3.20.30 \(\int \frac {(-4 b+a x^4) (b+a x^4)^{3/4}}{x^8 (4 b+a x^4)} \, dx\)

Optimal. Leaf size=134 \[ \frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt {2} b}+\frac {\left (6 b-a x^4\right ) \left (a x^4+b\right )^{3/4}}{42 b x^7} \]

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Rubi [A]  time = 0.19, antiderivative size = 143, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {580, 583, 12, 377, 212, 206, 203} \begin {gather*} \frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt {2} b}+\frac {\left (a x^4+b\right )^{3/4}}{7 x^7}-\frac {a \left (a x^4+b\right )^{3/4}}{42 b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-4*b + a*x^4)*(b + a*x^4)^(3/4))/(x^8*(4*b + a*x^4)),x]

[Out]

(b + a*x^4)^(3/4)/(7*x^7) - (a*(b + a*x^4)^(3/4))/(42*b*x^3) + (3^(3/4)*a^(7/4)*ArcTan[(3^(1/4)*a^(1/4)*x)/(Sq
rt[2]*(b + a*x^4)^(1/4))])/(8*Sqrt[2]*b) + (3^(3/4)*a^(7/4)*ArcTanh[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(
1/4))])/(8*Sqrt[2]*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

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

Rule 206

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

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 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (-4 b+a x^4\right ) \left (b+a x^4\right )^{3/4}}{x^8 \left (4 b+a x^4\right )} \, dx &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}+\frac {\int \frac {8 a b^2+44 a^2 b x^4}{x^4 \sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx}{28 b}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {a \left (b+a x^4\right )^{3/4}}{42 b x^3}-\frac {\int -\frac {504 a^2 b^3}{\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx}{336 b^3}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {a \left (b+a x^4\right )^{3/4}}{42 b x^3}+\frac {1}{2} \left (3 a^2\right ) \int \frac {1}{\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )} \, dx\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {a \left (b+a x^4\right )^{3/4}}{42 b x^3}+\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 b-3 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {a \left (b+a x^4\right )^{3/4}}{42 b x^3}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2-\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2+\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 b}\\ &=\frac {\left (b+a x^4\right )^{3/4}}{7 x^7}-\frac {a \left (b+a x^4\right )^{3/4}}{42 b x^3}+\frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{8 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{8 \sqrt {2} b}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 151, normalized size = 1.13 \begin {gather*} \frac {3^{3/4} a^{7/4} \left (-\log \left (2-\frac {\sqrt {2} \sqrt [4]{3} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac {\sqrt {2} \sqrt [4]{3} \sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}+2\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{a+b x^4}}\right )\right )}{16 \sqrt {2} b}+\left (\frac {1}{7 x^7}-\frac {a}{42 b x^3}\right ) \left (a x^4+b\right )^{3/4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-4*b + a*x^4)*(b + a*x^4)^(3/4))/(x^8*(4*b + a*x^4)),x]

[Out]

(1/(7*x^7) - a/(42*b*x^3))*(b + a*x^4)^(3/4) + (3^(3/4)*a^(7/4)*(2*ArcTan[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(a + b*
x^4)^(1/4))] - Log[2 - (Sqrt[2]*3^(1/4)*a^(1/4)*x)/(a + b*x^4)^(1/4)] + Log[2 + (Sqrt[2]*3^(1/4)*a^(1/4)*x)/(a
+ b*x^4)^(1/4)]))/(16*Sqrt[2]*b)

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IntegrateAlgebraic [A]  time = 0.56, size = 134, normalized size = 1.00 \begin {gather*} \frac {\left (6 b-a x^4\right ) \left (b+a x^4\right )^{3/4}}{42 b x^7}+\frac {3^{3/4} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{8 \sqrt {2} b}+\frac {3^{3/4} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [4]{a} x}{\sqrt {2} \sqrt [4]{b+a x^4}}\right )}{8 \sqrt {2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-4*b + a*x^4)*(b + a*x^4)^(3/4))/(x^8*(4*b + a*x^4)),x]

[Out]

((6*b - a*x^4)*(b + a*x^4)^(3/4))/(42*b*x^7) + (3^(3/4)*a^(7/4)*ArcTan[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4
)^(1/4))])/(8*Sqrt[2]*b) + (3^(3/4)*a^(7/4)*ArcTanh[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(8*Sqrt[
2]*b)

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fricas [B]  time = 86.52, size = 494, normalized size = 3.69 \begin {gather*} -\frac {84 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \arctan \left (-\frac {4 \, {\left (27 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} a^{9} b x^{3} + 12 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} {\left (a x^{4} + b\right )}^{\frac {3}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} a^{5} b^{3} x - \sqrt {\frac {3}{2}} \sqrt {\sqrt {3} \sqrt {\frac {a^{7}}{b^{4}}} a^{6} b^{2}} {\left (18 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} a^{4} b x^{2} + \left (\frac {27}{4}\right )^{\frac {3}{4}} {\left (7 \, a b^{3} x^{4} + 4 \, b^{4}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}}\right )}\right )}}{27 \, {\left (a^{11} x^{4} + 4 \, a^{10} b\right )}}\right ) - 21 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} + 8 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} + 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x + 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 \, a^{4} b x^{4} + 4 \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) + 21 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}} b x^{7} \log \left (\frac {18 \, \sqrt {3} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \sqrt {\frac {a^{7}}{b^{4}}} a^{2} b^{2} x^{3} - 8 \, \left (\frac {27}{4}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {3}{4}} b^{3} x^{2} + 36 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{5} x - 3 \, \left (\frac {27}{4}\right )^{\frac {1}{4}} {\left (7 \, a^{4} b x^{4} + 4 \, a^{3} b^{2}\right )} \left (\frac {a^{7}}{b^{4}}\right )^{\frac {1}{4}}}{4 \, {\left (a x^{4} + 4 \, b\right )}}\right ) + 16 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - 6 \, b\right )}}{672 \, b x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-4*b)*(a*x^4+b)^(3/4)/x^8/(a*x^4+4*b),x, algorithm="fricas")

[Out]

-1/672*(84*(27/4)^(1/4)*(a^7/b^4)^(1/4)*b*x^7*arctan(-4/27*(27*(27/4)^(1/4)*(a*x^4 + b)^(1/4)*(a^7/b^4)^(1/4)*
a^9*b*x^3 + 12*(27/4)^(3/4)*(a*x^4 + b)^(3/4)*(a^7/b^4)^(3/4)*a^5*b^3*x - sqrt(3/2)*sqrt(sqrt(3)*sqrt(a^7/b^4)
*a^6*b^2)*(18*(27/4)^(1/4)*sqrt(a*x^4 + b)*(a^7/b^4)^(1/4)*a^4*b*x^2 + (27/4)^(3/4)*(7*a*b^3*x^4 + 4*b^4)*(a^7
/b^4)^(3/4)))/(a^11*x^4 + 4*a^10*b)) - 21*(27/4)^(1/4)*(a^7/b^4)^(1/4)*b*x^7*log(1/4*(18*sqrt(3)*(a*x^4 + b)^(
1/4)*sqrt(a^7/b^4)*a^2*b^2*x^3 + 8*(27/4)^(3/4)*sqrt(a*x^4 + b)*(a^7/b^4)^(3/4)*b^3*x^2 + 36*(a*x^4 + b)^(3/4)
*a^5*x + 3*(27/4)^(1/4)*(7*a^4*b*x^4 + 4*a^3*b^2)*(a^7/b^4)^(1/4))/(a*x^4 + 4*b)) + 21*(27/4)^(1/4)*(a^7/b^4)^
(1/4)*b*x^7*log(1/4*(18*sqrt(3)*(a*x^4 + b)^(1/4)*sqrt(a^7/b^4)*a^2*b^2*x^3 - 8*(27/4)^(3/4)*sqrt(a*x^4 + b)*(
a^7/b^4)^(3/4)*b^3*x^2 + 36*(a*x^4 + b)^(3/4)*a^5*x - 3*(27/4)^(1/4)*(7*a^4*b*x^4 + 4*a^3*b^2)*(a^7/b^4)^(1/4)
)/(a*x^4 + 4*b)) + 16*(a*x^4 + b)^(3/4)*(a*x^4 - 6*b))/(b*x^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - 4 \, b\right )}}{{\left (a x^{4} + 4 \, b\right )} x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-4*b)*(a*x^4+b)^(3/4)/x^8/(a*x^4+4*b),x, algorithm="giac")

[Out]

integrate((a*x^4 + b)^(3/4)*(a*x^4 - 4*b)/((a*x^4 + 4*b)*x^8), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-4 b \right ) \left (a \,x^{4}+b \right )^{\frac {3}{4}}}{x^{8} \left (a \,x^{4}+4 b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4-4*b)*(a*x^4+b)^(3/4)/x^8/(a*x^4+4*b),x)

[Out]

int((a*x^4-4*b)*(a*x^4+b)^(3/4)/x^8/(a*x^4+4*b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - 4 \, b\right )}}{{\left (a x^{4} + 4 \, b\right )} x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-4*b)*(a*x^4+b)^(3/4)/x^8/(a*x^4+4*b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b)^(3/4)*(a*x^4 - 4*b)/((a*x^4 + 4*b)*x^8), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (a\,x^4+b\right )}^{3/4}\,\left (4\,b-a\,x^4\right )}{x^8\,\left (a\,x^4+4\,b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b + a*x^4)^(3/4)*(4*b - a*x^4))/(x^8*(4*b + a*x^4)),x)

[Out]

int(-((b + a*x^4)^(3/4)*(4*b - a*x^4))/(x^8*(4*b + a*x^4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{4} - 4 b\right ) \left (a x^{4} + b\right )^{\frac {3}{4}}}{x^{8} \left (a x^{4} + 4 b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4-4*b)*(a*x**4+b)**(3/4)/x**8/(a*x**4+4*b),x)

[Out]

Integral((a*x**4 - 4*b)*(a*x**4 + b)**(3/4)/(x**8*(a*x**4 + 4*b)), x)

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