3.6.47 \(\int \frac {1}{x^7 (a+b x^6) \sqrt {c+d x^6}} \, dx\)

Optimal. Leaf size=117 \[ -\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {b c-a d}}\right )}{3 a^2 \sqrt {b c-a d}}+\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{6 a^2 c^{3/2}}-\frac {\sqrt {c+d x^6}}{6 a c x^6} \]

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Rubi [A]  time = 0.12, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 103, 156, 63, 208} \begin {gather*} -\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {b c-a d}}\right )}{3 a^2 \sqrt {b c-a d}}+\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{6 a^2 c^{3/2}}-\frac {\sqrt {c+d x^6}}{6 a c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

-Sqrt[c + d*x^6]/(6*a*c*x^6) + ((2*b*c + a*d)*ArcTanh[Sqrt[c + d*x^6]/Sqrt[c]])/(6*a^2*c^(3/2)) - (b^(3/2)*Arc
Tanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(3*a^2*Sqrt[b*c - a*d])

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x^7 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x) \sqrt {c+d x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {c+d x^6}}{6 a c x^6}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (2 b c+a d)+\frac {b d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^6\right )}{6 a c}\\ &=-\frac {\sqrt {c+d x^6}}{6 a c x^6}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^6\right )}{6 a^2}-\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^6\right )}{12 a^2 c}\\ &=-\frac {\sqrt {c+d x^6}}{6 a c x^6}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^6}\right )}{3 a^2 d}-\frac {(2 b c+a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^6}\right )}{6 a^2 c d}\\ &=-\frac {\sqrt {c+d x^6}}{6 a c x^6}+\frac {(2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{6 a^2 c^{3/2}}-\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {b c-a d}}\right )}{3 a^2 \sqrt {b c-a d}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 151, normalized size = 1.29 \begin {gather*} \frac {b^{3/2} \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {b c-a d}}\right )}{3 a^2 (a d-b c)}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{3 a^2 \sqrt {c}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{6 a c^{3/2}}-\frac {\sqrt {c+d x^6}}{6 a c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.26, size = 127, normalized size = 1.09 \begin {gather*} -\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^6} \sqrt {a d-b c}}{b c-a d}\right )}{3 a^2 \sqrt {a d-b c}}+\frac {(a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{6 a^2 c^{3/2}}-\frac {\sqrt {c+d x^6}}{6 a c x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^7*(a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

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

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fricas [A]  time = 0.47, size = 565, normalized size = 4.83 \begin {gather*} \left [\frac {2 \, b c^{2} x^{6} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{6} + 2 \, b c - a d - 2 \, \sqrt {d x^{6} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{6} + a}\right ) + {\left (2 \, b c + a d\right )} \sqrt {c} x^{6} \log \left (\frac {d x^{6} + 2 \, \sqrt {d x^{6} + c} \sqrt {c} + 2 \, c}{x^{6}}\right ) - 2 \, \sqrt {d x^{6} + c} a c}{12 \, a^{2} c^{2} x^{6}}, -\frac {4 \, b c^{2} x^{6} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{6} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{6} + b c}\right ) - {\left (2 \, b c + a d\right )} \sqrt {c} x^{6} \log \left (\frac {d x^{6} + 2 \, \sqrt {d x^{6} + c} \sqrt {c} + 2 \, c}{x^{6}}\right ) + 2 \, \sqrt {d x^{6} + c} a c}{12 \, a^{2} c^{2} x^{6}}, \frac {b c^{2} x^{6} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{6} + 2 \, b c - a d - 2 \, \sqrt {d x^{6} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{6} + a}\right ) - {\left (2 \, b c + a d\right )} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {d x^{6} + c} \sqrt {-c}}{c}\right ) - \sqrt {d x^{6} + c} a c}{6 \, a^{2} c^{2} x^{6}}, -\frac {2 \, b c^{2} x^{6} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{6} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{6} + b c}\right ) + {\left (2 \, b c + a d\right )} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {d x^{6} + c} \sqrt {-c}}{c}\right ) + \sqrt {d x^{6} + c} a c}{6 \, a^{2} c^{2} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(2*b*c^2*x^6*sqrt(b/(b*c - a*d))*log((b*d*x^6 + 2*b*c - a*d - 2*sqrt(d*x^6 + c)*(b*c - a*d)*sqrt(b/(b*c
- a*d)))/(b*x^6 + a)) + (2*b*c + a*d)*sqrt(c)*x^6*log((d*x^6 + 2*sqrt(d*x^6 + c)*sqrt(c) + 2*c)/x^6) - 2*sqrt(
d*x^6 + c)*a*c)/(a^2*c^2*x^6), -1/12*(4*b*c^2*x^6*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^6 + c)*(b*c - a*d)*sqr
t(-b/(b*c - a*d))/(b*d*x^6 + b*c)) - (2*b*c + a*d)*sqrt(c)*x^6*log((d*x^6 + 2*sqrt(d*x^6 + c)*sqrt(c) + 2*c)/x
^6) + 2*sqrt(d*x^6 + c)*a*c)/(a^2*c^2*x^6), 1/6*(b*c^2*x^6*sqrt(b/(b*c - a*d))*log((b*d*x^6 + 2*b*c - a*d - 2*
sqrt(d*x^6 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^6 + a)) - (2*b*c + a*d)*sqrt(-c)*x^6*arctan(sqrt(d*x^6 +
c)*sqrt(-c)/c) - sqrt(d*x^6 + c)*a*c)/(a^2*c^2*x^6), -1/6*(2*b*c^2*x^6*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^
6 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^6 + b*c)) + (2*b*c + a*d)*sqrt(-c)*x^6*arctan(sqrt(d*x^6 + c)*s
qrt(-c)/c) + sqrt(d*x^6 + c)*a*c)/(a^2*c^2*x^6)]

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giac [A]  time = 0.18, size = 104, normalized size = 0.89 \begin {gather*} \frac {b^{2} \arctan \left (\frac {\sqrt {d x^{6} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{6} + c}}{\sqrt {-c}}\right )}{6 \, a^{2} \sqrt {-c} c} - \frac {\sqrt {d x^{6} + c}}{6 \, a c x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

1/3*b^2*arctan(sqrt(d*x^6 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^2) - 1/6*(2*b*c + a*d)*arctan(s
qrt(d*x^6 + c)/sqrt(-c))/(a^2*sqrt(-c)*c) - 1/6*sqrt(d*x^6 + c)/(a*c*x^6)

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maple [F]  time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}\, x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^7/(b*x^6+a)/(d*x^6+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^7), x)

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mupad [B]  time = 5.62, size = 396, normalized size = 3.38 \begin {gather*} \frac {\ln \left (\sqrt {d\,x^6+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}+b^6\,c^2+a^2\,b^4\,d^2-2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{6\,a^3\,d-6\,a^2\,b\,c}-\frac {\ln \left (\sqrt {d\,x^6+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}-b^6\,c^2-a^2\,b^4\,d^2+2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{6\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {\sqrt {d\,x^6+c}}{6\,a\,c\,x^6}-\frac {\mathrm {atan}\left (\frac {b^4\,d^4\,\sqrt {d\,x^6+c}\,1{}\mathrm {i}}{18\,\sqrt {c^3}\,\left (\frac {b^4\,d^4}{18\,c}+\frac {5\,a\,b^3\,d^5}{108\,c^2}+\frac {a^2\,b^2\,d^6}{108\,c^3}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^6+c}\,1{}\mathrm {i}}{108\,\sqrt {c^3}\,\left (\frac {5\,b^3\,d^5}{108\,a}+\frac {b^2\,d^6}{108\,c}+\frac {b^4\,c\,d^4}{18\,a^2}\right )}+\frac {b^3\,d^5\,\sqrt {d\,x^6+c}\,5{}\mathrm {i}}{108\,\sqrt {c^3}\,\left (\frac {b^4\,d^4}{18\,a}+\frac {5\,b^3\,d^5}{108\,c}+\frac {a\,b^2\,d^6}{108\,c^2}\right )}\right )\,\left (a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{6\,a^2\,\sqrt {c^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b*x^6)*(c + d*x^6)^(1/2)),x)

[Out]

(log((c + d*x^6)^(1/2)*(b^4*c - a*b^3*d)^(3/2) + b^6*c^2 + a^2*b^4*d^2 - 2*a*b^5*c*d)*(b^4*c - a*b^3*d)^(1/2))
/(6*a^3*d - 6*a^2*b*c) - (log((c + d*x^6)^(1/2)*(b^4*c - a*b^3*d)^(3/2) - b^6*c^2 - a^2*b^4*d^2 + 2*a*b^5*c*d)
*(b^4*c - a*b^3*d)^(1/2))/(6*(a^3*d - a^2*b*c)) - (c + d*x^6)^(1/2)/(6*a*c*x^6) - (atan((b^4*d^4*(c + d*x^6)^(
1/2)*1i)/(18*(c^3)^(1/2)*((b^4*d^4)/(18*c) + (5*a*b^3*d^5)/(108*c^2) + (a^2*b^2*d^6)/(108*c^3))) + (b^2*d^6*(c
+ d*x^6)^(1/2)*1i)/(108*(c^3)^(1/2)*((5*b^3*d^5)/(108*a) + (b^2*d^6)/(108*c) + (b^4*c*d^4)/(18*a^2))) + (b^3*
d^5*(c + d*x^6)^(1/2)*5i)/(108*(c^3)^(1/2)*((b^4*d^4)/(18*a) + (5*b^3*d^5)/(108*c) + (a*b^2*d^6)/(108*c^2))))*
(a*d + 2*b*c)*1i)/(6*a^2*(c^3)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{7} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(1/(x**7*(a + b*x**6)*sqrt(c + d*x**6)), x)

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