3.12.40 \(\int \frac {(-2 b+a x^4) (b+a x^4)^{3/4}}{x^8} \, dx\)

Optimal. Leaf size=85 \[ \frac {1}{2} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {\left (6 b-a x^4\right ) \left (a x^4+b\right )^{3/4}}{21 x^7} \]

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Rubi [A]  time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {451, 277, 240, 212, 206, 203} \begin {gather*} \frac {1}{2} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {2 \left (a x^4+b\right )^{7/4}}{7 x^7}-\frac {a \left (a x^4+b\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2*b + a*x^4)*(b + a*x^4)^(3/4))/x^8,x]

[Out]

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

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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 451

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))

Rubi steps

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

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Mathematica [C]  time = 0.06, size = 67, normalized size = 0.79 \begin {gather*} \frac {\left (a x^4+b\right )^{3/4} \left (6 \left (a x^4+b\right )-\frac {7 a x^4 \, _2F_1\left (-\frac {3}{4},-\frac {3}{4};\frac {1}{4};-\frac {a x^4}{b}\right )}{\left (\frac {a x^4}{b}+1\right )^{3/4}}\right )}{21 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2*b + a*x^4)*(b + a*x^4)^(3/4))/x^8,x]

[Out]

((b + a*x^4)^(3/4)*(6*(b + a*x^4) - (7*a*x^4*Hypergeometric2F1[-3/4, -3/4, 1/4, -((a*x^4)/b)])/(1 + (a*x^4)/b)
^(3/4)))/(21*x^7)

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IntegrateAlgebraic [A]  time = 0.27, size = 85, normalized size = 1.00 \begin {gather*} \frac {\left (6 b-a x^4\right ) \left (b+a x^4\right )^{3/4}}{21 x^7}+\frac {1}{2} a^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2*b + a*x^4)*(b + a*x^4)^(3/4))/x^8,x]

[Out]

((6*b - a*x^4)*(b + a*x^4)^(3/4))/(21*x^7) + (a^(7/4)*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2 + (a^(7/4)*ArcT
anh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2

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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((a*x^4-2*b)*(a*x^4+b)^(3/4)/x^8,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 {{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - 2 \, b\right )}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-2*b)*(a*x^4+b)^(3/4)/x^8,x, algorithm="giac")

[Out]

integrate((a*x^4 + b)^(3/4)*(a*x^4 - 2*b)/x^8, x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-2 b \right ) \left (a \,x^{4}+b \right )^{\frac {3}{4}}}{x^{8}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4-2*b)*(a*x^4+b)^(3/4)/x^8,x)

[Out]

int((a*x^4-2*b)*(a*x^4+b)^(3/4)/x^8,x)

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maxima [A]  time = 0.41, size = 103, normalized size = 1.21 \begin {gather*} -\frac {1}{12} \, {\left (3 \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} + \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{x^{3}}\right )} a + \frac {2 \, {\left (a x^{4} + b\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-2*b)*(a*x^4+b)^(3/4)/x^8,x, algorithm="maxima")

[Out]

-1/12*(3*a*(2*arctan((a*x^4 + b)^(1/4)/(a^(1/4)*x))/a^(1/4) + log(-(a^(1/4) - (a*x^4 + b)^(1/4)/x)/(a^(1/4) +
(a*x^4 + b)^(1/4)/x))/a^(1/4)) + 4*(a*x^4 + b)^(3/4)/x^3)*a + 2/7*(a*x^4 + b)^(7/4)/x^7

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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 (2\,b-a\,x^4\right )}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b + a*x^4)^(3/4)*(2*b - a*x^4))/x^8,x)

[Out]

-int(((b + a*x^4)^(3/4)*(2*b - a*x^4))/x^8, x)

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sympy [C]  time = 3.26, size = 114, normalized size = 1.34 \begin {gather*} - \frac {a^{\frac {7}{4}} \left (1 + \frac {b}{a x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (- \frac {3}{4}\right )} - \frac {a^{\frac {3}{4}} b \left (1 + \frac {b}{a x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 x^{4} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4-2*b)*(a*x**4+b)**(3/4)/x**8,x)

[Out]

-a**(7/4)*(1 + b/(a*x**4))**(3/4)*gamma(-7/4)/(2*gamma(-3/4)) - a**(3/4)*b*(1 + b/(a*x**4))**(3/4)*gamma(-7/4)
/(2*x**4*gamma(-3/4)) + a*b**(3/4)*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), a*x**4*exp_polar(I*pi)/b)/(4*x**3*g
amma(1/4))

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