3.887 \(\int \frac {x^{23}}{(a+b x^8) \sqrt {c+d x^8}} \, dx\)

Optimal. Leaf size=104 \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac {\left (c+d x^8\right )^{3/2}}{12 b d^2} \]

[Out]

1/12*(d*x^8+c)^(3/2)/b/d^2-1/4*a^2*arctanh(b^(1/2)*(d*x^8+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(5/2)/(-a*d+b*c)^(1/2)-
1/4*(a*d+b*c)*(d*x^8+c)^(1/2)/b^2/d^2

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {446, 88, 63, 208} \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^8} (a d+b c)}{4 b^2 d^2}+\frac {\left (c+d x^8\right )^{3/2}}{12 b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^23/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 208

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

Rule 446

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

Rubi steps

\begin {align*} \int \frac {x^{23}}{\left (a+b x^8\right ) \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \left (\frac {-b c-a d}{b^2 d \sqrt {c+d x}}+\frac {a^2}{b^2 (a+b x) \sqrt {c+d x}}+\frac {\sqrt {c+d x}}{b d}\right ) \, dx,x,x^8\right )\\ &=-\frac {(b c+a d) \sqrt {c+d x^8}}{4 b^2 d^2}+\frac {\left (c+d x^8\right )^{3/2}}{12 b d^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{8 b^2}\\ &=-\frac {(b c+a d) \sqrt {c+d x^8}}{4 b^2 d^2}+\frac {\left (c+d x^8\right )^{3/2}}{12 b d^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^8}\right )}{4 b^2 d}\\ &=-\frac {(b c+a d) \sqrt {c+d x^8}}{4 b^2 d^2}+\frac {\left (c+d x^8\right )^{3/2}}{12 b d^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.22, size = 91, normalized size = 0.88 \[ \frac {\sqrt {c+d x^8} \left (-3 a d-2 b c+b d x^8\right )}{12 b^2 d^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^23/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.22, size = 288, normalized size = 2.77 \[ \left [\frac {3 \, \sqrt {b^{2} c - a b d} a^{2} d^{2} \log \left (\frac {b d x^{8} + 2 \, b c - a d - 2 \, \sqrt {d x^{8} + c} \sqrt {b^{2} c - a b d}}{b x^{8} + a}\right ) + 2 \, {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{8} - 2 \, b^{3} c^{2} - a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \sqrt {d x^{8} + c}}{24 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac {3 \, \sqrt {-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{8} + c} \sqrt {-b^{2} c + a b d}}{b d x^{8} + b c}\right ) + {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} x^{8} - 2 \, b^{3} c^{2} - a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \sqrt {d x^{8} + c}}{12 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="fricas")

[Out]

[1/24*(3*sqrt(b^2*c - a*b*d)*a^2*d^2*log((b*d*x^8 + 2*b*c - a*d - 2*sqrt(d*x^8 + c)*sqrt(b^2*c - a*b*d))/(b*x^
8 + a)) + 2*((b^3*c*d - a*b^2*d^2)*x^8 - 2*b^3*c^2 - a*b^2*c*d + 3*a^2*b*d^2)*sqrt(d*x^8 + c))/(b^4*c*d^2 - a*
b^3*d^3), 1/12*(3*sqrt(-b^2*c + a*b*d)*a^2*d^2*arctan(sqrt(d*x^8 + c)*sqrt(-b^2*c + a*b*d)/(b*d*x^8 + b*c)) +
((b^3*c*d - a*b^2*d^2)*x^8 - 2*b^3*c^2 - a*b^2*c*d + 3*a^2*b*d^2)*sqrt(d*x^8 + c))/(b^4*c*d^2 - a*b^3*d^3)]

________________________________________________________________________________________

giac [A]  time = 0.16, size = 106, normalized size = 1.02 \[ \frac {a^{2} \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{8} + c\right )}^{\frac {3}{2}} b^{2} d^{4} - 3 \, \sqrt {d x^{8} + c} b^{2} c d^{4} - 3 \, \sqrt {d x^{8} + c} a b d^{5}}{12 \, b^{3} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

1/4*a^2*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2) + 1/12*((d*x^8 + c)^(3/2)*b^
2*d^4 - 3*sqrt(d*x^8 + c)*b^2*c*d^4 - 3*sqrt(d*x^8 + c)*a*b*d^5)/(b^3*d^6)

________________________________________________________________________________________

maple [F]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {x^{23}}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c positive or negative?

________________________________________________________________________________________

mupad [B]  time = 4.68, size = 103, normalized size = 0.99 \[ \frac {{\left (d\,x^8+c\right )}^{3/2}}{12\,b\,d^2}-\left (\frac {c}{2\,b\,d^2}+\frac {4\,a\,d^3-4\,b\,c\,d^2}{16\,b^2\,d^4}\right )\,\sqrt {d\,x^8+c}+\frac {a^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^8+c}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{5/2}\,\sqrt {a\,d-b\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^23/((a + b*x^8)*(c + d*x^8)^(1/2)),x)

[Out]

(c + d*x^8)^(3/2)/(12*b*d^2) - (c/(2*b*d^2) + (4*a*d^3 - 4*b*c*d^2)/(16*b^2*d^4))*(c + d*x^8)^(1/2) + (a^2*ata
n((b^(1/2)*(c + d*x^8)^(1/2))/(a*d - b*c)^(1/2)))/(4*b^(5/2)*(a*d - b*c)^(1/2))

________________________________________________________________________________________

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(x**23/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________