3.109 \(\int \frac{x^6}{(a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=296 \[ -\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{4/3} (b c-a d)}+\frac{x}{b d} \]

[Out]

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

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Rubi [A]  time = 0.266874, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {479, 522, 200, 31, 634, 617, 204, 628} \[ -\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{4/3} (b c-a d)}+\frac{x}{b d} \]

Antiderivative was successfully verified.

[In]

Int[x^6/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(2*n
- 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q) + 1)), x] - Dist[e^(2*n)
/(b*d*(m + n*(p + q) + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m +
n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d
, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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{x^6}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=\frac{x}{b d}-\frac{\int \frac{a c+(b c+a d) x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{b d}\\ &=\frac{x}{b d}+\frac{a^2 \int \frac{1}{a+b x^3} \, dx}{b (b c-a d)}-\frac{c^2 \int \frac{1}{c+d x^3} \, dx}{d (b c-a d)}\\ &=\frac{x}{b d}+\frac{a^{4/3} \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b (b c-a d)}+\frac{a^{4/3} \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b (b c-a d)}-\frac{c^{4/3} \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 d (b c-a d)}-\frac{c^{4/3} \int \frac{2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 d (b c-a d)}\\ &=\frac{x}{b d}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}-\frac{a^{4/3} \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{4/3} (b c-a d)}+\frac{a^{5/3} \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b (b c-a d)}+\frac{c^{4/3} \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 d^{4/3} (b c-a d)}-\frac{c^{5/3} \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 d (b c-a d)}\\ &=\frac{x}{b d}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)}+\frac{a^{4/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{4/3} (b c-a d)}-\frac{c^{4/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{d^{4/3} (b c-a d)}\\ &=\frac{x}{b d}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} (b c-a d)}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{4/3} (b c-a d)}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} (b c-a d)}-\frac{c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{4/3} (b c-a d)}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} (b c-a d)}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{4/3} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.122466, size = 238, normalized size = 0.8 \[ \frac{-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{4/3}}+\frac{2 a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{4/3}}-\frac{2 \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{4/3}}-\frac{6 a x}{b}+\frac{c^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{4/3}}-\frac{2 c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{4/3}}+\frac{2 \sqrt{3} c^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{4/3}}+\frac{6 c x}{d}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 266, normalized size = 0.9 \begin{align*}{\frac{x}{bd}}+{\frac{{c}^{2}}{3\,{d}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{c}^{2}}{6\,{d}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{c}^{2}\sqrt{3}}{3\,{d}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{3\,{b}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}}{6\,{b}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}\sqrt{3}}{3\,{b}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68465, size = 555, normalized size = 1.88 \begin{align*} -\frac{2 \, \sqrt{3} a d \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 2 \, \sqrt{3} b c \left (\frac{c}{d}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} d x \left (\frac{c}{d}\right )^{\frac{2}{3}} - \sqrt{3} c}{3 \, c}\right ) - a d \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - b c \left (\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{c}{d}\right )^{\frac{1}{3}} + \left (\frac{c}{d}\right )^{\frac{2}{3}}\right ) + 2 \, a d \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 2 \, b c \left (\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{c}{d}\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b c - a d\right )} x}{6 \,{\left (b^{2} c d - a b d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 90.2173, size = 452, normalized size = 1.53 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} d^{7} - 81 a^{2} b c d^{6} + 81 a b^{2} c^{2} d^{5} - 27 b^{3} c^{3} d^{4}\right ) - c^{4}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{4} a^{5} b^{4} d^{9} - 243 t^{4} a^{4} b^{5} c d^{8} + 162 t^{4} a^{3} b^{6} c^{2} d^{7} + 162 t^{4} a^{2} b^{7} c^{3} d^{6} - 243 t^{4} a b^{8} c^{4} d^{5} + 81 t^{4} b^{9} c^{5} d^{4} + 3 t a^{6} d^{6} - 3 t a^{5} b c d^{5} - 3 t a b^{5} c^{5} d + 3 t b^{6} c^{6}}{a^{5} c d^{4} + a b^{4} c^{5}} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} b^{4} d^{3} - 81 a^{2} b^{5} c d^{2} + 81 a b^{6} c^{2} d - 27 b^{7} c^{3}\right ) + a^{4}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{4} a^{5} b^{4} d^{9} - 243 t^{4} a^{4} b^{5} c d^{8} + 162 t^{4} a^{3} b^{6} c^{2} d^{7} + 162 t^{4} a^{2} b^{7} c^{3} d^{6} - 243 t^{4} a b^{8} c^{4} d^{5} + 81 t^{4} b^{9} c^{5} d^{4} + 3 t a^{6} d^{6} - 3 t a^{5} b c d^{5} - 3 t a b^{5} c^{5} d + 3 t b^{6} c^{6}}{a^{5} c d^{4} + a b^{4} c^{5}} \right )} \right )\right )} + \frac{x}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**3+a)/(d*x**3+c),x)

[Out]

RootSum(_t**3*(27*a**3*d**7 - 81*a**2*b*c*d**6 + 81*a*b**2*c**2*d**5 - 27*b**3*c**3*d**4) - c**4, Lambda(_t, _
t*log(x + (81*_t**4*a**5*b**4*d**9 - 243*_t**4*a**4*b**5*c*d**8 + 162*_t**4*a**3*b**6*c**2*d**7 + 162*_t**4*a*
*2*b**7*c**3*d**6 - 243*_t**4*a*b**8*c**4*d**5 + 81*_t**4*b**9*c**5*d**4 + 3*_t*a**6*d**6 - 3*_t*a**5*b*c*d**5
- 3*_t*a*b**5*c**5*d + 3*_t*b**6*c**6)/(a**5*c*d**4 + a*b**4*c**5)))) + RootSum(_t**3*(27*a**3*b**4*d**3 - 81
*a**2*b**5*c*d**2 + 81*a*b**6*c**2*d - 27*b**7*c**3) + a**4, Lambda(_t, _t*log(x + (81*_t**4*a**5*b**4*d**9 -
243*_t**4*a**4*b**5*c*d**8 + 162*_t**4*a**3*b**6*c**2*d**7 + 162*_t**4*a**2*b**7*c**3*d**6 - 243*_t**4*a*b**8*
c**4*d**5 + 81*_t**4*b**9*c**5*d**4 + 3*_t*a**6*d**6 - 3*_t*a**5*b*c*d**5 - 3*_t*a*b**5*c**5*d + 3*_t*b**6*c**
6)/(a**5*c*d**4 + a*b**4*c**5)))) + x/(b*d)

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Giac [A]  time = 1.15153, size = 416, normalized size = 1.41 \begin{align*} -\frac{a^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{2} c - a^{2} b d\right )}} + \frac{c^{2} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} d - a c d^{2}\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} a \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{3} c - \sqrt{3} a b^{2} d} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} c \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{2} - \sqrt{3} a d^{3}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} a \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{3} c - a b^{2} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} c \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c d^{2} - a d^{3}\right )}} + \frac{x}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")

[Out]

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