3.4.12 \(\int \frac {x^7}{a+b x^4+c x^8} \, dx\) [312]

Optimal. Leaf size=63 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^4+c x^8\right )}{8 c} \]

[Out]

1/8*ln(c*x^8+b*x^4+a)/c+1/4*b*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/c/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1371, 648, 632, 212, 642} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^4+c x^8\right )}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^4 + c*x^8),x]

[Out]

(b*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*c*Sqrt[b^2 - 4*a*c]) + Log[a + b*x^4 + c*x^8]/(8*c)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1371

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^7}{a+b x^4+c x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,x^4\right )\\ &=\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^4\right )}{8 c}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^4\right )}{8 c}\\ &=\frac {\log \left (a+b x^4+c x^8\right )}{8 c}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )}{4 c}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^4+c x^8\right )}{8 c}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 62, normalized size = 0.98 \begin {gather*} \frac {-\frac {2 b \tan ^{-1}\left (\frac {b+2 c x^4}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log \left (a+b x^4+c x^8\right )}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^4 + c*x^8),x]

[Out]

((-2*b*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*x^4 + c*x^8])/(8*c)

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Maple [A]
time = 0.03, size = 60, normalized size = 0.95

method	result	size
default	\(\frac {\ln \left (c \,x^{8}+b \,x^{4}+a \right )}{8 c}-\frac {b \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{4 c \sqrt {4 a c -b^{2}}}\)	\(60\)
risch	\(\frac {\ln \left (\left (-4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{8 a c -2 b^{2}}-\frac {\ln \left (\left (-4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{8 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{8 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{8 a c -2 b^{2}}-\frac {\ln \left (\left (-4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{8 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) x^{4}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{8 c \left (4 a c -b^{2}\right )}\)	\(467\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^8+b*x^4+a),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(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 0.36, size = 197, normalized size = 3.13 \begin {gather*} \left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^8+b*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (54) = 108\).
time = 1.32, size = 223, normalized size = 3.54 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) \log {\left (x^{4} + \frac {- 16 a c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) + 2 a + 4 b^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right )}{b} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) \log {\left (x^{4} + \frac {- 16 a c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right ) + 2 a + 4 b^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{8 c \left (4 a c - b^{2}\right )} + \frac {1}{8 c}\right )}{b} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**8+b*x**4+a),x)

[Out]

(-b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c))*log(x**4 + (-16*a*c*(-b*sqrt(-4*a*c + b**2)/(8*c*(4*a*
c - b**2)) + 1/(8*c)) + 2*a + 4*b**2*(-b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c)))/b) + (b*sqrt(-4*
a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c))*log(x**4 + (-16*a*c*(b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) +
1/(8*c)) + 2*a + 4*b**2*(b*sqrt(-4*a*c + b**2)/(8*c*(4*a*c - b**2)) + 1/(8*c)))/b)

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Giac [A]
time = 8.36, size = 59, normalized size = 0.94 \begin {gather*} -\frac {b \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} c} + \frac {\log \left (c x^{8} + b x^{4} + a\right )}{8 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^8+b*x^4+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.61, size = 2654, normalized size = 42.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^4 + c*x^8),x)

[Out]

(log(a + b*x^4 + c*x^8)*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - (b*atan((8*x^4*(((a*c - b^2)*(((((16*a*c
- 4*b^2)*((b*(448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2))
- (32*b^4*c^3*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(2*(64*a*c^2 - 16*b^2*c)) - (b*(
144*b^3*c^2 - ((448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*(64*a
*c^2 - 16*b^2*c))))/(8*c*(4*a*c - b^2)^(1/2)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - (b*((b*((b*(448*b
^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (32*b^4*c^3*(16*a*
c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c - b^2)^(1/2)) - (4*b^5*c^2*(16*a*c - 4*b
^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2))))/(8*c*(4*a*c - b^2)^(1/2)) + (b*(20*b^3*c - ((144*b^3*c^2 - ((448*
b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)))*(
16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c))))/(8*c*(4*a*c - b^2)^(1/2)) + (b^6*c*(16*a*c - 4*b^2))/(2*(64*a*c^2
- 16*b^2*c)*(4*a*c - b^2)^(3/2))))/(8*a^3*c^2) + ((b^3 - 3*a*b*c)*(b^7/(8*(4*a*c - b^2)^2) + b^3 - ((20*b^3*c
- ((144*b^3*c^2 - ((448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*
(64*a*c^2 - 16*b^2*c)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c
)) + ((16*a*c - 4*b^2)*((b*((b*(448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a
*c - b^2)^(1/2)) - (32*b^4*c^3*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c - b
^2)^(1/2)) - (4*b^5*c^2*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2))))/(2*(64*a*c^2 - 16*b^2*c)) +
(b*(((16*a*c - 4*b^2)*((b*(448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c -
b^2)^(1/2)) - (32*b^4*c^3*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(2*(64*a*c^2 - 16*b^
2*c)) - (b*(144*b^3*c^2 - ((448*b^3*c^3 - (256*b^3*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^
2))/(2*(64*a*c^2 - 16*b^2*c))))/(8*c*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c - b^2)^(1/2))))/(8*a^3*c^2*(4*a*c - b^
2)^(1/2)))*(4*a*c - b^2)^2)/b^4 + ((4*a*c - b^2)^(3/2)*(b^3 - 3*a*b*c)*(a*b^2 + (((b*((b*(768*a*b^2*c^3 - (512
*a*b^2*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (64*a*b^3*c^3*(16*a*c - 4*b^2
))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c - b^2)^(1/2)) - (8*a*b^4*c^2*(16*a*c - 4*b^2))/((
64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) + (a*b^6)/(4*(4*a*c - b^2)^2)
- ((16*a*c - 4*b^2)*(((16*a*c - 4*b^2)*(((768*a*b^2*c^3 - (512*a*b^2*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2
*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - 208*a*b^2*c^2))/(2*(64*a*c^2 - 16*b^2*c)) + 24*a*b^2*c))/(2
*(64*a*c^2 - 16*b^2*c)) + (b*(((16*a*c - 4*b^2)*((b*(768*a*b^2*c^3 - (512*a*b^2*c^4*(16*a*c - 4*b^2))/(64*a*c^
2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (64*a*b^3*c^3*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*c)*(4*a*c - b^
2)^(1/2))))/(2*(64*a*c^2 - 16*b^2*c)) + (b*(((768*a*b^2*c^3 - (512*a*b^2*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*
b^2*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - 208*a*b^2*c^2))/(8*c*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c
- b^2)^(1/2))))/(a^3*b^4*c^2) + ((a*c - b^2)*(4*a*c - b^2)^2*(((((16*a*c - 4*b^2)*((b*(768*a*b^2*c^3 - (512*a*
b^2*c^4*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (64*a*b^3*c^3*(16*a*c - 4*b^2))/
((64*a*c^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(2*(64*a*c^2 - 16*b^2*c)) + (b*(((768*a*b^2*c^3 - (512*a*b^2*c^4
*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - 208*a*b^2*c^2))/(8*c*(
4*a*c - b^2)^(1/2)))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - (b*((b*((b*(768*a*b^2*c^3 - (512*a*b^2*c^4*
(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c)))/(8*c*(4*a*c - b^2)^(1/2)) - (64*a*b^3*c^3*(16*a*c - 4*b^2))/((64*a*c
^2 - 16*b^2*c)*(4*a*c - b^2)^(1/2))))/(8*c*(4*a*c - b^2)^(1/2)) - (8*a*b^4*c^2*(16*a*c - 4*b^2))/((64*a*c^2 -
16*b^2*c)*(4*a*c - b^2))))/(8*c*(4*a*c - b^2)^(1/2)) + (b*(((16*a*c - 4*b^2)*(((768*a*b^2*c^3 - (512*a*b^2*c^4
*(16*a*c - 4*b^2))/(64*a*c^2 - 16*b^2*c))*(16*a*c - 4*b^2))/(2*(64*a*c^2 - 16*b^2*c)) - 208*a*b^2*c^2))/(2*(64
*a*c^2 - 16*b^2*c)) + 24*a*b^2*c))/(8*c*(4*a*c - b^2)^(1/2)) + (a*b^5*c*(16*a*c - 4*b^2))/((64*a*c^2 - 16*b^2*
c)*(4*a*c - b^2)^(3/2))))/(a^3*b^4*c^2)))/(4*c*(4*a*c - b^2)^(1/2))

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