3.861 \(\int \frac {1}{x^4 (a+b x^6) \sqrt {c+d x^6}} \, dx\)

Optimal. Leaf size=80 \[ -\frac {b \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^6}}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {465, 480, 12, 377, 205} \[ -\frac {b \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^6}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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 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]

Rubi steps

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

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Mathematica [C]  time = 1.17, size = 179, normalized size = 2.24 \[ -\frac {\left (\frac {d x^6}{c}+1\right ) \left (\frac {4 x^6 \left (c+d x^6\right ) (b c-a d) \, _2F_1\left (2,2;\frac {5}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )}{3 c^2 \left (a+b x^6\right )}+\frac {\left (c+2 d x^6\right ) \sin ^{-1}\left (\sqrt {\frac {x^6 (b c-a d)}{c \left (a+b x^6\right )}}\right )}{c \sqrt {\frac {a x^6 \left (c+d x^6\right ) (b c-a d)}{c^2 \left (a+b x^6\right )^2}}}\right )}{3 x^3 \left (a+b x^6\right ) \sqrt {c+d x^6}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

-1/3*((1 + (d*x^6)/c)*(((c + 2*d*x^6)*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]])/(c*Sqrt[(a*(b*c - a*d)*
x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)]) + (4*(b*c - a*d)*x^6*(c + d*x^6)*Hypergeometric2F1[2, 2, 5/2, ((b*c - a
*d)*x^6)/(c*(a + b*x^6))])/(3*c^2*(a + b*x^6))))/(x^3*(a + b*x^6)*Sqrt[c + d*x^6])

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fricas [B]  time = 1.08, size = 332, normalized size = 4.15 \[ \left [-\frac {\sqrt {-a b c + a^{2} d} b c x^{3} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 4 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )}}{12 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}}, -\frac {\sqrt {a b c - a^{2} d} b c x^{3} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right ) + 2 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )}}{6 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/12*(sqrt(-a*b*c + a^2*d)*b*c*x^3*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*sqrt(d*x^6 + c)*(a*b*c - a^2*d))/((a^2*b*c^2 - a^3*c*d)*x^3), -1/6*(sqrt(a*b*c - a^2*d)*b*c*x^3*arcta
n(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*sqrt(d*x^6 + c)*(a*b*c - a^2*d))/((a^2*b*c^2 - a^3*c*d)*x^3)]

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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^4/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}\, x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^6)*(c + d*x^6)^(1/2)),x)

[Out]

int(1/(x^4*(a + b*x^6)*(c + d*x^6)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(1/(x**4*(a + b*x**6)*sqrt(c + d*x**6)), x)

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