3.1033 \(\int \frac{x^4}{(a+b x^2)^{7/6}} \, dx\)

Optimal. Leaf size=630 \[ \frac{27\ 3^{3/4} a^2 \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{8 \sqrt{2} b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{81 a^2 x}{16 b^2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}-\frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^2 \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{32 b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}} \]

[Out]

(-81*a*x)/(16*b^2*(a + b*x^2)^(1/6)) - (3*x^3)/(b*(a + b*x^2)^(1/6)) + (27*x*(a + b*x^2)^(5/6))/(8*b^2) - (81*
a^2*x)/(16*b^2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) - (81*3^(1/4)*Sq
rt[2 + Sqrt[3]]*a^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]*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]])/(32*b^3*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)]) + (27*3^(3/4)*a^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]])/(8*Sqrt[2]*b^3*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)])

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Rubi [A]  time = 0.591613, antiderivative size = 630, normalized size of antiderivative = 1., 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 = {288, 321, 238, 198, 235, 304, 219, 1879} \[ -\frac{81 a^2 x}{16 b^2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )}+\frac{27\ 3^{3/4} a^2 \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{8 \sqrt{2} b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}-\frac{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^2 \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{32 b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^2)^(7/6),x]

[Out]

(-81*a*x)/(16*b^2*(a + b*x^2)^(1/6)) - (3*x^3)/(b*(a + b*x^2)^(1/6)) + (27*x*(a + b*x^2)^(5/6))/(8*b^2) - (81*
a^2*x)/(16*b^2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))) - (81*3^(1/4)*Sq
rt[2 + Sqrt[3]]*a^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]*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]])/(32*b^3*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)]) + (27*3^(3/4)*a^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]])/(8*Sqrt[2]*b^3*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 288

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

Rule 321

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

Rule 238

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 198

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 235

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 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*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 219

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

Rule 1879

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]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), 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{x^4}{\left (a+b x^2\right )^{7/6}} \, dx &=-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{9 \int \frac{x^2}{\sqrt [6]{a+b x^2}} \, dx}{b}\\ &=-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac{(27 a) \int \frac{1}{\sqrt [6]{a+b x^2}} \, dx}{8 b^2}\\ &=-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac{\left (27 a^2\right ) \int \frac{1}{\left (a+b x^2\right )^{7/6}} \, dx}{16 b^2}\\ &=-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac{\left (27 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 b^2 \left (\frac{a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac{\left (81 a^2 \sqrt{-\frac{b x^2}{a+b x^2}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{32 b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac{\left (81 a^2 \sqrt{-\frac{b x^2}{a+b x^2}}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{32 b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac{\left (81 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} a^2 \sqrt{-\frac{b x^2}{a+b x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{\frac{a}{a+b x^2}}\right )}{16 b^3 x \left (\frac{a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac{81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac{3 x^3}{b \sqrt [6]{a+b x^2}}+\frac{27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac{81 a^2 \sqrt{-\frac{b x^2}{a+b x^2}} \sqrt{-1+\frac{a}{a+b x^2}}}{16 b^3 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{81 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^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}} 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 )}{32 b^3 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{27\ 3^{3/4} a^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 )}{8 \sqrt{2} b^3 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*}

Mathematica [C]  time = 0.0201148, size = 65, normalized size = 0.1 \[ \frac{3 x \left (9 a \sqrt [6]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};-\frac{b x^2}{a}\right )-9 a+2 b x^2\right )}{16 b^2 \sqrt [6]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^2)^(7/6),x]

[Out]

(3*x*(-9*a + 2*b*x^2 + 9*a*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, -((b*x^2)/a)]))/(16*b^2*(a +
b*x^2)^(1/6))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{6}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2+a)^(7/6),x)

[Out]

int(x^4/(b*x^2+a)^(7/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{7}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(7/6),x, algorithm="maxima")

[Out]

integrate(x^4/(b*x^2 + a)^(7/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{5}{6}} x^{4}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(7/6),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.11077, size = 27, normalized size = 0.04 \begin{align*} \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{6}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{7}{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**2+a)**(7/6),x)

[Out]

x**5*hyper((7/6, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(7/6))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{7}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)^(7/6),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^2 + a)^(7/6), x)