3.742 \(\int \frac{1}{x^5 (a+b x^3)^{2/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=215 \[ \frac{\sqrt [3]{a+b x^3} (4 a d+3 b c)}{4 a^2 c^2 x}+\frac{d^2 \log \left (c+d x^3\right )}{6 c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{7/3} (b c-a d)^{2/3}}-\frac{\sqrt [3]{a+b x^3}}{4 a c x^4} \]

[Out]

-(a + b*x^3)^(1/3)/(4*a*c*x^4) + ((3*b*c + 4*a*d)*(a + b*x^3)^(1/3))/(4*a^2*c^2*x) - (d^2*ArcTan[(1 + (2*(b*c
- a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(7/3)*(b*c - a*d)^(2/3)) + (d^2*Log[c + d*x^
3])/(6*c^(7/3)*(b*c - a*d)^(2/3)) - (d^2*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(7/3)*(b
*c - a*d)^(2/3))

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Rubi [A]  time = 0.296604, antiderivative size = 269, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {494, 461, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{a+b x^3} (a d+b c)}{a^2 c^2 x}-\frac{\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac{d^2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac{d^2 \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{6 c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{7/3} (b c-a d)^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^3)^(2/3)*(c + d*x^3)),x]

[Out]

((b*c + a*d)*(a + b*x^3)^(1/3))/(a^2*c^2*x) - (a + b*x^3)^(4/3)/(4*a^2*c*x^4) - (d^2*ArcTan[(c^(1/3) + (2*(b*c
- a*d)^(1/3)*x)/(a + b*x^3)^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(7/3)*(b*c - a*d)^(2/3)) - (d^2*Log[c^(1/3)
- ((b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)])/(3*c^(7/3)*(b*c - a*d)^(2/3)) + (d^2*Log[c^(2/3) + ((b*c - a*d)^(
2/3)*x^2)/(a + b*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)])/(6*c^(7/3)*(b*c - a*d)^(2/3))

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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{1}{x^5 \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-b x^3\right )^2}{x^5 \left (c-(b c-a d) x^3\right )} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{c x^5}+\frac{-b c-a d}{c^2 x^2}+\frac{a^2 d^2 x}{c^2 \left (c-(b c-a d) x^3\right )}\right ) \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{a^2}\\ &=\frac{(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac{\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}+\frac{d^2 \operatorname{Subst}\left (\int \frac{x}{c-(b c-a d) x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{c^2}\\ &=\frac{(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac{\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{c}-\sqrt [3]{b c-a d} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} \sqrt [3]{b c-a d}}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\sqrt [3]{c}-\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} \sqrt [3]{b c-a d}}\\ &=\frac{(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac{\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac{d^2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{2 c^2 \sqrt [3]{b c-a d}}\\ &=\frac{(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac{\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac{d^2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac{d^2 \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{7/3} (b c-a d)^{2/3}}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{7/3} (b c-a d)^{2/3}}\\ &=\frac{(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac{\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac{d^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c^{7/3} (b c-a d)^{2/3}}-\frac{d^2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac{d^2 \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{7/3} (b c-a d)^{2/3}}\\ \end{align*}

Mathematica [C]  time = 1.5319, size = 267, normalized size = 1.24 \[ -\frac{216 d x^6 \left (c+d x^3\right ) (a d-b c) \text{HypergeometricPFQ}\left (\left \{\frac{2}{3},2,2\right \},\left \{1,\frac{8}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )-81 x^3 \left (c+d x^3\right )^2 (b c-a d) \text{HypergeometricPFQ}\left (\left \{\frac{2}{3},2,2,2\right \},\left \{1,1,\frac{8}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )-5 \left (\left (a \left (17 c^2 d x^3-8 c^3+46 c d^2 x^6+9 d^3 x^9\right )+3 b c x^3 \left (-3 c^2+2 c d x^3+9 d^2 x^6\right )\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+2 c \left (a+b x^3\right ) \left (c^2+10 c d x^3+9 d^2 x^6\right )\right )}{120 c^4 x^4 \left (a+b x^3\right )^{5/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^5*(a + b*x^3)^(2/3)*(c + d*x^3)),x]

[Out]

-(-5*(2*c*(a + b*x^3)*(c^2 + 10*c*d*x^3 + 9*d^2*x^6) + (3*b*c*x^3*(-3*c^2 + 2*c*d*x^3 + 9*d^2*x^6) + a*(-8*c^3
+ 17*c^2*d*x^3 + 46*c*d^2*x^6 + 9*d^3*x^9))*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
) + 216*d*(-(b*c) + a*d)*x^6*(c + d*x^3)*HypergeometricPFQ[{2/3, 2, 2}, {1, 8/3}, ((b*c - a*d)*x^3)/(c*(a + b*
x^3))] - 81*(b*c - a*d)*x^3*(c + d*x^3)^2*HypergeometricPFQ[{2/3, 2, 2, 2}, {1, 1, 8/3}, ((b*c - a*d)*x^3)/(c*
(a + b*x^3))])/(120*c^4*x^4*(a + b*x^3)^(5/3))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^3+a)^(2/3)/(d*x^3+c),x)

[Out]

int(1/x^5/(b*x^3+a)^(2/3)/(d*x^3+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)*x^5), 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(1/x^5/(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \left (a + b x^{3}\right )^{\frac{2}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**3+a)**(2/3)/(d*x**3+c),x)

[Out]

Integral(1/(x**5*(a + b*x**3)**(2/3)*(c + d*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)*x^5), x)