3.879 \(\int \frac {1}{x^{10} (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\)

Optimal. Leaf size=208 \[ \frac {b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{18 a^2 c x^9 (b c-a d)}+\frac {\sqrt {c+d x^6} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{18 a^3 c^2 x^3 (b c-a d)}+\frac {b \sqrt {c+d x^6}}{6 a x^9 \left (a+b x^6\right ) (b c-a d)} \]

[Out]

1/6*b^2*(-6*a*d+5*b*c)*arctan(x^3*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^6+c)^(1/2))/a^(7/2)/(-a*d+b*c)^(3/2)-1/18*(-2*
a*d+5*b*c)*(d*x^6+c)^(1/2)/a^2/c/(-a*d+b*c)/x^9+1/18*(-4*a^2*d^2-8*a*b*c*d+15*b^2*c^2)*(d*x^6+c)^(1/2)/a^3/c^2
/(-a*d+b*c)/x^3+1/6*b*(d*x^6+c)^(1/2)/a/(-a*d+b*c)/x^9/(b*x^6+a)

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Rubi [A]  time = 0.31, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 472, 583, 12, 377, 205} \[ \frac {\sqrt {c+d x^6} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{18 a^3 c^2 x^3 (b c-a d)}+\frac {b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{18 a^2 c x^9 (b c-a d)}+\frac {b \sqrt {c+d x^6}}{6 a x^9 \left (a+b x^6\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^10*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^6])/(18*a^2*c*(b*c - a*d)*x^9) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*Sqrt[c +
d*x^6])/(18*a^3*c^2*(b*c - a*d)*x^3) + (b*Sqrt[c + d*x^6])/(6*a*(b*c - a*d)*x^9*(a + b*x^6)) + (b^2*(5*b*c -
6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(6*a^(7/2)*(b*c - a*d)^(3/2))

Rule 12

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

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 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]

Rubi steps

\begin {align*} \int \frac {1}{x^{10} \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^3\right )\\ &=\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\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^3\right )}{6 a (b c-a d)}\\ &=-\frac {(5 b c-2 a d) \sqrt {c+d x^6}}{18 a^2 c (b c-a d) x^9}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\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^3\right )}{18 a^2 c (b c-a d)}\\ &=-\frac {(5 b c-2 a d) \sqrt {c+d x^6}}{18 a^2 c (b c-a d) x^9}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^6}}{18 a^3 c^2 (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\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^3\right )}{18 a^3 c^2 (b c-a d)}\\ &=-\frac {(5 b c-2 a d) \sqrt {c+d x^6}}{18 a^2 c (b c-a d) x^9}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^6}}{18 a^3 c^2 (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\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^3\right )}{6 a^3 (b c-a d)}\\ &=-\frac {(5 b c-2 a d) \sqrt {c+d x^6}}{18 a^2 c (b c-a d) x^9}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^6}}{18 a^3 c^2 (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\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^3}{\sqrt {c+d x^6}}\right )}{6 a^3 (b c-a d)}\\ &=-\frac {(5 b c-2 a d) \sqrt {c+d x^6}}{18 a^2 c (b c-a d) x^9}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^6}}{18 a^3 c^2 (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\right )}+\frac {b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{7/2} (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 5.86, size = 175, normalized size = 0.84 \[ \frac {a^2 \left (c+d x^6\right ) \left (\frac {3 b^3 x^{12}}{\left (a+b x^6\right ) (b c-a d)}+\frac {4 x^6 (a d+3 b c)}{c^2}-\frac {2 a}{c}\right )+\frac {3 b^2 x^{18} \sqrt {\frac {d x^6}{c}+1} (5 b c-6 a d) \sin ^{-1}\left (\frac {\sqrt {x^6 \left (\frac {b}{a}-\frac {d}{c}\right )}}{\sqrt {\frac {b x^6}{a}+1}}\right )}{c \left (\frac {x^6 (b c-a d)}{a c}\right )^{3/2}}}{18 a^5 x^9 \sqrt {c+d x^6}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^10*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(a^2*(c + d*x^6)*((-2*a)/c + (4*(3*b*c + a*d)*x^6)/c^2 + (3*b^3*x^12)/((b*c - a*d)*(a + b*x^6))) + (3*b^2*(5*b
*c - 6*a*d)*x^18*Sqrt[1 + (d*x^6)/c]*ArcSin[Sqrt[(b/a - d/c)*x^6]/Sqrt[1 + (b*x^6)/a]])/(c*(((b*c - a*d)*x^6)/
(a*c))^(3/2)))/(18*a^5*x^9*Sqrt[c + d*x^6])

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fricas [A]  time = 1.37, size = 760, normalized size = 3.65 \[ \left [-\frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{15} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{9}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, {\left ({\left (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}\right )} x^{12} - 2 \, a^{3} b^{2} c^{3} + 4 \, a^{4} b c^{2} d - 2 \, a^{5} c d^{2} + 2 \, {\left (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}\right )} x^{6}\right )} \sqrt {d x^{6} + c}}{72 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{15} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{9}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{15} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{9}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right ) + 2 \, {\left ({\left (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}\right )} x^{12} - 2 \, a^{3} b^{2} c^{3} + 4 \, a^{4} b c^{2} d - 2 \, a^{5} c d^{2} + 2 \, {\left (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}\right )} x^{6}\right )} \sqrt {d x^{6} + c}}{36 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{15} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{9}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/72*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^15 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^9)*sqrt(-a*b*c + a^2*d)*log(((
b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 - 4*((b*c - 2*a*d)*x^9 - a*c*x
^3)*sqrt(d*x^6 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6 + 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^12 - 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^6)*sqrt(d*x^6 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x
^15 + (a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^9), 1/36*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^15 + (5*a*b^3*c
^3 - 6*a^2*b^2*c^2*d)*x^9)*sqrt(a*b*c - a^2*d)*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a*b*c
- a^2*d)/((a*b*c*d - a^2*d^2)*x^9 + (a*b*c^2 - a^2*c*d)*x^3)) + 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^12 - 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^6)*sqrt(d*x^6 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^15 + (a^
5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^9)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}\, x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^10/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^10), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^{10}\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(a + b*x^6)^2*(c + d*x^6)^(1/2)),x)

[Out]

int(1/(x^10*(a + b*x^6)^2*(c + d*x^6)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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