3.771 \(\int \frac{(c x)^{19/3}}{(a+b x^2)^{2/3}} \, dx\)

Optimal. Leaf size=198 \[ \frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}+\frac{10 a^3 c^{19/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{27 b^{11/3}}+\frac{20 a^3 c^{19/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{11/3}}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b} \]

[Out]

(10*a^2*c^5*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(27*b^3) - (2*a*c^3*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(9*b^2) + (c*(c
*x)^(16/3)*(a + b*x^2)^(1/3))/(6*b) + (20*a^3*c^(19/3)*ArcTan[(1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2
)^(1/3)))/Sqrt[3]])/(27*Sqrt[3]*b^(11/3)) + (10*a^3*c^(19/3)*Log[b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/
3)])/(27*b^(11/3))

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Rubi [A]  time = 0.337918, antiderivative size = 278, normalized size of antiderivative = 1.4, number of steps used = 12, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {321, 329, 275, 331, 292, 31, 634, 617, 204, 628} \[ \frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}+\frac{20 a^3 c^{19/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}-\frac{10 a^3 c^{19/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{81 b^{11/3}}+\frac{20 a^3 c^{19/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{27 \sqrt{3} b^{11/3}}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(19/3)/(a + b*x^2)^(2/3),x]

[Out]

(10*a^2*c^5*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(27*b^3) - (2*a*c^3*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(9*b^2) + (c*(c
*x)^(16/3)*(a + b*x^2)^(1/3))/(6*b) + (20*a^3*c^(19/3)*ArcTan[(c^(2/3) + (2*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(
1/3))/(Sqrt[3]*c^(2/3))])/(27*Sqrt[3]*b^(11/3)) + (20*a^3*c^(19/3)*Log[c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*
x^2)^(1/3)])/(81*b^(11/3)) - (10*a^3*c^(19/3)*Log[c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)
*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)])/(81*b^(11/3))

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 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 331

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

Rule 292

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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 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]

Rubi steps

\begin{align*} \int \frac{(c x)^{19/3}}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac{\left (8 a c^2\right ) \int \frac{(c x)^{13/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{9 b}\\ &=-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac{\left (20 a^2 c^4\right ) \int \frac{(c x)^{7/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{27 b^2}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac{\left (40 a^3 c^6\right ) \int \frac{\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{81 b^3}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac{\left (40 a^3 c^5\right ) \operatorname{Subst}\left (\int \frac{x^3}{\left (a+\frac{b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{27 b^3}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac{\left (20 a^3 c^5\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a+\frac{b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{27 b^3}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac{\left (20 a^3 c^5\right ) \operatorname{Subst}\left (\int \frac{x}{1-\frac{b x^3}{c^2}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{27 b^3}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac{\left (20 a^3 c^{17/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{\sqrt [3]{b} x}{c^{2/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{10/3}}+\frac{\left (20 a^3 c^{17/3}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt [3]{b} x}{c^{2/3}}}{1+\frac{\sqrt [3]{b} x}{c^{2/3}}+\frac{b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{10/3}}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac{20 a^3 c^{19/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}+\frac{\left (10 a^3 c^{17/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{\sqrt [3]{b} x}{c^{2/3}}+\frac{b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{27 b^{10/3}}-\frac{\left (10 a^3 c^{19/3}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt [3]{b}}{c^{2/3}}+\frac{2 b^{2/3} x}{c^{4/3}}}{1+\frac{\sqrt [3]{b} x}{c^{2/3}}+\frac{b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac{20 a^3 c^{19/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}-\frac{10 a^3 c^{19/3} \log \left (c^{4/3}+\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}-\frac{\left (20 a^3 c^{19/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}\right )}{27 b^{11/3}}\\ &=\frac{10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac{20 a^3 c^{19/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt{3}}\right )}{27 \sqrt{3} b^{11/3}}+\frac{20 a^3 c^{19/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}-\frac{10 a^3 c^{19/3} \log \left (c^{4/3}+\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}\\ \end{align*}

Mathematica [C]  time = 0.0276539, size = 87, normalized size = 0.44 \[ \frac{c^5 (c x)^{4/3} \left (-20 a^3 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{b x^2}{b x^2+a}\right )+8 a^2 b x^2+20 a^3-3 a b^2 x^4+9 b^3 x^6\right )}{54 b^3 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(19/3)/(a + b*x^2)^(2/3),x]

[Out]

(c^5*(c*x)^(4/3)*(20*a^3 + 8*a^2*b*x^2 - 3*a*b^2*x^4 + 9*b^3*x^6 - 20*a^3*Hypergeometric2F1[2/3, 1, 5/3, (b*x^
2)/(a + b*x^2)]))/(54*b^3*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{{\frac{19}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(19/3)/(b*x^2+a)^(2/3),x)

[Out]

int((c*x)^(19/3)/(b*x^2+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{19}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(19/3)/(b*x^2+a)^(2/3),x, algorithm="maxima")

[Out]

integrate((c*x)^(19/3)/(b*x^2 + a)^(2/3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(19/3)/(b*x^2+a)^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(19/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{19}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(19/3)/(b*x^2+a)^(2/3),x, algorithm="giac")

[Out]

integrate((c*x)^(19/3)/(b*x^2 + a)^(2/3), x)