3.4.42 \(\int \frac {x^7}{(d+e x^2) (a+b x^2+c x^4)^{3/2}} \, dx\) [342]

Optimal. Leaf size=236 \[ \frac {a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x^2}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2} e}-\frac {d^3 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1265, 1660, 857, 635, 212, 738} \begin {gather*} \frac {x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2} e}-\frac {d^3 \tanh ^{-1}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{2 e \left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1265

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

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.97, size = 251, normalized size = 1.06 \begin {gather*} \frac {-b^3 d x^2+a b \left (-b d+3 c d x^2+b e x^2\right )+a^2 \left (b e+2 c \left (d-e x^2\right )\right )}{c \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+b x^2+c x^4}}-\frac {d^3 \sqrt {-c d^2+e (b d-a e)} \tan ^{-1}\left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{e \left (c d^2+e (-b d+a e)\right )^2}-\frac {\log \left (c e \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{2 c^{3/2} e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(695\) vs. \(2(214)=428\).
time = 0.14, size = 696, normalized size = 2.95

method	result	size
elliptic	\(\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}} e}-\frac {2 c \,d^{3} \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {\left (-a c \sqrt {-4 a c +b^{2}}+b^{2} \sqrt {-4 a c +b^{2}}+3 a b c -b^{3}\right ) \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{c^{2} \left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {\left (a c \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-4 a c +b^{2}}+3 a b c -b^{3}\right ) \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{c^{2} \left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\)	\(577\)
default	\(\frac {-\frac {x^{2}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b}{4 c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b^{2} x^{2}}{2 c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b^{3}}{4 c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}}{e}+\frac {d \left (b \,x^{2}+2 a \right )}{e^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {d^{2} \left (2 c \,x^{2}+b \right )}{e^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-\frac {d^{3} \left (\frac {2 c e \ln \left (\frac {\frac {2 a \,e^{2}-2 d e b +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {\frac {a \,e^{2}-d e b +c \,d^{2}}{e^{2}}}}-\frac {2 c \sqrt {\left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \left (x^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}+\frac {2 c \sqrt {\left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (-4 a c +b^{2}\right ) \left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \left (x^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}\right )}{e^{3}}\)	\(696\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^7/((c*x^4 + b*x^2 + a)^(3/2)*(x^2*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (217) = 434\).
time = 38.18, size = 4905, normalized size = 20.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(((b^2*c^3 - 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 -
4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^4 + (a
*b^4 - 4*a^2*b^2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^4 + (b^5 -
2*a*b^3*c - 8*a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*
x^4 + (b^4*c - 4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e)*sqrt(c)*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2
- 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + ((b^2*c^3 - 4*a*c^4)*d^3*x^4 + (b^3*c^2 - 4*a*b*
c^3)*d^3*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^3)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^4 + 8*b*c*d^2*x^2 +
(b^2 + 4*a*c)*d^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*d*x^2 + b*d - (b*x^2 + 2*a)*e)*sqrt(c*d^2 - b*d*e + a*e^2)
+ ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^4 + (3*b^2 + 4*a*c)*d*x^2 + 4*a*b*d)*e)/(x^4*e^2
+ 2*d*x^2*e + d^2)) - 4*sqrt(c*x^4 + b*x^2 + a)*((a^3*b*c + (a^2*b^2*c - 2*a^3*c^2)*x^2)*e^4 - ((2*a*b^3*c - 5
*a^2*b*c^2)*d*x^2 + 2*(a^2*b^2*c - a^3*c^2)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 2*a^2*c^3)*d^2*x^2 + (a*b^3*c - a
^2*b*c^2)*d^2)*e^2 - ((b^3*c^2 - 3*a*b*c^3)*d^3*x^2 + (a*b^2*c^2 - 2*a^2*c^3)*d^3)*e))/((a^3*b^2*c^2 - 4*a^4*c
^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*x^4 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*x^2)*e^5 - 2*((a*b^3*c^3 - 4*a^2*b*c^4)*d*x^4
+ (a*b^4*c^2 - 4*a^2*b^2*c^3)*d*x^2 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*d)*e^4 + ((b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^
5)*d^2*x^4 + (b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^2*x^2 + (a*b^4*c^2 - 2*a^2*b^2*c^3 - 8*a^3*c^4)*d^2)*e^3
- 2*((b^3*c^4 - 4*a*b*c^5)*d^3*x^4 + (b^4*c^3 - 4*a*b^2*c^4)*d^3*x^2 + (a*b^3*c^3 - 4*a^2*b*c^4)*d^3)*e^2 + ((
b^2*c^5 - 4*a*c^6)*d^4*x^4 + (b^3*c^4 - 4*a*b*c^5)*d^4*x^2 + (a*b^2*c^4 - 4*a^2*c^5)*d^4)*e), -1/4*(2*((b^2*c^
3 - 4*a*c^4)*d^3*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^3*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^3)*sqrt(-c*d^2 + b*d*e - a*e^
2)*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*d*x^2 + b*d - (b*x^2 + 2*a)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)/(c^2*d
^2*x^4 + b*c*d^2*x^2 + a*c*d^2 + (a*c*x^4 + a*b*x^2 + a^2)*e^2 - (b*c*d*x^4 + b^2*d*x^2 + a*b*d)*e)) - ((b^2*c
^3 - 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a^3*b^2 - 4*a^4*c + (a^
2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^2)*d*x^4 + (a*b^4 - 4*a^2*
b^2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*x^4 + (b^5 - 2*a*b^3*c -
8*a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*a*b*c^3)*d^3*x^4 + (b^4*c
- 4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e)*sqrt(c)*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x
^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + 4*sqrt(c*x^4 + b*x^2 + a)*((a^3*b*c + (a^2*b^2*c - 2*a^3*c^2)
*x^2)*e^4 - ((2*a*b^3*c - 5*a^2*b*c^2)*d*x^2 + 2*(a^2*b^2*c - a^3*c^2)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 2*a^2*
c^3)*d^2*x^2 + (a*b^3*c - a^2*b*c^2)*d^2)*e^2 - ((b^3*c^2 - 3*a*b*c^3)*d^3*x^2 + (a*b^2*c^2 - 2*a^2*c^3)*d^3)*
e))/((a^3*b^2*c^2 - 4*a^4*c^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*x^4 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*x^2)*e^5 - 2*((a*b
^3*c^3 - 4*a^2*b*c^4)*d*x^4 + (a*b^4*c^2 - 4*a^2*b^2*c^3)*d*x^2 + (a^2*b^3*c^2 - 4*a^3*b*c^3)*d)*e^4 + ((b^4*c
^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*x^4 + (b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^2*x^2 + (a*b^4*c^2 - 2*a^2*b^2
*c^3 - 8*a^3*c^4)*d^2)*e^3 - 2*((b^3*c^4 - 4*a*b*c^5)*d^3*x^4 + (b^4*c^3 - 4*a*b^2*c^4)*d^3*x^2 + (a*b^3*c^3 -
4*a^2*b*c^4)*d^3)*e^2 + ((b^2*c^5 - 4*a*c^6)*d^4*x^4 + (b^3*c^4 - 4*a*b*c^5)*d^4*x^2 + (a*b^2*c^4 - 4*a^2*c^5
)*d^4)*e), -1/4*(2*((b^2*c^3 - 4*a*c^4)*d^4*x^4 + (b^3*c^2 - 4*a*b*c^3)*d^4*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^4
+ (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2)*e^4 - 2*((a*b^3*c - 4*a^2*b*c^
2)*d*x^4 + (a*b^4 - 4*a^2*b^2*c)*d*x^2 + (a^2*b^3 - 4*a^3*b*c)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2
*x^4 + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*x^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2)*e^2 - 2*((b^3*c^2 - 4*
a*b*c^3)*d^3*x^4 + (b^4*c - 4*a*b^2*c^2)*d^3*x^2 + (a*b^3*c - 4*a^2*b*c^2)*d^3)*e)*sqrt(-c)*arctan(1/2*sqrt(c*
x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 + b*c*x^2 + a*c)) - ((b^2*c^3 - 4*a*c^4)*d^3*x^4 + (b^3*c^2 -
4*a*b*c^3)*d^3*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d^3)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^4 + 8*b*c*d^2
*x^2 + (b^2 + 4*a*c)*d^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*d*x^2 + b*d - (b*x^2 + 2*a)*e)*sqrt(c*d^2 - b*d*e +
a*e^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^4 + (3*b^2 + 4*a*c)*d*x^2 + 4*a*b*d)*e)/(x
^4*e^2 + 2*d*x^2*e + d^2)) + 4*sqrt(c*x^4 + b*x^2 + a)*((a^3*b*c + (a^2*b^2*c - 2*a^3*c^2)*x^2)*e^4 - ((2*a*b^
3*c - 5*a^2*b*c^2)*d*x^2 + 2*(a^2*b^2*c - a^3*c^2)*d)*e^3 + ((b^4*c - 2*a*b^2*c^2 - 2*a^2*c^3)*d^2*x^2 + (a*b^
3*c - a^2*b*c^2)*d^2)*e^2 - ((b^3*c^2 - 3*a*b*c^3)*d^3*x^2 + (a*b^2*c^2 - 2*a^2*c^3)*d^3)*e))/((a^3*b^2*c^2 -
4*a^4*c^3 + (a^2*b^2*c^3 - 4*a^3*c^4)*x^4 + (a^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7}}{\left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(e*x**2+d)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x**7/((d + e*x**2)*(a + b*x**2 + c*x**4)**(3/2)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^7}{\left (e\,x^2+d\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x)

[Out]

int(x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)), x)

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