3.232 \(\int \frac {x^3}{(d+e x^2) (a+c x^4)} \, dx\)

Optimal. Leaf size=96 \[ \frac {d \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )}-\frac {d \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {c} \left (a e^2+c d^2\right )} \]

[Out]

-1/2*d*ln(e*x^2+d)/(a*e^2+c*d^2)+1/4*d*ln(c*x^4+a)/(a*e^2+c*d^2)+1/2*e*arctan(x^2*c^(1/2)/a^(1/2))*a^(1/2)/(a*
e^2+c*d^2)/c^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1252, 801, 635, 205, 260} \[ -\frac {d \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac {d \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )}+\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {c} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((d + e*x^2)*(a + c*x^4)),x]

[Out]

(Sqrt[a]*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[c]*(c*d^2 + a*e^2)) - (d*Log[d + e*x^2])/(2*(c*d^2 + a*e^2))
+ (d*Log[a + c*x^4])/(4*(c*d^2 + a*e^2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 66, normalized size = 0.69 \[ \frac {d \log \left (a+c x^4\right )+\frac {2 \sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {c}}-2 d \log \left (d+e x^2\right )}{4 a e^2+4 c d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/((d + e*x^2)*(a + c*x^4)),x]

[Out]

((2*Sqrt[a]*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/Sqrt[c] - 2*d*Log[d + e*x^2] + d*Log[a + c*x^4])/(4*c*d^2 + 4*a*e
^2)

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fricas [A]  time = 0.84, size = 145, normalized size = 1.51 \[ \left [\frac {e \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} + 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right ) + d \log \left (c x^{4} + a\right ) - 2 \, d \log \left (e x^{2} + d\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}}, \frac {2 \, e \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right ) + d \log \left (c x^{4} + a\right ) - 2 \, d \log \left (e x^{2} + d\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x^2+d)/(c*x^4+a),x, algorithm="fricas")

[Out]

[1/4*(e*sqrt(-a/c)*log((c*x^4 + 2*c*x^2*sqrt(-a/c) - a)/(c*x^4 + a)) + d*log(c*x^4 + a) - 2*d*log(e*x^2 + d))/
(c*d^2 + a*e^2), 1/4*(2*e*sqrt(a/c)*arctan(c*x^2*sqrt(a/c)/a) + d*log(c*x^4 + a) - 2*d*log(e*x^2 + d))/(c*d^2
+ a*e^2)]

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giac [A]  time = 0.39, size = 86, normalized size = 0.90 \[ -\frac {d e \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{2} e + a e^{3}\right )}} + \frac {a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {a c}} + \frac {d \log \left (c x^{4} + a\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x^2+d)/(c*x^4+a),x, algorithm="giac")

[Out]

-1/2*d*e*log(abs(x^2*e + d))/(c*d^2*e + a*e^3) + 1/2*a*arctan(c*x^2/sqrt(a*c))*e/((c*d^2 + a*e^2)*sqrt(a*c)) +
1/4*d*log(c*x^4 + a)/(c*d^2 + a*e^2)

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maple [A]  time = 0.01, size = 83, normalized size = 0.86 \[ \frac {a e \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {a c}}+\frac {d \ln \left (c \,x^{4}+a \right )}{4 a \,e^{2}+4 c \,d^{2}}-\frac {d \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(e*x^2+d)/(c*x^4+a),x)

[Out]

1/4*d*ln(c*x^4+a)/(a*e^2+c*d^2)+1/2/(a*e^2+c*d^2)*a*e/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x^2)-1/2*d*ln(e*x^2+d
)/(a*e^2+c*d^2)

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maxima [A]  time = 1.99, size = 82, normalized size = 0.85 \[ \frac {a e \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {a c}} + \frac {d \log \left (c x^{4} + a\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}} - \frac {d \log \left (e x^{2} + d\right )}{2 \, {\left (c d^{2} + a e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x^2+d)/(c*x^4+a),x, algorithm="maxima")

[Out]

1/2*a*e*arctan(c*x^2/sqrt(a*c))/((c*d^2 + a*e^2)*sqrt(a*c)) + 1/4*d*log(c*x^4 + a)/(c*d^2 + a*e^2) - 1/2*d*log
(e*x^2 + d)/(c*d^2 + a*e^2)

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mupad [B]  time = 1.94, size = 944, normalized size = 9.83 \[ \frac {c\,d\,\ln \left (a^4\,e^6-9\,a\,c^3\,d^6-39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}+79\,a^2\,c^2\,d^4\,e^2-42\,c\,d^5\,e\,{\left (-a\,c\right )}^{3/2}+76\,a\,d^3\,e^3\,{\left (-a\,c\right )}^{3/2}+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+76\,a^2\,c^2\,d^3\,e^3\,x^2-42\,a\,c^3\,d^5\,e\,x^2-10\,a^3\,c\,d\,e^5\,x^2+39\,a\,d^2\,e^4\,x^2\,{\left (-a\,c\right )}^{3/2}-79\,c\,d^4\,e^2\,x^2\,{\left (-a\,c\right )}^{3/2}\right )}{4\,c^2\,d^2+4\,a\,c\,e^2}-\frac {d\,\ln \left (e\,x^2+d\right )}{2\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,d\,\ln \left (9\,a\,c^3\,d^6-a^4\,e^6+39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}-79\,a^2\,c^2\,d^4\,e^2+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+42\,a\,c^2\,d^5\,e\,\sqrt {-a\,c}-76\,a^2\,c^2\,d^3\,e^3\,x^2+42\,a\,c^3\,d^5\,e\,x^2+10\,a^3\,c\,d\,e^5\,x^2-76\,a^2\,c\,d^3\,e^3\,\sqrt {-a\,c}+79\,a\,c^2\,d^4\,e^2\,x^2\,\sqrt {-a\,c}-39\,a^2\,c\,d^2\,e^4\,x^2\,\sqrt {-a\,c}\right )}{4\,c^2\,d^2+4\,a\,c\,e^2}-\frac {e\,\ln \left (a^4\,e^6-9\,a\,c^3\,d^6-39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}+79\,a^2\,c^2\,d^4\,e^2-42\,c\,d^5\,e\,{\left (-a\,c\right )}^{3/2}+76\,a\,d^3\,e^3\,{\left (-a\,c\right )}^{3/2}+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+76\,a^2\,c^2\,d^3\,e^3\,x^2-42\,a\,c^3\,d^5\,e\,x^2-10\,a^3\,c\,d\,e^5\,x^2+39\,a\,d^2\,e^4\,x^2\,{\left (-a\,c\right )}^{3/2}-79\,c\,d^4\,e^2\,x^2\,{\left (-a\,c\right )}^{3/2}\right )\,\sqrt {-a\,c}}{4\,c^2\,d^2+4\,a\,c\,e^2}+\frac {e\,\ln \left (9\,a\,c^3\,d^6-a^4\,e^6+39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}-79\,a^2\,c^2\,d^4\,e^2+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+42\,a\,c^2\,d^5\,e\,\sqrt {-a\,c}-76\,a^2\,c^2\,d^3\,e^3\,x^2+42\,a\,c^3\,d^5\,e\,x^2+10\,a^3\,c\,d\,e^5\,x^2-76\,a^2\,c\,d^3\,e^3\,\sqrt {-a\,c}+79\,a\,c^2\,d^4\,e^2\,x^2\,\sqrt {-a\,c}-39\,a^2\,c\,d^2\,e^4\,x^2\,\sqrt {-a\,c}\right )\,\sqrt {-a\,c}}{4\,c^2\,d^2+4\,a\,c\,e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + c*x^4)*(d + e*x^2)),x)

[Out]

(c*d*log(a^4*e^6 - 9*a*c^3*d^6 - 39*a^3*c*d^2*e^4 + a^3*e^6*x^2*(-a*c)^(1/2) - 9*c^3*d^6*x^2*(-a*c)^(1/2) + 79
*a^2*c^2*d^4*e^2 - 42*c*d^5*e*(-a*c)^(3/2) + 76*a*d^3*e^3*(-a*c)^(3/2) + 10*a^3*d*e^5*(-a*c)^(1/2) + 76*a^2*c^
2*d^3*e^3*x^2 - 42*a*c^3*d^5*e*x^2 - 10*a^3*c*d*e^5*x^2 + 39*a*d^2*e^4*x^2*(-a*c)^(3/2) - 79*c*d^4*e^2*x^2*(-a
*c)^(3/2)))/(4*c^2*d^2 + 4*a*c*e^2) - (d*log(d + e*x^2))/(2*(a*e^2 + c*d^2)) + (c*d*log(9*a*c^3*d^6 - a^4*e^6
+ 39*a^3*c*d^2*e^4 + a^3*e^6*x^2*(-a*c)^(1/2) - 9*c^3*d^6*x^2*(-a*c)^(1/2) - 79*a^2*c^2*d^4*e^2 + 10*a^3*d*e^5
*(-a*c)^(1/2) + 42*a*c^2*d^5*e*(-a*c)^(1/2) - 76*a^2*c^2*d^3*e^3*x^2 + 42*a*c^3*d^5*e*x^2 + 10*a^3*c*d*e^5*x^2
- 76*a^2*c*d^3*e^3*(-a*c)^(1/2) + 79*a*c^2*d^4*e^2*x^2*(-a*c)^(1/2) - 39*a^2*c*d^2*e^4*x^2*(-a*c)^(1/2)))/(4*
c^2*d^2 + 4*a*c*e^2) - (e*log(a^4*e^6 - 9*a*c^3*d^6 - 39*a^3*c*d^2*e^4 + a^3*e^6*x^2*(-a*c)^(1/2) - 9*c^3*d^6*
x^2*(-a*c)^(1/2) + 79*a^2*c^2*d^4*e^2 - 42*c*d^5*e*(-a*c)^(3/2) + 76*a*d^3*e^3*(-a*c)^(3/2) + 10*a^3*d*e^5*(-a
*c)^(1/2) + 76*a^2*c^2*d^3*e^3*x^2 - 42*a*c^3*d^5*e*x^2 - 10*a^3*c*d*e^5*x^2 + 39*a*d^2*e^4*x^2*(-a*c)^(3/2) -
79*c*d^4*e^2*x^2*(-a*c)^(3/2))*(-a*c)^(1/2))/(4*c^2*d^2 + 4*a*c*e^2) + (e*log(9*a*c^3*d^6 - a^4*e^6 + 39*a^3*
c*d^2*e^4 + a^3*e^6*x^2*(-a*c)^(1/2) - 9*c^3*d^6*x^2*(-a*c)^(1/2) - 79*a^2*c^2*d^4*e^2 + 10*a^3*d*e^5*(-a*c)^(
1/2) + 42*a*c^2*d^5*e*(-a*c)^(1/2) - 76*a^2*c^2*d^3*e^3*x^2 + 42*a*c^3*d^5*e*x^2 + 10*a^3*c*d*e^5*x^2 - 76*a^2
*c*d^3*e^3*(-a*c)^(1/2) + 79*a*c^2*d^4*e^2*x^2*(-a*c)^(1/2) - 39*a^2*c*d^2*e^4*x^2*(-a*c)^(1/2))*(-a*c)^(1/2))
/(4*c^2*d^2 + 4*a*c*e^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(e*x**2+d)/(c*x**4+a),x)

[Out]

Timed out

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