3.11.36 \(\int \frac {1}{x^2 (a+b x^2)^{7/6}} \, dx\) [1036]

Optimal. Leaf size=614 \[ \frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {4 b x}{a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}+\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}}-\frac {4 \sqrt {2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]

[Out]

3/a/x/(b*x^2+a)^(1/6)+4*b*x/a^2/(b*x^2+a)^(1/6)-4*(b*x^2+a)^(5/6)/a^2/x+4*b*x/a/(a/(b*x^2+a))^(2/3)/(b*x^2+a)^
(7/6)/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))-4/3*(1-(a/(b*x^2+a))^(1/3))*EllipticF((1-(a/(b*x^2+a))^(1/3)+3^(1/2))/(1
-(a/(b*x^2+a))^(1/3)-3^(1/2)),2*I-I*3^(1/2))*2^(1/2)*((1+(a/(b*x^2+a))^(1/3)+(a/(b*x^2+a))^(2/3))/(1-(a/(b*x^2
+a))^(1/3)-3^(1/2))^2)^(1/2)*3^(3/4)/a/x/(a/(b*x^2+a))^(2/3)/(b*x^2+a)^(1/6)/((-1+(a/(b*x^2+a))^(1/3))/(1-(a/(
b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)+2*3^(1/4)*(1-(a/(b*x^2+a))^(1/3))*EllipticE((1-(a/(b*x^2+a))^(1/3)+3^(1/2))/
(1-(a/(b*x^2+a))^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+(a/(b*x^2+a))^(1/3)+(a/(b*x^2+a))^(2/3))/(1-(a/(b*x^2+a))^(
1/3)-3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/a/x/(a/(b*x^2+a))^(2/3)/(b*x^2+a)^(1/6)/((-1+(a/(b*x^2+a))^(1
/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 614, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {296, 331, 244, 204, 241, 310, 225, 1893} \begin {gather*} \frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {4 \sqrt {2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\text {ArcSin}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {4 b x}{a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {3}{a x \sqrt [6]{a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)^(7/6)),x]

[Out]

3/(a*x*(a + b*x^2)^(1/6)) + (4*b*x)/(a^2*(a + b*x^2)^(1/6)) - (4*(a + b*x^2)^(5/6))/(a^2*x) + (4*b*x)/(a*(a/(a
+ b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) + (2*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 -
(a/(a + b*x^2))^(1/3))*Sqrt[(1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))
^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 +
4*Sqrt[3]])/(a*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*Sqrt[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a
/(a + b*x^2))^(1/3))^2)]) - (4*Sqrt[2]*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1 + (a/(a + b*x^2))^(1/3) + (a/(a + b
*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/
(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*a*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*
Sqrt[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-7/6), x_Symbol] :> Dist[1/((a + b*x^2)^(2/3)*(a/(a + b*x^2))^(2/3)), Subst[Int[1/(
1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b}, x]

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r
*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 241

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[3*(Sqrt[b*x^2]/(2*b*x)), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 244

Int[((a_) + (b_.)*(x_)^2)^(-1/6), x_Symbol] :> Simp[3*(x/(2*(a + b*x^2)^(1/6))), x] - Dist[a/2, Int[1/(a + b*x
^2)^(7/6), x], x] /; FreeQ[{a, b}, x]

Rule 296

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

Rule 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]
, x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 331

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

Rule 1893

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 + Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 + Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x
] + Simp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^{7/6}} \, dx &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 \int \frac {1}{x^2 \sqrt [6]{a+b x^2}} \, dx}{a}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {(8 b) \int \frac {1}{\sqrt [6]{a+b x^2}} \, dx}{3 a^2}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {(4 b) \int \frac {1}{\left (a+b x^2\right )^{7/6}} \, dx}{3 a}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{3 a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {\left (2 \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {\left (2 \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {\left (2 \sqrt {2 \left (2+\sqrt {3}\right )} \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {4 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt {-1+\frac {a}{a+b x^2}}}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}+\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {-\frac {b x^2}{a+b x^2}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}-\frac {4 \sqrt {2} \sqrt {-\frac {b x^2}{a+b x^2}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 9.18, size = 52, normalized size = 0.08 \begin {gather*} -\frac {\sqrt [6]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{2},\frac {7}{6};\frac {1}{2};-\frac {b x^2}{a}\right )}{a x \sqrt [6]{a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)^(7/6)),x]

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2 + a)^(5/6)/(b^2*x^6 + 2*a*b*x^4 + a^2*x^2), x)

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Sympy [A]
time = 0.59, size = 27, normalized size = 0.04 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {7}{6}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-hyper((-1/2, 7/6), (1/2,), b*x**2*exp_polar(I*pi)/a)/(a**(7/6)*x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.12, size = 40, normalized size = 0.07 \begin {gather*} -\frac {3\,{\left (\frac {a}{b\,x^2}+1\right )}^{7/6}\,{{}}_2{\mathrm {F}}_1\left (\frac {7}{6},\frac {5}{3};\ \frac {8}{3};\ -\frac {a}{b\,x^2}\right )}{10\,x\,{\left (b\,x^2+a\right )}^{7/6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^2)^(7/6)),x)

[Out]

-(3*(a/(b*x^2) + 1)^(7/6)*hypergeom([7/6, 5/3], 8/3, -a/(b*x^2)))/(10*x*(a + b*x^2)^(7/6))

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