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

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

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Rubi [A]  time = 0.15, antiderivative size = 288, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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 31

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

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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 0.13, size = 224, normalized size = 0.78 \begin {gather*} \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} \end {gather*}

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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.05, size = 244, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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)

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giac [A]  time = 0.25, size = 286, normalized size = 0.99 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 246, normalized size = 0.85 \begin {gather*} \frac {\sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} b}-\frac {a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} b}+\frac {a \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} b}-\frac {\sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} d}+\frac {c \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} d}-\frac {c \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} d} \end {gather*}

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

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maxima [A]  time = 1.27, size = 289, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {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 )}{3 \, {\left (b^{2} c - a b d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {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 )}{3 \, {\left (b c d - a d^{2}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {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^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {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 \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {c \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} \end {gather*}

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]

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

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mupad [B]  time = 9.05, size = 1364, normalized size = 4.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x^3)*(c + d*x^3)),x)

[Out]

log(a*x + b^3*c^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3) + a^2*b*d^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3) - 2*a*b^2*c*d*
(-a^2/(b^2*(a*d - b*c)^3))^(2/3))*(a^2/(27*b^5*c^3 - 27*a^3*b^2*d^3 + 81*a^2*b^3*c*d^2 - 81*a*b^4*c^2*d))^(1/3
) + log(c*x + a^2*d^3*(c^2/(d^2*(a*d - b*c)^3))^(2/3) + b^2*c^2*d*(c^2/(d^2*(a*d - b*c)^3))^(2/3) - 2*a*b*c*d^
2*(c^2/(d^2*(a*d - b*c)^3))^(2/3))*(c^2/(27*a^3*d^5 - 27*b^3*c^3*d^2 + 81*a*b^2*c^2*d^3 - 81*a^2*b*c*d^4))^(1/
3) + (log(((3^(1/2)*1i - 1)^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i - 1)*(54*a^2*b^3*c^2*d^3*x*(a*d -
b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-a^2/(b^2*(a*d - b*c)^3))^(2/3))/4)*(-
a^2/(b^2*(a*d - b*c)^3))^(1/3))/6 - 9*a^2*b^4*c^4*d^2 - 9*a^4*b^2*c^2*d^4 + 9*a*b^5*c^5*d + 9*a^5*b*c*d^5))/36
+ a^2*b*c^2*d*x*(a*d + b*c))*(a^2/(27*b^5*c^3 - 27*a^3*b^2*d^3 + 81*a^2*b^3*c*d^2 - 81*a*b^4*c^2*d))^(1/3)*(3
^(1/2)*1i - 1))/2 - (log(((3^(1/2)*1i + 1)^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i + 1)*(54*a^2*b^3*c
^2*d^3*x*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-a^2/(b^2*(a*d - b*c)^3
))^(2/3))/4)*(-a^2/(b^2*(a*d - b*c)^3))^(1/3))/6 + 9*a^2*b^4*c^4*d^2 + 9*a^4*b^2*c^2*d^4 - 9*a*b^5*c^5*d - 9*a
^5*b*c*d^5))/36 - a^2*b*c^2*d*x*(a*d + b*c))*(a^2/(27*b^5*c^3 - 27*a^3*b^2*d^3 + 81*a^2*b^3*c*d^2 - 81*a*b^4*c
^2*d))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(((3^(1/2)*1i - 1)^2*(c^2/(d^2*(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i - 1)
*(54*a^2*b^3*c^2*d^3*x*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(a*d - b*c)^4*(c^2/(d^2*
(a*d - b*c)^3))^(2/3))/4)*(c^2/(d^2*(a*d - b*c)^3))^(1/3))/6 - 9*a^2*b^4*c^4*d^2 - 9*a^4*b^2*c^2*d^4 + 9*a*b^5
*c^5*d + 9*a^5*b*c*d^5))/36 + a^2*b*c^2*d*x*(a*d + b*c))*(c^2/(27*a^3*d^5 - 27*b^3*c^3*d^2 + 81*a*b^2*c^2*d^3
- 81*a^2*b*c*d^4))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(((3^(1/2)*1i + 1)^2*(c^2/(d^2*(a*d - b*c)^3))^(2/3)*(((3^(
1/2)*1i + 1)*(54*a^2*b^3*c^2*d^3*x*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^
4*(c^2/(d^2*(a*d - b*c)^3))^(2/3))/4)*(c^2/(d^2*(a*d - b*c)^3))^(1/3))/6 + 9*a^2*b^4*c^4*d^2 + 9*a^4*b^2*c^2*d
^4 - 9*a*b^5*c^5*d - 9*a^5*b*c*d^5))/36 - a^2*b*c^2*d*x*(a*d + b*c))*(c^2/(27*a^3*d^5 - 27*b^3*c^3*d^2 + 81*a*
b^2*c^2*d^3 - 81*a^2*b*c*d^4))^(1/3)*(3^(1/2)*1i + 1))/2

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sympy [B]  time = 123.98, size = 573, normalized size = 1.99 \begin {gather*} \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 {gather*}

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))))

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