1 + I)*c^(1/4)*(a + b*x^4)^(1/4))/
(b*c - a*d)^(1/4))/(2*x)])/(b*c - a*d)^(13/4)))/c^(7/4)

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (d \,x^{4}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x)

[Out]

int(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{4}\right )^{\frac {9}{4}} \left (c + d x^{4}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(9/4)/(d*x**4+c)**2,x)

[Out]

Integral(1/((a + b*x**4)**(9/4)*(c + d*x**4)**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (b\,x^4+a\right )}^{9/4}\,{\left (d\,x^4+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x)

[Out]

int(1/((a + b*x^4)^(9/4)*(c + d*x^4)^2), x)

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