3.6.70 \(\int \frac {x^3}{(a+b x^8) \sqrt {c+d x^8}} \, dx\)

Optimal. Leaf size=54 \[ \frac {\tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 \sqrt {a} \sqrt {b c-a d}} \]

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Rubi [A]  time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {465, 377, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {x^4 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^8}}\right )}{4 \sqrt {a} \sqrt {b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])]/(4*Sqrt[a]*Sqrt[b*c - a*d])

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]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 95, normalized size = 1.76 \begin {gather*} \frac {x^4 \sqrt {\frac {d x^8}{c}+1} \tanh ^{-1}\left (\frac {\sqrt {\frac {d x^8}{c}-\frac {b x^8}{a}}}{\sqrt {\frac {d x^8}{c}+1}}\right )}{4 a \sqrt {c+d x^8} \sqrt {\frac {d x^8}{c}-\frac {b x^8}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

(x^4*Sqrt[1 + (d*x^8)/c]*ArcTanh[Sqrt[-((b*x^8)/a) + (d*x^8)/c]/Sqrt[1 + (d*x^8)/c]])/(4*a*Sqrt[c + d*x^8]*Sqr
t[-((b*x^8)/a) + (d*x^8)/c])

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IntegrateAlgebraic [A]  time = 0.49, size = 106, normalized size = 1.96 \begin {gather*} \frac {\tan ^{-1}\left (\frac {b \sqrt {d} x^8}{\sqrt {a} \sqrt {b c-a d}}+\frac {b x^4 \sqrt {c+d x^8}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{4 \sqrt {a} \sqrt {b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

ArcTan[(Sqrt[a]*Sqrt[d])/Sqrt[b*c - a*d] + (b*Sqrt[d]*x^8)/(Sqrt[a]*Sqrt[b*c - a*d]) + (b*x^4*Sqrt[c + d*x^8])
/(Sqrt[a]*Sqrt[b*c - a*d])]/(4*Sqrt[a]*Sqrt[b*c - a*d])

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fricas [B]  time = 0.49, size = 245, normalized size = 4.54 \begin {gather*} \left [-\frac {\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^{16} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{12} - a c x^{4}\right )} \sqrt {d x^{8} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{16 \, {\left (a b c - a^{2} d\right )}}, \frac {\arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{8} - a c\right )} \sqrt {d x^{8} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{12} + {\left (a b c^{2} - a^{2} c d\right )} x^{4}\right )}}\right )}{8 \, \sqrt {a b c - a^{2} d}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*
c^2 - 4*((b*c - 2*a*d)*x^12 - a*c*x^4)*sqrt(d*x^8 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8 + a^2))/(a*
b*c - a^2*d), 1/8*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)*sqrt(d*x^8 + c)*sqrt(a*b*c - a^2*d)/((a*b*c*d - a^2*d^2
)*x^12 + (a*b*c^2 - a^2*c*d)*x^4))/sqrt(a*b*c - a^2*d)]

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giac [A]  time = 0.20, size = 72, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x^{4} - \sqrt {d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{4 \, \sqrt {a b c d - a^{2} d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (b \,x^{8}+a \right ) \sqrt {d \,x^{8}+c}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (b x^{8} + a\right )} \sqrt {d x^{8} + c}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^3/((b*x^8 + a)*sqrt(d*x^8 + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^3}{\left (b\,x^8+a\right )\,\sqrt {d\,x^8+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x^8)*(c + d*x^8)^(1/2)),x)

[Out]

int(x^3/((a + b*x^8)*(c + d*x^8)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b x^{8}\right ) \sqrt {c + d x^{8}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Integral(x**3/((a + b*x**8)*sqrt(c + d*x**8)), x)

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