\(\int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [343]

Optimal result

Integrand size = 22, antiderivative size = 204 \[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=-\frac {5 d \sqrt {a+c x^2}}{2 c e^3}-\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}+\frac {\left (6 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac {d^3 \left (3 c d^2+4 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

1/2*(-a*e^2+6*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^4+d^3*(4*a*e^2+3*c*d^2)*arctanh((-c*d*x+a*e)
/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^4/(a*e^2+c*d^2)^(3/2)-5/2*d*(c*x^2+a)^(1/2)/c/e^3-d^4*(c*x^2+a)^(1/2)/
e^3/(a*e^2+c*d^2)/(e*x+d)+1/2*(e*x+d)*(c*x^2+a)^(1/2)/c/e^3

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1665, 1668, 858, 223, 212, 739} \[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (6 c d^2-a e^2\right )}{2 c^{3/2} e^4}+\frac {d^3 \left (4 a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac {d^4 \sqrt {a+c x^2}}{e^3 (d+e x) \left (a e^2+c d^2\right )}-\frac {5 d \sqrt {a+c x^2}}{2 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)}{2 c e^3} \]

[In]

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

[Out]

(-5*d*Sqrt[a + c*x^2])/(2*c*e^3) - (d^4*Sqrt[a + c*x^2])/(e^3*(c*d^2 + a*e^2)*(d + e*x)) + ((d + e*x)*Sqrt[a +
c*x^2])/(2*c*e^3) + ((6*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2)*e^4) + (d^3*(3*c*d^2
+ 4*a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^4*(c*d^2 + 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 223

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

Rule 739

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

Rule 858

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

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {\frac {a d^3}{e^2}-\frac {d^2 \left (c d^2+a e^2\right ) x}{e^3}+d \left (a+\frac {c d^2}{e^2}\right ) x^2-\frac {\left (c d^2+a e^2\right ) x^3}{e}}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2} \\ & = -\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}-\frac {\int \frac {a d e \left (3 c d^2+a e^2\right )-\left (c^2 d^4-a^2 e^4\right ) x+5 c d e \left (c d^2+a e^2\right ) x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c e^3 \left (c d^2+a e^2\right )} \\ & = -\frac {5 d \sqrt {a+c x^2}}{2 c e^3}-\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}-\frac {\int \frac {a c d e^3 \left (3 c d^2+a e^2\right )-c e^2 \left (6 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c^2 e^5 \left (c d^2+a e^2\right )} \\ & = -\frac {5 d \sqrt {a+c x^2}}{2 c e^3}-\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}+\frac {\left (6 c d^2-a e^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c e^4}-\frac {\left (d^3 \left (3 c d^2+4 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^4 \left (c d^2+a e^2\right )} \\ & = -\frac {5 d \sqrt {a+c x^2}}{2 c e^3}-\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}+\frac {\left (6 c d^2-a e^2\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c e^4}+\frac {\left (d^3 \left (3 c d^2+4 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^4 \left (c d^2+a e^2\right )} \\ & = -\frac {5 d \sqrt {a+c x^2}}{2 c e^3}-\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}+\frac {\left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac {d^3 \left (3 c d^2+4 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \left (c d^2+a e^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.46 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09 \[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (c d^2 \left (-6 d^2-3 d e x+e^2 x^2\right )+a e^2 \left (-4 d^2-3 d e x+e^2 x^2\right )\right )}{c \left (c d^2+a e^2\right ) (d+e x)}-\frac {4 d^3 \left (3 c d^2+4 a e^2\right ) \arctan \left (\frac {\sqrt {-c d^2-a e^2} x}{\sqrt {a} (d+e x)-d \sqrt {a+c x^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 \left (6 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+c x^2}}\right )}{c^{3/2}}}{2 e^4} \]

[In]

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

[Out]

((e*Sqrt[a + c*x^2]*(c*d^2*(-6*d^2 - 3*d*e*x + e^2*x^2) + a*e^2*(-4*d^2 - 3*d*e*x + e^2*x^2)))/(c*(c*d^2 + a*e
^2)*(d + e*x)) - (4*d^3*(3*c*d^2 + 4*a*e^2)*ArcTan[(Sqrt[-(c*d^2) - a*e^2]*x)/(Sqrt[a]*(d + e*x) - d*Sqrt[a +
c*x^2])])/(-(c*d^2) - a*e^2)^(3/2) + (2*(6*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + c*x^2])])/c
^(3/2))/(2*e^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(180)=360\).

Time = 0.45 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.06

method	result	size
risch	\(-\frac {\left (-e x +4 d \right ) \sqrt {c \,x^{2}+a}}{2 c \,e^{3}}-\frac {\frac {\left (e^{2} a -6 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {8 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {2 c \,d^{4} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}}{2 c \,e^{3}}\)	\(420\)
default	\(\frac {\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}}{e^{2}}+\frac {3 d^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}-\frac {2 d \sqrt {c \,x^{2}+a}}{c \,e^{3}}+\frac {d^{4} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{6}}+\frac {4 d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\)	\(435\)

[In]

int(x^4/(e*x+d)^2/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-e*x+4*d)*(c*x^2+a)^(1/2)/c/e^3-1/2/c/e^3*((a*e^2-6*c*d^2)/e*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-8*c*d
^3/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2/e*c*d*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^
2*c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))-2*c*d^4/e^3*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c-
2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-e*c*d/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2
-2/e*c*d*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (181) = 362\).

Time = 46.20 (sec) , antiderivative size = 1786, normalized size of antiderivative = 8.75 \[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]

[In]

integrate(x^4/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/4*((6*c^3*d^7 + 11*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - a^3*d*e^6 + (6*c^3*d^6*e + 11*a*c^2*d^4*e^3 + 4*a^2*c
*d^2*e^5 - a^3*e^7)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(3*c^3*d^6 + 4*a*c^2*d^4*e^
2 + (3*c^3*d^5*e + 4*a*c^2*d^3*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2
+ a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*c^3*d
^6*e + 10*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - (c^3*d^4*e^3 + 2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^2 + 3*(c^3*d^5*e^2 +
2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8 + (c^4*d^4*
e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x), 1/4*(4*(3*c^3*d^6 + 4*a*c^2*d^4*e^2 + (3*c^3*d^5*e + 4*a*c^2*d^3*e^3)
*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d
^2 + a*c*e^2)*x^2)) - (6*c^3*d^7 + 11*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - a^3*d*e^6 + (6*c^3*d^6*e + 11*a*c^2*d^
4*e^3 + 4*a^2*c*d^2*e^5 - a^3*e^7)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(6*c^3*d^6*e
+ 10*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - (c^3*d^4*e^3 + 2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^2 + 3*(c^3*d^5*e^2 + 2*a
*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8 + (c^4*d^4*e^5
+ 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x), -1/2*((6*c^3*d^7 + 11*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - a^3*d*e^6 + (6*c^
3*d^6*e + 11*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - a^3*e^7)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (3*c^
3*d^6 + 4*a*c^2*d^4*e^2 + (3*c^3*d^5*e + 4*a*c^2*d^3*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 -
2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*
x + d^2)) + (6*c^3*d^6*e + 10*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - (c^3*d^4*e^3 + 2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^
2 + 3*(c^3*d^5*e^2 + 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c
^2*d*e^8 + (c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x), 1/2*(2*(3*c^3*d^6 + 4*a*c^2*d^4*e^2 + (3*c^3*d^5*
e + 4*a*c^2*d^3*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^
2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - (6*c^3*d^7 + 11*a*c^2*d^5*e^2 + 4*a^2*c*d^3*e^4 - a^3*d*e^6 + (6*c^3
*d^6*e + 11*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - a^3*e^7)*x)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (6*c^3
*d^6*e + 10*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - (c^3*d^4*e^3 + 2*a*c^2*d^2*e^5 + a^2*c*e^7)*x^2 + 3*(c^3*d^5*e^2
+ 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8 + (c^4*d^
4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x)]

Sympy [F]

\[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^{4}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]

[In]

integrate(x**4/(e*x+d)**2/(c*x**2+a)**(1/2),x)

[Out]

Integral(x**4/(sqrt(a + c*x**2)*(d + e*x)**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {x^{4}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \]

[In]

integrate(x^4/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^4}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

int(x^4/((a + c*x^2)^(1/2)*(d + e*x)^2),x)

[Out]

int(x^4/((a + c*x^2)^(1/2)*(d + e*x)^2), x)