3.778 \(\int \frac{1}{x^7 (a+b x^4) (c+d x^4)} \, dx\)

Optimal. Leaf size=112 \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)}-\frac{1}{6 a c x^6} \]

[Out]

-1/(6*a*c*x^6) + (b*c + a*d)/(2*a^2*c^2*x^2) + (b^(5/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*a^(5/2)*(b*c - a*d))
- (d^(5/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d))

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Rubi [A]  time = 0.218099, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {465, 480, 583, 522, 205} \[ \frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)}+\frac{a d+b c}{2 a^2 c^2 x^2}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)}-\frac{1}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^4)*(c + d*x^4)),x]

[Out]

-1/(6*a*c*x^6) + (b*c + a*d)/(2*a^2*c^2*x^2) + (b^(5/2)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*a^(5/2)*(b*c - a*d))
- (d^(5/2)*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*c^(5/2)*(b*c - a*d))

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 480

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

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 522

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

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}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 a c x^6}+\frac{\operatorname{Subst}\left (\int \frac{-3 (b c+a d)-3 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 a c}\\ &=-\frac{1}{6 a c x^6}+\frac{b c+a d}{2 a^2 c^2 x^2}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (b^2 c^2+a b c d+a^2 d^2\right )-3 b d (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )}{6 a^2 c^2}\\ &=-\frac{1}{6 a c x^6}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 a^2 (b c-a d)}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{c+d x^2} \, dx,x,x^2\right )}{2 c^2 (b c-a d)}\\ &=-\frac{1}{6 a c x^6}+\frac{b c+a d}{2 a^2 c^2 x^2}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 a^{5/2} (b c-a d)}-\frac{d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.213169, size = 193, normalized size = 1.72 \[ \frac{-\frac{3 b^2 x^4}{a^2}+\frac{3 b^{5/2} x^6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/2}}+\frac{3 b^{5/2} x^6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{5/2}}+\frac{b}{a}+\frac{3 d^2 x^4}{c^2}-\frac{3 d^{5/2} x^6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/2}}-\frac{3 d^{5/2} x^6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{5/2}}-\frac{d}{c}}{6 x^6 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^4)*(c + d*x^4)),x]

[Out]

(b/a - d/c - (3*b^2*x^4)/a^2 + (3*d^2*x^4)/c^2 + (3*b^(5/2)*x^6*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/
2) + (3*b^(5/2)*x^6*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/2) - (3*d^(5/2)*x^6*ArcTan[1 - (Sqrt[2]*d^(1
/4)*x)/c^(1/4)])/c^(5/2) - (3*d^(5/2)*x^6*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/c^(5/2))/(6*(-(b*c) + a*d)*
x^6)

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Maple [A]  time = 0.012, size = 105, normalized size = 0.9 \begin{align*}{\frac{{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({{x}^{2}d{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}}{2\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{6\,ac{x}^{6}}}+{\frac{d}{2\,{c}^{2}a{x}^{2}}}+{\frac{b}{2\,{a}^{2}c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^4+a)/(d*x^4+c),x)

[Out]

1/2*d^3/c^2/(a*d-b*c)/(c*d)^(1/2)*arctan(x^2*d/(c*d)^(1/2))-1/2*b^3/a^2/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*
b)^(1/2))-1/6/a/c/x^6+1/2/a/c^2/x^2*d+1/2/a^2/c/x^2*b

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Maxima [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/x^7/(b*x^4+a)/(d*x^4+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.4613, size = 1191, normalized size = 10.63 \begin{align*} \left [-\frac{3 \, b^{2} c^{2} x^{6} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} - 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} + 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, \frac{6 \, a^{2} d^{2} x^{6} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - 3 \, b^{2} c^{2} x^{6} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{4} - 2 \, a x^{2} \sqrt{-\frac{b}{a}} - a}{b x^{4} + a}\right ) + 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} - 2 \, a b c^{2} + 2 \, a^{2} c d}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac{6 \, b^{2} c^{2} x^{6} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{4} + 2 \, c x^{2} \sqrt{-\frac{d}{c}} - c}{d x^{4} + c}\right ) - 6 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac{3 \, b^{2} c^{2} x^{6} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b x^{2}}\right ) - 3 \, a^{2} d^{2} x^{6} \sqrt{\frac{d}{c}} \arctan \left (\frac{c \sqrt{\frac{d}{c}}}{d x^{2}}\right ) - 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + a b c^{2} - a^{2} c d}{6 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)/(d*x^4+c),x, algorithm="fricas")

[Out]

[-1/12*(3*b^2*c^2*x^6*sqrt(-b/a)*log((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + 3*a^2*d^2*x^6*sqrt(-d/c)*
log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) - 6*(b^2*c^2 - a^2*d^2)*x^4 + 2*a*b*c^2 - 2*a^2*c*d)/((a^2*b
*c^3 - a^3*c^2*d)*x^6), 1/12*(6*a^2*d^2*x^6*sqrt(d/c)*arctan(c*sqrt(d/c)/(d*x^2)) - 3*b^2*c^2*x^6*sqrt(-b/a)*l
og((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + a)) + 6*(b^2*c^2 - a^2*d^2)*x^4 - 2*a*b*c^2 + 2*a^2*c*d)/((a^2*b*
c^3 - a^3*c^2*d)*x^6), -1/12*(6*b^2*c^2*x^6*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*x^2)) + 3*a^2*d^2*x^6*sqrt(-d/c)*l
og((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) - 6*(b^2*c^2 - a^2*d^2)*x^4 + 2*a*b*c^2 - 2*a^2*c*d)/((a^2*b*
c^3 - a^3*c^2*d)*x^6), -1/6*(3*b^2*c^2*x^6*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*x^2)) - 3*a^2*d^2*x^6*sqrt(d/c)*arc
tan(c*sqrt(d/c)/(d*x^2)) - 3*(b^2*c^2 - a^2*d^2)*x^4 + a*b*c^2 - a^2*c*d)/((a^2*b*c^3 - a^3*c^2*d)*x^6)]

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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(1/x**7/(b*x**4+a)/(d*x**4+c),x)

[Out]

Timed out

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Giac [B]  time = 1.69473, size = 736, normalized size = 6.57 \begin{align*} -\frac{{\left (\sqrt{c d} a^{2} b^{3} c^{3} d x^{4}{\left | d \right |} + \sqrt{c d} a^{3} b^{2} c^{2} d^{2} x^{4}{\left | d \right |} + \sqrt{c d} a^{2} b^{3} c^{4}{\left | d \right |} + \sqrt{c d} a^{3} b^{2} c^{3} d{\left | d \right |} + \sqrt{c d} a^{4} b c^{2} d^{2}{\left | d \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a^{2} b c^{3} + 2 \, a^{3} c^{2} d + \sqrt{-16 \, a^{5} b c^{5} d + 4 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b c^{3} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} c^{2} d^{2}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} +{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} d} + \frac{{\left (\sqrt{a b} a^{2} b^{2} c^{3} d^{2} x^{4}{\left | b \right |} + \sqrt{a b} a^{3} b c^{2} d^{3} x^{4}{\left | b \right |} + \sqrt{a b} a^{2} b^{2} c^{4} d{\left | b \right |} + \sqrt{a b} a^{3} b c^{3} d^{2}{\left | b \right |} + \sqrt{a b} a^{4} c^{2} d^{3}{\left | b \right |}\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, a^{2} b c^{3} + 2 \, a^{3} c^{2} d - \sqrt{-16 \, a^{5} b c^{5} d + 4 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )}^{2}}}{a^{2} b c^{2} d}}}\right )}{a^{2} b^{2} c^{3}{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} + a^{3} b c^{2} d{\left | a^{2} b c^{3} - a^{3} c^{2} d \right |} -{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )}^{2} b} + \frac{3 \, b c x^{4} + 3 \, a d x^{4} - a c}{6 \, a^{2} c^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)/(d*x^4+c),x, algorithm="giac")

[Out]

-(sqrt(c*d)*a^2*b^3*c^3*d*x^4*abs(d) + sqrt(c*d)*a^3*b^2*c^2*d^2*x^4*abs(d) + sqrt(c*d)*a^2*b^3*c^4*abs(d) + s
qrt(c*d)*a^3*b^2*c^3*d*abs(d) + sqrt(c*d)*a^4*b*c^2*d^2*abs(d))*arctan(2*x^2/sqrt((2*a^2*b*c^3 + 2*a^3*c^2*d +
sqrt(-16*a^5*b*c^5*d + 4*(a^2*b*c^3 + a^3*c^2*d)^2))/(a^2*b*c^2*d)))/(a^2*b*c^3*d*abs(a^2*b*c^3 - a^3*c^2*d)
+ a^3*c^2*d^2*abs(a^2*b*c^3 - a^3*c^2*d) + (a^2*b*c^3 - a^3*c^2*d)^2*d) + (sqrt(a*b)*a^2*b^2*c^3*d^2*x^4*abs(b
) + sqrt(a*b)*a^3*b*c^2*d^3*x^4*abs(b) + sqrt(a*b)*a^2*b^2*c^4*d*abs(b) + sqrt(a*b)*a^3*b*c^3*d^2*abs(b) + sqr
t(a*b)*a^4*c^2*d^3*abs(b))*arctan(2*x^2/sqrt((2*a^2*b*c^3 + 2*a^3*c^2*d - sqrt(-16*a^5*b*c^5*d + 4*(a^2*b*c^3
+ a^3*c^2*d)^2))/(a^2*b*c^2*d)))/(a^2*b^2*c^3*abs(a^2*b*c^3 - a^3*c^2*d) + a^3*b*c^2*d*abs(a^2*b*c^3 - a^3*c^2
*d) - (a^2*b*c^3 - a^3*c^2*d)^2*b) + 1/6*(3*b*c*x^4 + 3*a*d*x^4 - a*c)/(a^2*c^2*x^6)