3.3.30 \(\int \frac {x^7}{(d+e x^2) (a+c x^4)} \, dx\) [230]

Optimal. Leaf size=118 \[ \frac {x^2}{2 c e}-\frac {a^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 c^{3/2} \left (c d^2+a e^2\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2+a e^2\right )}-\frac {a d \log \left (a+c x^4\right )}{4 c \left (c d^2+a e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 118, 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 = {1266, 1643, 649, 211, 266} \begin {gather*} -\frac {a^{3/2} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}-\frac {a d \log \left (a+c x^4\right )}{4 c \left (a e^2+c d^2\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac {x^2}{2 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/((d + e*x^2)*(a + c*x^4)),x]

[Out]

x^2/(2*c*e) - (a^(3/2)*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(3/2)*(c*d^2 + a*e^2)) - (d^3*Log[d + e*x^2])/(2*
e^2*(c*d^2 + a*e^2)) - (a*d*Log[a + c*x^4])/(4*c*(c*d^2 + a*e^2))

Rule 211

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

Rule 266

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 649

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 1266

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]

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{c e}-\frac {d^3}{e \left (c d^2+a e^2\right ) (d+e x)}-\frac {a (a e+c d x)}{c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 c e}-\frac {d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2+a e^2\right )}-\frac {a \text {Subst}\left (\int \frac {a e+c d x}{a+c x^2} \, dx,x,x^2\right )}{2 c \left (c d^2+a e^2\right )}\\ &=\frac {x^2}{2 c e}-\frac {d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2+a e^2\right )}-\frac {(a d) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )}-\frac {\left (a^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 c \left (c d^2+a e^2\right )}\\ &=\frac {x^2}{2 c e}-\frac {a^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 c^{3/2} \left (c d^2+a e^2\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2+a e^2\right )}-\frac {a d \log \left (a+c x^4\right )}{4 c \left (c d^2+a e^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 99, normalized size = 0.84 \begin {gather*} \frac {-\frac {2 a^{3/2} e^3 \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{c^{3/2}}-2 d^3 \log \left (d+e x^2\right )+\frac {e \left (2 \left (c d^2+a e^2\right ) x^2-a d e \log \left (a+c x^4\right )\right )}{c}}{4 e^2 \left (c d^2+a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/((d + e*x^2)*(a + c*x^4)),x]

[Out]

((-2*a^(3/2)*e^3*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/c^(3/2) - 2*d^3*Log[d + e*x^2] + (e*(2*(c*d^2 + a*e^2)*x^2 - a
*d*e*Log[a + c*x^4]))/c)/(4*e^2*(c*d^2 + a*e^2))

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Maple [A]
time = 0.17, size = 92, normalized size = 0.78

method	result	size
default	\(\frac {x^{2}}{2 c e}-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 e^{2} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {a \left (\frac {d \ln \left (c \,x^{4}+a \right )}{2}+\frac {a e \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) c}\)	\(92\)
risch	\(\frac {x^{2}}{2 c e}+\frac {a \ln \left (\left (-\sqrt {-a c}\, a^{2} e^{5}+3 \sqrt {-a c}\, a c \,d^{2} e^{3}-4 \sqrt {-a c}\, c^{2} d^{4} e +3 a^{2} c d \,e^{4}-3 a \,c^{2} d^{3} e^{2}+2 c^{3} d^{5}\right ) x^{2}-3 \sqrt {-a c}\, a^{2} d \,e^{4}+3 \sqrt {-a c}\, a c \,d^{3} e^{2}-2 \sqrt {-a c}\, c^{2} d^{5}-a^{3} e^{5}+3 d^{2} e^{3} a^{2} c -4 d^{4} e a \,c^{2}\right ) \sqrt {-a c}\, e}{4 c^{2} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {a \ln \left (\left (-\sqrt {-a c}\, a^{2} e^{5}+3 \sqrt {-a c}\, a c \,d^{2} e^{3}-4 \sqrt {-a c}\, c^{2} d^{4} e +3 a^{2} c d \,e^{4}-3 a \,c^{2} d^{3} e^{2}+2 c^{3} d^{5}\right ) x^{2}-3 \sqrt {-a c}\, a^{2} d \,e^{4}+3 \sqrt {-a c}\, a c \,d^{3} e^{2}-2 \sqrt {-a c}\, c^{2} d^{5}-a^{3} e^{5}+3 d^{2} e^{3} a^{2} c -4 d^{4} e a \,c^{2}\right ) d}{4 c \left (a \,e^{2}+c \,d^{2}\right )}-\frac {a \ln \left (\left (\sqrt {-a c}\, a^{2} e^{5}-3 \sqrt {-a c}\, a c \,d^{2} e^{3}+4 \sqrt {-a c}\, c^{2} d^{4} e +3 a^{2} c d \,e^{4}-3 a \,c^{2} d^{3} e^{2}+2 c^{3} d^{5}\right ) x^{2}+3 \sqrt {-a c}\, a^{2} d \,e^{4}-3 \sqrt {-a c}\, a c \,d^{3} e^{2}+2 \sqrt {-a c}\, c^{2} d^{5}-a^{3} e^{5}+3 d^{2} e^{3} a^{2} c -4 d^{4} e a \,c^{2}\right ) \sqrt {-a c}\, e}{4 c^{2} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {a \ln \left (\left (\sqrt {-a c}\, a^{2} e^{5}-3 \sqrt {-a c}\, a c \,d^{2} e^{3}+4 \sqrt {-a c}\, c^{2} d^{4} e +3 a^{2} c d \,e^{4}-3 a \,c^{2} d^{3} e^{2}+2 c^{3} d^{5}\right ) x^{2}+3 \sqrt {-a c}\, a^{2} d \,e^{4}-3 \sqrt {-a c}\, a c \,d^{3} e^{2}+2 \sqrt {-a c}\, c^{2} d^{5}-a^{3} e^{5}+3 d^{2} e^{3} a^{2} c -4 d^{4} e a \,c^{2}\right ) d}{4 c \left (a \,e^{2}+c \,d^{2}\right )}-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 e^{2} \left (a \,e^{2}+c \,d^{2}\right )}\)	\(760\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(e*x^2+d)/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2/c/e-1/2*d^3*ln(e*x^2+d)/e^2/(a*e^2+c*d^2)-1/2*a/(a*e^2+c*d^2)/c*(1/2*d*ln(c*x^4+a)+a*e/(a*c)^(1/2)*arc
tan(c*x^2/(a*c)^(1/2)))

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Maxima [A]
time = 0.49, size = 104, normalized size = 0.88 \begin {gather*} -\frac {d^{3} \log \left (x^{2} e + d\right )}{2 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} - \frac {a^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (c^{2} d^{2} + a c e^{2}\right )} \sqrt {a c}} + \frac {x^{2} e^{\left (-1\right )}}{2 \, c} - \frac {a d \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{2} + a c e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.39, size = 211, normalized size = 1.79 \begin {gather*} \left [\frac {2 \, c d^{2} x^{2} e - 2 \, c d^{3} \log \left (x^{2} e + d\right ) + 2 \, a x^{2} e^{3} - a d e^{2} \log \left (c x^{4} + a\right ) + a \sqrt {-\frac {a}{c}} e^{3} \log \left (\frac {c x^{4} - 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right )}{4 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}, \frac {2 \, c d^{2} x^{2} e - 2 \, c d^{3} \log \left (x^{2} e + d\right ) + 2 \, a x^{2} e^{3} - a d e^{2} \log \left (c x^{4} + a\right ) - 2 \, a \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right ) e^{3}}{4 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(e*x**2+d)/(c*x**4+a),x)

[Out]

Timed out

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Giac [A]
time = 4.39, size = 105, normalized size = 0.89 \begin {gather*} -\frac {d^{3} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{2} e^{2} + a e^{4}\right )}} - \frac {a^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (c^{2} d^{2} + a c e^{2}\right )} \sqrt {a c}} + \frac {x^{2} e^{\left (-1\right )}}{2 \, c} - \frac {a d \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{2} + a c e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.72, size = 166, normalized size = 1.41 \begin {gather*} \frac {x^2}{2\,c\,e}-\frac {d^3\,\ln \left (e\,x^2+d\right )}{2\,c\,d^2\,e^2+2\,a\,e^4}-\frac {\ln \left (\sqrt {-a^3\,c^3}+a\,c^2\,x^2\right )\,\left (e\,\sqrt {-a^3\,c^3}+a\,c^2\,d\right )}{4\,\left (c^4\,d^2+a\,c^3\,e^2\right )}+\frac {\ln \left (\sqrt {-a^3\,c^3}-a\,c^2\,x^2\right )\,\left (e\,\sqrt {-a^3\,c^3}-a\,c^2\,d\right )}{4\,c^4\,d^2+4\,a\,c^3\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/((a + c*x^4)*(d + e*x^2)),x)

[Out]

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

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