3.21.77 \(\int \frac {(-b+a x^6)^{3/4}}{x^7} \, dx\)

Optimal. Leaf size=150 \[ -\frac {\left (a x^6-b\right )^{3/4}}{6 x^6}-\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^6-b}}{\sqrt {a x^6-b}-\sqrt {b}}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {a \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^6-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^6-b}}\right )}{4 \sqrt {2} \sqrt [4]{b}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 225, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {266, 47, 63, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (a x^6-b\right )^{3/4}}{6 x^6}+\frac {a \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^6-b}+\sqrt {a x^6-b}+\sqrt {b}\right )}{8 \sqrt {2} \sqrt [4]{b}}-\frac {a \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^6-b}+\sqrt {a x^6-b}+\sqrt {b}\right )}{8 \sqrt {2} \sqrt [4]{b}}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^6-b}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^6-b}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b + a*x^6)^(3/4)/x^7,x]

[Out]

-1/6*(-b + a*x^6)^(3/4)/x^6 - (a*ArcTan[1 - (Sqrt[2]*(-b + a*x^6)^(1/4))/b^(1/4)])/(4*Sqrt[2]*b^(1/4)) + (a*Ar
cTan[1 + (Sqrt[2]*(-b + a*x^6)^(1/4))/b^(1/4)])/(4*Sqrt[2]*b^(1/4)) + (a*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*(-b + a
*x^6)^(1/4) + Sqrt[-b + a*x^6]])/(8*Sqrt[2]*b^(1/4)) - (a*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4) + S
qrt[-b + a*x^6]])/(8*Sqrt[2]*b^(1/4))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

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

Rule 266

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

Rule 297

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\left (-b+a x^6\right )^{3/4}}{x^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(-b+a x)^{3/4}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}+\frac {1}{8} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^6\right )\\ &=-\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^6}\right )\\ &=-\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^6}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^6}\right )\\ &=-\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}+\frac {1}{8} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )+\frac {1}{8} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{8 \sqrt {2} \sqrt [4]{b}}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{8 \sqrt {2} \sqrt [4]{b}}\\ &=-\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}+\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{8 \sqrt {2} \sqrt [4]{b}}-\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{8 \sqrt {2} \sqrt [4]{b}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b}}\\ &=-\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b}}+\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} \sqrt [4]{b}}+\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{8 \sqrt {2} \sqrt [4]{b}}-\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}+\sqrt {-b+a x^6}\right )}{8 \sqrt {2} \sqrt [4]{b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 40, normalized size = 0.27 \begin {gather*} \frac {2 a \left (a x^6-b\right )^{7/4} \, _2F_1\left (\frac {7}{4},2;\frac {11}{4};1-\frac {a x^6}{b}\right )}{21 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b + a*x^6)^(3/4)/x^7,x]

[Out]

(2*a*(-b + a*x^6)^(7/4)*Hypergeometric2F1[7/4, 2, 11/4, 1 - (a*x^6)/b])/(21*b^2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.21, size = 149, normalized size = 0.99 \begin {gather*} -\frac {\left (-b+a x^6\right )^{3/4}}{6 x^6}+\frac {a \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^6}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^6}}\right )}{4 \sqrt {2} \sqrt [4]{b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^6}}{\sqrt {b}+\sqrt {-b+a x^6}}\right )}{4 \sqrt {2} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 201, normalized size = 1.34 \begin {gather*} -\frac {12 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{6} \arctan \left (-\frac {{\left (a x^{6} - b\right )}^{\frac {1}{4}} \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} a^{3} - \sqrt {\sqrt {a x^{6} - b} a^{6} - \sqrt {-\frac {a^{4}}{b}} a^{4} b} \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}}}{a^{4}}\right ) - 3 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{6} \log \left ({\left (a x^{6} - b\right )}^{\frac {1}{4}} a^{3} + \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 3 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{6} \log \left ({\left (a x^{6} - b\right )}^{\frac {1}{4}} a^{3} - \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 4 \, {\left (a x^{6} - b\right )}^{\frac {3}{4}}}{24 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6-b)^(3/4)/x^7,x, algorithm="fricas")

[Out]

-1/24*(12*(-a^4/b)^(1/4)*x^6*arctan(-((a*x^6 - b)^(1/4)*(-a^4/b)^(1/4)*a^3 - sqrt(sqrt(a*x^6 - b)*a^6 - sqrt(-
a^4/b)*a^4*b)*(-a^4/b)^(1/4))/a^4) - 3*(-a^4/b)^(1/4)*x^6*log((a*x^6 - b)^(1/4)*a^3 + (-a^4/b)^(3/4)*b) + 3*(-
a^4/b)^(1/4)*x^6*log((a*x^6 - b)^(1/4)*a^3 - (-a^4/b)^(3/4)*b) + 4*(a*x^6 - b)^(3/4))/x^6

________________________________________________________________________________________

giac [A]  time = 0.29, size = 196, normalized size = 1.31 \begin {gather*} \frac {\frac {6 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {3 \, \sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} - \frac {8 \, {\left (a x^{6} - b\right )}^{\frac {3}{4}} a}{x^{6}}}{48 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6-b)^(3/4)/x^7,x, algorithm="giac")

[Out]

1/48*(6*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^6 - b)^(1/4))/b^(1/4))/b^(1/4) + 6*sqrt(2)*a^
2*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^6 - b)^(1/4))/b^(1/4))/b^(1/4) - 3*sqrt(2)*a^2*log(sqrt(2)*(a*
x^6 - b)^(1/4)*b^(1/4) + sqrt(a*x^6 - b) + sqrt(b))/b^(1/4) + 3*sqrt(2)*a^2*log(-sqrt(2)*(a*x^6 - b)^(1/4)*b^(
1/4) + sqrt(a*x^6 - b) + sqrt(b))/b^(1/4) - 8*(a*x^6 - b)^(3/4)*a/x^6)/a

________________________________________________________________________________________

maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{6}-b \right )^{\frac {3}{4}}}{x^{7}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6-b)^(3/4)/x^7,x)

[Out]

int((a*x^6-b)^(3/4)/x^7,x)

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 181, normalized size = 1.21 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{6} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{6} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{6} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} a - \frac {{\left (a x^{6} - b\right )}^{\frac {3}{4}}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6-b)^(3/4)/x^7,x, algorithm="maxima")

[Out]

1/16*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^6 - b)^(1/4))/b^(1/4))/b^(1/4) + 2*sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^6 - b)^(1/4))/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(2)*(a*x^6 - b)^(1/4)
*b^(1/4) + sqrt(a*x^6 - b) + sqrt(b))/b^(1/4) + sqrt(2)*log(-sqrt(2)*(a*x^6 - b)^(1/4)*b^(1/4) + sqrt(a*x^6 -
b) + sqrt(b))/b^(1/4))*a - 1/6*(a*x^6 - b)^(3/4)/x^6

________________________________________________________________________________________

mupad [B]  time = 1.24, size = 69, normalized size = 0.46 \begin {gather*} \frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^6-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{1/4}}-\frac {{\left (a\,x^6-b\right )}^{3/4}}{6\,x^6}-\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^6-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6 - b)^(3/4)/x^7,x)

[Out]

(a*atan((a*x^6 - b)^(1/4)/(-b)^(1/4)))/(4*(-b)^(1/4)) - (a*x^6 - b)^(3/4)/(6*x^6) - (a*atanh((a*x^6 - b)^(1/4)
/(-b)^(1/4)))/(4*(-b)^(1/4))

________________________________________________________________________________________

sympy [C]  time = 1.43, size = 44, normalized size = 0.29 \begin {gather*} - \frac {a^{\frac {3}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{6}}} \right )}}{6 x^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**6-b)**(3/4)/x**7,x)

[Out]

-a**(3/4)*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), b*exp_polar(2*I*pi)/(a*x**6))/(6*x**(3/2)*gamma(5/4))

________________________________________________________________________________________