3.21.55 \(\int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} (-2 f+e x^8)} \, dx\)

Optimal. Leaf size=147 \[ \frac {\text {RootSum}\left [-2 \text {$\#$1}^8 f+4 \text {$\#$1}^4 a f-2 a^2 f+b^2 e\& ,\frac {-2 \text {$\#$1}^4 d \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 d \log (x)+b c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+2 a d \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-2 a d \log (x)-b c \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ]}{16 f} \]

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Rubi [B]  time = 1.15, antiderivative size = 459, normalized size of antiderivative = 3.12, number of steps used = 10, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {6725, 377, 212, 208, 205} \begin {gather*} -\frac {\left (c \sqrt {f}+\sqrt {2} d \sqrt {e}\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}-\frac {\left (\sqrt {2} d \sqrt {e}-c \sqrt {f}\right ) \tan ^{-1}\left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}}-\frac {\left (c \sqrt {f}+\sqrt {2} d \sqrt {e}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}-\frac {\left (\sqrt {2} d \sqrt {e}-c \sqrt {f}\right ) \tanh ^{-1}\left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*d + c*x^4)/((-b + a*x^4)^(1/4)*(-2*f + e*x^8)),x]

[Out]

-1/4*((Sqrt[2]*d*Sqrt[e] + c*Sqrt[f])*ArcTan[((-(b*Sqrt[e]) + Sqrt[2]*a*Sqrt[f])^(1/4)*x)/(2^(1/8)*f^(1/8)*(-b
+ a*x^4)^(1/4))])/(2^(3/8)*Sqrt[e]*(-(b*Sqrt[e]) + Sqrt[2]*a*Sqrt[f])^(1/4)*f^(7/8)) - ((Sqrt[2]*d*Sqrt[e] -
c*Sqrt[f])*ArcTan[((b*Sqrt[e] + Sqrt[2]*a*Sqrt[f])^(1/4)*x)/(2^(1/8)*f^(1/8)*(-b + a*x^4)^(1/4))])/(4*2^(3/8)*
Sqrt[e]*(b*Sqrt[e] + Sqrt[2]*a*Sqrt[f])^(1/4)*f^(7/8)) - ((Sqrt[2]*d*Sqrt[e] + c*Sqrt[f])*ArcTanh[((-(b*Sqrt[e
]) + Sqrt[2]*a*Sqrt[f])^(1/4)*x)/(2^(1/8)*f^(1/8)*(-b + a*x^4)^(1/4))])/(4*2^(3/8)*Sqrt[e]*(-(b*Sqrt[e]) + Sqr
t[2]*a*Sqrt[f])^(1/4)*f^(7/8)) - ((Sqrt[2]*d*Sqrt[e] - c*Sqrt[f])*ArcTanh[((b*Sqrt[e] + Sqrt[2]*a*Sqrt[f])^(1/
4)*x)/(2^(1/8)*f^(1/8)*(-b + a*x^4)^(1/4))])/(4*2^(3/8)*Sqrt[e]*(b*Sqrt[e] + Sqrt[2]*a*Sqrt[f])^(1/4)*f^(7/8))

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 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [F]  time = 0.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2*d + c*x^4)/((-b + a*x^4)^(1/4)*(-2*f + e*x^8)),x]

[Out]

Integrate[(2*d + c*x^4)/((-b + a*x^4)^(1/4)*(-2*f + e*x^8)), x]

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IntegrateAlgebraic [A]  time = 2.22, size = 146, normalized size = 0.99 \begin {gather*} \frac {\text {RootSum}\left [b^2 e-2 a^2 f+4 a f \text {$\#$1}^4-2 f \text {$\#$1}^8\&,\frac {b c \log (x)+2 a d \log (x)-b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 a d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 d \log (x) \text {$\#$1}^4+2 d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ]}{16 f} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2*d + c*x^4)/((-b + a*x^4)^(1/4)*(-2*f + e*x^8)),x]

[Out]

RootSum[b^2*e - 2*a^2*f + 4*a*f*#1^4 - 2*f*#1^8 & , (b*c*Log[x] + 2*a*d*Log[x] - b*c*Log[(-b + a*x^4)^(1/4) -
x*#1] - 2*a*d*Log[(-b + a*x^4)^(1/4) - x*#1] - 2*d*Log[x]*#1^4 + 2*d*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a
*#1) + #1^5) & ]/(16*f)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+2*d)/(a*x^4-b)^(1/4)/(e*x^8-2*f),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + 2 \, d}{{\left (e x^{8} - 2 \, f\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+2*d)/(a*x^4-b)^(1/4)/(e*x^8-2*f),x, algorithm="giac")

[Out]

integrate((c*x^4 + 2*d)/((e*x^8 - 2*f)*(a*x^4 - b)^(1/4)), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{4}+2 d}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (e \,x^{8}-2 f \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+2*d)/(a*x^4-b)^(1/4)/(e*x^8-2*f),x)

[Out]

int((c*x^4+2*d)/(a*x^4-b)^(1/4)/(e*x^8-2*f),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + 2 \, d}{{\left (e x^{8} - 2 \, f\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+2*d)/(a*x^4-b)^(1/4)/(e*x^8-2*f),x, algorithm="maxima")

[Out]

integrate((c*x^4 + 2*d)/((e*x^8 - 2*f)*(a*x^4 - b)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {c\,x^4+2\,d}{{\left (a\,x^4-b\right )}^{1/4}\,\left (2\,f-e\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*d + c*x^4)/((a*x^4 - b)^(1/4)*(2*f - e*x^8)),x)

[Out]

int(-(2*d + c*x^4)/((a*x^4 - b)^(1/4)*(2*f - e*x^8)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + 2 d}{\sqrt [4]{a x^{4} - b} \left (e x^{8} - 2 f\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+2*d)/(a*x**4-b)**(1/4)/(e*x**8-2*f),x)

[Out]

Integral((c*x**4 + 2*d)/((a*x**4 - b)**(1/4)*(e*x**8 - 2*f)), x)

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