3.833 \(\int \frac{1}{x^7 (a+b x^4)^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=208 \[ \frac{\sqrt{c+d x^4} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{12 a^3 c^2 x^2 (b c-a d)}+\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (5 b c-2 a d)}{12 a^2 c x^6 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^6 \left (a+b x^4\right ) (b c-a d)} \]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^4])/(12*a^2*c*(b*c - a*d)*x^6) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*Sqrt[c +
d*x^4])/(12*a^3*c^2*(b*c - a*d)*x^2) + (b*Sqrt[c + d*x^4])/(4*a*(b*c - a*d)*x^6*(a + b*x^4)) + (b^2*(5*b*c -
6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*a^(7/2)*(b*c - a*d)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.328009, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 472, 583, 12, 377, 205} \[ \frac{\sqrt{c+d x^4} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{12 a^3 c^2 x^2 (b c-a d)}+\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^4} (5 b c-2 a d)}{12 a^2 c x^6 (b c-a d)}+\frac{b \sqrt{c+d x^4}}{4 a x^6 \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^4])/(12*a^2*c*(b*c - a*d)*x^6) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*Sqrt[c +
d*x^4])/(12*a^3*c^2*(b*c - a*d)*x^2) + (b*Sqrt[c + d*x^4])/(4*a*(b*c - a*d)*x^6*(a + b*x^4)) + (b^2*(5*b*c -
6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*a^(7/2)*(b*c - a*d)^(3/2))

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 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[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 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 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 )^2 \sqrt{c+d x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^2\right )\\ &=\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-5 b c+2 a d-4 b d x^2}{x^4 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{4 a (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^6}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{-15 b^2 c^2+8 a b c d+4 a^2 d^2-2 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{12 a^2 c (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^6}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^4}}{12 a^3 c^2 (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}-\frac{\operatorname{Subst}\left (\int -\frac{3 b^2 c^2 (5 b c-6 a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{12 a^3 c^2 (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^6}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^4}}{12 a^3 c^2 (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}+\frac{\left (b^2 (5 b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{4 a^3 (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^6}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^4}}{12 a^3 c^2 (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}+\frac{\left (b^2 (5 b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^2}{\sqrt{c+d x^4}}\right )}{4 a^3 (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^4}}{12 a^2 c (b c-a d) x^6}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^4}}{12 a^3 c^2 (b c-a d) x^2}+\frac{b \sqrt{c+d x^4}}{4 a (b c-a d) x^6 \left (a+b x^4\right )}+\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 a^{7/2} (b c-a d)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 5.63878, size = 175, normalized size = 0.84 \[ \frac{a^2 \left (c+d x^4\right ) \left (\frac{3 b^3 x^8}{\left (a+b x^4\right ) (b c-a d)}+\frac{4 x^4 (a d+3 b c)}{c^2}-\frac{2 a}{c}\right )+\frac{3 b^2 x^{12} \sqrt{\frac{d x^4}{c}+1} (5 b c-6 a d) \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{c \left (\frac{x^4 (b c-a d)}{a c}\right )^{3/2}}}{12 a^5 x^6 \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^7*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

(a^2*(c + d*x^4)*((-2*a)/c + (4*(3*b*c + a*d)*x^4)/c^2 + (3*b^3*x^8)/((b*c - a*d)*(a + b*x^4))) + (3*b^2*(5*b*
c - 6*a*d)*x^12*Sqrt[1 + (d*x^4)/c]*ArcSin[Sqrt[(b/a - d/c)*x^4]/Sqrt[1 + (b*x^4)/a]])/(c*(((b*c - a*d)*x^4)/(
a*c))^(3/2)))/(12*a^5*x^6*Sqrt[c + d*x^4])

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 923, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

1/a^3*b*(d*x^4+c)^(1/2)/x^2/c-5/8/a^3*b^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2
)/b*(x^2-(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((x^2-(-a*b)^(1/2)/b)^2*d+2*d*(-a*b)^(1/2)/b*(x^2-(-a*b)^(1/2)
/b)-(a*d-b*c)/b)^(1/2))/(x^2-(-a*b)^(1/2)/b))+5/8/a^3*b^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b
-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((x^2+(-a*b)^(1/2)/b)^2*d-2*d*(-a*b)^(1/2)/b*(
x^2+(-a*b)^(1/2)/b)-(a*d-b*c)/b)^(1/2))/(x^2+(-a*b)^(1/2)/b))-1/8*b^2/a^3/(a*d-b*c)/(x^2+(-a*b)^(1/2)/b)*((x^2
+(-a*b)^(1/2)/b)^2*d-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)-(a*d-b*c)/b)^(1/2)-1/8*b/a^3*d*(-a*b)^(1/2)/(a*d-
b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((
x^2+(-a*b)^(1/2)/b)^2*d-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)-(a*d-b*c)/b)^(1/2))/(x^2+(-a*b)^(1/2)/b))-1/8*
b^2/a^3/(a*d-b*c)/(x^2-(-a*b)^(1/2)/b)*((x^2-(-a*b)^(1/2)/b)^2*d+2*d*(-a*b)^(1/2)/b*(x^2-(-a*b)^(1/2)/b)-(a*d-
b*c)/b)^(1/2)+1/8*b/a^3*d*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x
^2-(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((x^2-(-a*b)^(1/2)/b)^2*d+2*d*(-a*b)^(1/2)/b*(x^2-(-a*b)^(1/2)/b)-(a
*d-b*c)/b)^(1/2))/(x^2-(-a*b)^(1/2)/b))-1/6/a^2*(d*x^4+c)^(1/2)*(-2*d*x^4+c)/x^6/c^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^7), x)

________________________________________________________________________________________

Fricas [A]  time = 2.65047, size = 1547, normalized size = 7.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/48*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^10 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^6)*sqrt(-a*b*c + a^2*d)*log(((
b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 - 4*((b*c - 2*a*d)*x^6 - a*c*x^
2)*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + a^2)) - 4*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d +
4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^8 - 2*a^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2 + 2*(5*a^2*b^3*c^3 - 8*a^3*b^
2*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^4)*sqrt(d*x^4 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^10
+ (a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^6), 1/24*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^10 + (5*a*b^3*c^3
- 6*a^2*b^2*c^2*d)*x^6)*sqrt(a*b*c - a^2*d)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt(a*b*c -
a^2*d)/((a*b*c*d - a^2*d^2)*x^6 + (a*b*c^2 - a^2*c*d)*x^2)) + 2*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*
c*d^2 + 4*a^4*b*d^3)*x^8 - 2*a^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2 + 2*(5*a^2*b^3*c^3 - 8*a^3*b^2*c^2*d +
a^4*b*c*d^2 + 2*a^5*d^3)*x^4)*sqrt(d*x^4 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^10 + (a^5*b^
2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^6)]

________________________________________________________________________________________

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)**2/(d*x**4+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.57454, size = 244, normalized size = 1.17 \begin{align*} \frac{b^{3} c \sqrt{d + \frac{c}{x^{4}}}}{4 \,{\left (a^{3} b c - a^{4} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}} - \frac{{\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{4 \,{\left (a^{3} b c - a^{4} d\right )} \sqrt{a b c - a^{2} d}} + \frac{6 \, a^{3} b c^{5} \sqrt{d + \frac{c}{x^{4}}} - a^{4} c^{4}{\left (d + \frac{c}{x^{4}}\right )}^{\frac{3}{2}} + 3 \, a^{4} c^{4} \sqrt{d + \frac{c}{x^{4}}} d}{6 \, a^{6} c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

1/4*b^3*c*sqrt(d + c/x^4)/((a^3*b*c - a^4*d)*(b*c + a*(d + c/x^4) - a*d)) - 1/4*(5*b^3*c - 6*a*b^2*d)*arctan(a
*sqrt(d + c/x^4)/sqrt(a*b*c - a^2*d))/((a^3*b*c - a^4*d)*sqrt(a*b*c - a^2*d)) + 1/6*(6*a^3*b*c^5*sqrt(d + c/x^
4) - a^4*c^4*(d + c/x^4)^(3/2) + 3*a^4*c^4*sqrt(d + c/x^4)*d)/(a^6*c^6)