3.25 \(\int \frac {b x+c x^2}{d+e x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1593, 1871, 12, 292, 31, 634, 617, 204, 628, 260} \[ \frac {b \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 \sqrt [3]{d} e^{2/3}}-\frac {b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac {b \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} \sqrt [3]{d} e^{2/3}}+\frac {c \log \left (d+e x^3\right )}{3 e} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)/(d + e*x^3),x]

[Out]

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

Rule 12

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

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 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]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 122, normalized size = 0.91 \[ \frac {b \sqrt [3]{e} \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 b \sqrt [3]{e} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-2 \sqrt {3} b \sqrt [3]{e} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )+2 c \sqrt [3]{d} \log \left (d+e x^3\right )}{6 \sqrt [3]{d} e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [C]  time = 1.92, size = 1043, normalized size = 7.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*sqrt(1/3)*e*sqrt(((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3
))^(1/3) - 2*c/e)^2*e^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3
))^(1/3) - 2*c/e)*c*e + 4*c^2)/e^2)*arctan(1/8*sqrt(1/3)*((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2)
+ 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*d*e^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^
2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*c*d*e - 8*b^2*e*x + 4*b^2*e*sqrt(-((3*(I*sqrt(3) + 1)*(-1/54
*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*d*e^2*x - 4*b^2*e*x^2 + 4*c^2*d*x
- 4*b*c*d + 2*(2*c*d*e*x - b*d*e)*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)
/(d*e^3))^(1/3) - 2*c/e))/(b^2*e)) + 4*c^2*d)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/5
4*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*e^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/5
4*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*c*e + 4*c^2)/e^2)/b^3) + 2*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*
b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*e*log(1/4*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*
b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*d*e^2 + (3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*
b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*c*d*e + b^2*e*x + c^2*d) - ((3*(I*sqrt(3) + 1)*(-1/
54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)*e + 6*c)*log(-1/4*(3*(I*sqrt(3) +
1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d - b^3*e)/(d*e^3))^(1/3) - 2*c/e)^2*d*e^2*x + b^2*e*x^2 - c
^2*d*x + b*c*d - 1/2*(2*c*d*e*x - b*d*e)*(3*(I*sqrt(3) + 1)*(-1/54*c^3/e^3 + 1/54*b^3/(d*e^2) + 1/54*(c^3*d -
b^3*e)/(d*e^3))^(1/3) - 2*c/e)))/e

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giac [A]  time = 0.18, size = 110, normalized size = 0.82 \[ \frac {1}{3} \, c e^{\left (-1\right )} \log \left ({\left | x^{3} e + d \right |}\right ) + \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {1}{3}}} - \frac {\left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}} b \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 108, normalized size = 0.81 \[ \frac {\sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {d}{e}\right )^{\frac {1}{3}} e}-\frac {b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {d}{e}\right )^{\frac {1}{3}} e}+\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {d}{e}\right )^{\frac {1}{3}} e}+\frac {c \ln \left (e \,x^{3}+d \right )}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.92, size = 145, normalized size = 1.08 \[ \frac {{\left (2 \, c \left (\frac {d}{e}\right )^{\frac {1}{3}} + b\right )} \log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, e \left (\frac {d}{e}\right )^{\frac {1}{3}}} + \frac {{\left (c \left (\frac {d}{e}\right )^{\frac {1}{3}} - b\right )} \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 \, e \left (\frac {d}{e}\right )^{\frac {1}{3}}} - \frac {\sqrt {3} {\left (2 \, c d - {\left (3 \, b \left (\frac {d}{e}\right )^{\frac {2}{3}} + \frac {2 \, c d}{e}\right )} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{9 \, d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*c*(d/e)^(1/3) + b)*log(x^2 - x*(d/e)^(1/3) + (d/e)^(2/3))/(e*(d/e)^(1/3)) + 1/3*(c*(d/e)^(1/3) - b)*log
(x + (d/e)^(1/3))/(e*(d/e)^(1/3)) - 1/9*sqrt(3)*(2*c*d - (3*b*(d/e)^(2/3) + 2*c*d/e)*e)*arctan(1/3*sqrt(3)*(2*
x - (d/e)^(1/3))/(d/e)^(1/3))/(d*e)

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mupad [B]  time = 0.19, size = 158, normalized size = 1.18 \[ \sum _{k=1}^3\ln \left (-\mathrm {root}\left (27\,d\,e^3\,z^3-27\,c\,d\,e^2\,z^2+9\,c^2\,d\,e\,z+b^3\,e-c^3\,d,z,k\right )\,\left (6\,c\,d\,e-\mathrm {root}\left (27\,d\,e^3\,z^3-27\,c\,d\,e^2\,z^2+9\,c^2\,d\,e\,z+b^3\,e-c^3\,d,z,k\right )\,d\,e^2\,9\right )+c^2\,d+b^2\,e\,x\right )\,\mathrm {root}\left (27\,d\,e^3\,z^3-27\,c\,d\,e^2\,z^2+9\,c^2\,d\,e\,z+b^3\,e-c^3\,d,z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)/(d + e*x^3),x)

[Out]

symsum(log(c^2*d - root(27*d*e^3*z^3 - 27*c*d*e^2*z^2 + 9*c^2*d*e*z + b^3*e - c^3*d, z, k)*(6*c*d*e - 9*root(2
7*d*e^3*z^3 - 27*c*d*e^2*z^2 + 9*c^2*d*e*z + b^3*e - c^3*d, z, k)*d*e^2) + b^2*e*x)*root(27*d*e^3*z^3 - 27*c*d
*e^2*z^2 + 9*c^2*d*e*z + b^3*e - c^3*d, z, k), k, 1, 3)

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sympy [A]  time = 0.71, size = 75, normalized size = 0.56 \[ \operatorname {RootSum} {\left (27 t^{3} d e^{3} - 27 t^{2} c d e^{2} + 9 t c^{2} d e + b^{3} e - c^{3} d, \left (t \mapsto t \log {\left (x + \frac {9 t^{2} d e^{2} - 6 t c d e + c^{2} d}{b^{2} e} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*d*e**3 - 27*_t**2*c*d*e**2 + 9*_t*c**2*d*e + b**3*e - c**3*d, Lambda(_t, _t*log(x + (9*_t**2*
d*e**2 - 6*_t*c*d*e + c**2*d)/(b**2*e))))

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