3.26 \(\int \frac{a+c x^2}{d-e x^3} \, dx\)

Optimal. Leaf size=134 \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0891273, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1871, 12, 200, 31, 634, 617, 204, 628, 260} \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(d - e*x^3),x]

[Out]

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

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0299887, size = 123, normalized size = 0.92 \[ \frac{a e^{2/3} \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 a e^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \sqrt{3} a e^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )-2 c d^{2/3} \log \left (d-e x^3\right )}{6 d^{2/3} e} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(d - e*x^3),x]

[Out]

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

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Maple [A]  time = 0.041, size = 111, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\ln \left ( e{x}^{3}-d \right ) }{3\,e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(-e*x^3+d),x)

[Out]

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

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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((c*x^2+a)/(-e*x^3+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 6.29991, size = 2820, normalized size = 21.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(-e*x^3+d),x, algorithm="fricas")

[Out]

1/12*(12*sqrt(1/3)*e*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3)
)^(1/3) + 2*c/e)^2*e^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3)
)^(1/3) + 2*c/e)*c*e + 4*c^2)/e^2)*arctan(-1/8*(2*sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*
e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*a*d*e^2 - 2*a*c*d*e)*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^
3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*e^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^
3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*c*e + 4*c^2)/e^2)*sqrt((((1/2)^(1/3)*(I*sq
rt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*d^2*e^2 + 4*a^2*e^2*x^2 -
4*a*c*d*e*x + 4*c^2*d^2 + 2*(a*d*e^2*x - 2*c*d^2*e)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3
*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e))/(a^2*e^2)) - sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3
/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*d^2*e^2 - 8*a*c*d*e*x + 4*c^2*d^2 + 4*(a*d*e^2*x -
c*d^2*e)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e))*
sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*e^
2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)*c*e
+ 4*c^2)/e^2))/(a^3*e)) - 2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3
))^(1/3) + 2*c/e)*e*log(-1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^
3))^(1/3) + 2*c/e)*d*e + a*e*x + c*d) + (((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*
e^2)/(d^2*e^3))^(1/3) + 2*c/e)*e - 6*c)*log(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2
+ a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)^2*d^2*e^2 + a^2*e^2*x^2 - a*c*d*e*x + c^2*d^2 + 1/2*(a*d*e^2*x - 2*c*d^2
*e)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(c^3/e^3 + a^3/(d^2*e) - (c^3*d^2 + a^3*e^2)/(d^2*e^3))^(1/3) + 2*c/e)))/e

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Sympy [A]  time = 0.413526, size = 70, normalized size = 0.52 \begin{align*} - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{3} - 27 t^{2} c d^{2} e^{2} + 9 t c^{2} d^{2} e - a^{3} e^{2} - c^{3} d^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t d e + c d}{a e} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(-e*x**3+d),x)

[Out]

-RootSum(27*_t**3*d**2*e**3 - 27*_t**2*c*d**2*e**2 + 9*_t*c**2*d**2*e - a**3*e**2 - c**3*d**2, Lambda(_t, _t*l
og(x + (-3*_t*d*e + c*d)/(a*e))))

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Giac [A]  time = 1.09519, size = 128, normalized size = 0.96 \begin{align*} -\frac{1}{3} \, c e^{\left (-1\right )} \log \left ({\left | x^{3} e - d \right |}\right ) + \frac{\sqrt{3} a \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-\frac{1}{3}\right )}}{3 \, d^{\frac{2}{3}}} + \frac{a e^{\left (-\frac{1}{3}\right )} \log \left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d^{\frac{2}{3}}} - \frac{a e^{\left (-\frac{1}{3}\right )} \log \left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(-e*x^3+d),x, algorithm="giac")

[Out]

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