3.111 \(\int \frac{x^4}{(a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=288 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} (b c-a d)}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} (b c-a d)}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} (b c-a d)}+\frac{c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{2/3} (b c-a d)}-\frac{c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{2/3} (b c-a d)}-\frac{c^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{2/3} (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.15416, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {481, 292, 31, 634, 617, 204, 628} \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} (b c-a d)}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} (b c-a d)}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} (b c-a d)}+\frac{c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{2/3} (b c-a d)}-\frac{c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{2/3} (b c-a d)}-\frac{c^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{2/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

Rule 481

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

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

Mathematica [A]  time = 0.0898973, size = 224, normalized size = 0.78 \[ \frac{-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{2/3}}-\frac{2 c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{2/3}}-\frac{2 \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{2/3}}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 246, normalized size = 0.9 \begin{align*}{\frac{c}{ \left ( 3\,ad-3\,bc \right ) d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{c}{ \left ( 6\,ad-6\,bc \right ) d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{c\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{a}{ \left ( 3\,ad-3\,bc \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a}{ \left ( 6\,ad-6\,bc \right ) b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

________________________________________________________________________________________

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^4/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.67034, size = 590, normalized size = 2.05 \begin{align*} -\frac{2 \, \sqrt{3} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} + \sqrt{3} a}{3 \, a}\right ) - 2 \, \sqrt{3} \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} d x \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} - \sqrt{3} c}{3 \, c}\right ) - \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x^{2} - d x \left (\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}} + c \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) + 2 \, \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x + d \left (\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c - a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 11.714, size = 573, normalized size = 1.99 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} d^{5} - 81 a^{2} b c d^{4} + 81 a b^{2} c^{2} d^{3} - 27 b^{3} c^{3} d^{2}\right ) - c^{2}, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{6} b^{2} d^{8} - 1458 t^{5} a^{5} b^{3} c d^{7} + 3645 t^{5} a^{4} b^{4} c^{2} d^{6} - 4860 t^{5} a^{3} b^{5} c^{3} d^{5} + 3645 t^{5} a^{2} b^{6} c^{4} d^{4} - 1458 t^{5} a b^{7} c^{5} d^{3} + 243 t^{5} b^{8} c^{6} d^{2} + 9 t^{2} a^{5} d^{5} - 18 t^{2} a^{4} b c d^{4} + 9 t^{2} a^{3} b^{2} c^{2} d^{3} + 9 t^{2} a^{2} b^{3} c^{3} d^{2} - 18 t^{2} a b^{4} c^{4} d + 9 t^{2} b^{5} c^{5}}{a^{3} c d^{2} + a b^{2} c^{3}} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} b^{2} d^{3} - 81 a^{2} b^{3} c d^{2} + 81 a b^{4} c^{2} d - 27 b^{5} c^{3}\right ) + a^{2}, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{6} b^{2} d^{8} - 1458 t^{5} a^{5} b^{3} c d^{7} + 3645 t^{5} a^{4} b^{4} c^{2} d^{6} - 4860 t^{5} a^{3} b^{5} c^{3} d^{5} + 3645 t^{5} a^{2} b^{6} c^{4} d^{4} - 1458 t^{5} a b^{7} c^{5} d^{3} + 243 t^{5} b^{8} c^{6} d^{2} + 9 t^{2} a^{5} d^{5} - 18 t^{2} a^{4} b c d^{4} + 9 t^{2} a^{3} b^{2} c^{2} d^{3} + 9 t^{2} a^{2} b^{3} c^{3} d^{2} - 18 t^{2} a b^{4} c^{4} d + 9 t^{2} b^{5} c^{5}}{a^{3} c d^{2} + a b^{2} c^{3}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**3+a)/(d*x**3+c),x)

[Out]

RootSum(_t**3*(27*a**3*d**5 - 81*a**2*b*c*d**4 + 81*a*b**2*c**2*d**3 - 27*b**3*c**3*d**2) - c**2, Lambda(_t, _
t*log(x + (243*_t**5*a**6*b**2*d**8 - 1458*_t**5*a**5*b**3*c*d**7 + 3645*_t**5*a**4*b**4*c**2*d**6 - 4860*_t**
5*a**3*b**5*c**3*d**5 + 3645*_t**5*a**2*b**6*c**4*d**4 - 1458*_t**5*a*b**7*c**5*d**3 + 243*_t**5*b**8*c**6*d**
2 + 9*_t**2*a**5*d**5 - 18*_t**2*a**4*b*c*d**4 + 9*_t**2*a**3*b**2*c**2*d**3 + 9*_t**2*a**2*b**3*c**3*d**2 - 1
8*_t**2*a*b**4*c**4*d + 9*_t**2*b**5*c**5)/(a**3*c*d**2 + a*b**2*c**3)))) + RootSum(_t**3*(27*a**3*b**2*d**3 -
81*a**2*b**3*c*d**2 + 81*a*b**4*c**2*d - 27*b**5*c**3) + a**2, Lambda(_t, _t*log(x + (243*_t**5*a**6*b**2*d**
8 - 1458*_t**5*a**5*b**3*c*d**7 + 3645*_t**5*a**4*b**4*c**2*d**6 - 4860*_t**5*a**3*b**5*c**3*d**5 + 3645*_t**5
*a**2*b**6*c**4*d**4 - 1458*_t**5*a*b**7*c**5*d**3 + 243*_t**5*b**8*c**6*d**2 + 9*_t**2*a**5*d**5 - 18*_t**2*a
**4*b*c*d**4 + 9*_t**2*a**3*b**2*c**2*d**3 + 9*_t**2*a**2*b**3*c**3*d**2 - 18*_t**2*a*b**4*c**4*d + 9*_t**2*b*
*5*c**5)/(a**3*c*d**2 + a*b**2*c**3))))

________________________________________________________________________________________

Giac [A]  time = 1.13683, size = 386, normalized size = 1.34 \begin{align*} \frac{a \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} - \frac{c \left (-\frac{c}{d}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \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{2}{3}} \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{2}{3}} \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{2}{3}} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")

[Out]

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