\(\int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx\) [326]

Optimal result

Integrand size = 22, antiderivative size = 195 \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d \left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac {d^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \sqrt {c d^2+a e^2}} \]

[Out]

-1/2*d*(-a*e^2+2*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^4-d^4*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)^(1/2)+1/6*(-4*a*e^2+11*c*d^2)*(c*x^2+a)^(1/2)/c^2/e^3-7/6*d*(e*x+d)*(
c*x^2+a)^(1/2)/c/e^3+1/3*(e*x+d)^2*(c*x^2+a)^(1/2)/c/e^3

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1668, 858, 223, 212, 739} \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{2 c^{3/2} e^4}-\frac {d^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4 \sqrt {a e^2+c d^2}}+\frac {\sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 c^2 e^3}-\frac {7 d \sqrt {a+c x^2} (d+e x)}{6 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^3} \]

[In]

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

[Out]

((11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*c^2*e^3) - (7*d*(d + e*x)*Sqrt[a + c*x^2])/(6*c*e^3) + ((d + e*x)^2*
Sqrt[a + c*x^2])/(3*c*e^3) - (d*(2*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2)*e^4) - (d^4
*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^4*Sqrt[c*d^2 + a*e^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 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+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {\int \frac {-2 a d^2 e^2-d e \left (c d^2+4 a e^2\right ) x-e^2 \left (5 c d^2+2 a e^2\right ) x^2-7 c d e^3 x^3}{(d+e x) \sqrt {a+c x^2}} \, dx}{3 c e^4} \\ & = -\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {\int \frac {3 a c d^2 e^5+c d e^4 \left (5 c d^2-a e^2\right ) x+c e^5 \left (11 c d^2-4 a e^2\right ) x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^2 e^7} \\ & = \frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {\int \frac {3 a c^2 d^2 e^7-3 c^2 d e^6 \left (2 c d^2-a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^3 e^9} \\ & = \frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}+\frac {d^4 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^4}-\frac {\left (d \left (2 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c e^4} \\ & = \frac {\left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d^4 \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}-\frac {\left (d \left (2 c d^2-a e^2\right )\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 (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 c^2 e^3}-\frac {7 d (d+e x) \sqrt {a+c x^2}}{6 c e^3}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^3}-\frac {d \left (2 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^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \sqrt {c d^2+a e^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (-4 a e^2+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )}{c^2}-\frac {12 d^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {3 d \left (2 c d^2-a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{6 e^4} \]

[In]

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

[Out]

((e*Sqrt[a + c*x^2]*(-4*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)))/c^2 - (12*d^4*ArcTan[(Sqrt[c]*(d + e*x) - e*
Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/Sqrt[-(c*d^2) - a*e^2] + (3*d*(2*c*d^2 - a*e^2)*Log[-(Sqrt[c]*x) + S
qrt[a + c*x^2]])/c^(3/2))/(6*e^4)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {\left (-2 c \,e^{2} x^{2}+3 c d e x +4 e^{2} a -6 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{2} e^{3}}+\frac {d \left (\frac {\left (e^{2} a -2 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {2 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 e^{3} c}\)	\(222\)
default	\(\frac {\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}}{e}+\frac {d^{2} \sqrt {c \,x^{2}+a}}{e^{3} c}-\frac {d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}-\frac {d \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{e^{2}}-\frac {d^{4} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\)	\(258\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 2.39 (sec) , antiderivative size = 1060, normalized size of antiderivative = 5.44 \[ \int \frac {x^4}{(d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {6 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, -\frac {12 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, \frac {3 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}, -\frac {6 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (2 \, c^{2} d^{5} + a c d^{3} e^{2} - a^{2} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (6 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5} + 2 \, {\left (c^{2} d^{2} e^{3} + a c e^{5}\right )} x^{2} - 3 \, {\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )}}\right ] \]

[In]

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

[Out]

[1/12*(6*sqrt(c*d^2 + a*e^2)*c^2*d^4*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*sq
rt(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)) - 3*(2*c^2*d^5 + a*c*d^3*e^2 - a^2
*d*e^4)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(6*c^2*d^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 +
2*(c^2*d^2*e^3 + a*c*e^5)*x^2 - 3*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^4 + a*c^2*e^6), -1
/12*(12*sqrt(-c*d^2 - a*e^2)*c^2*d^4*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)) + 3*(2*c^2*d^5 + a*c*d^3*e^2 - a^2*d*e^4)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2
+ a)*sqrt(c)*x - a) - 2*(6*c^2*d^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 - 3*(c^2*d^3*
e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^4 + a*c^2*e^6), 1/6*(3*sqrt(c*d^2 + a*e^2)*c^2*d^4*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)) + 3*(2*c^2*d^5 + a*c*d^3*e^2 - a^2*d*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 +
a)) + (6*c^2*d^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 + 2*(c^2*d^2*e^3 + a*c*e^5)*x^2 - 3*(c^2*d^3*e^2 + a*c*d*e^4)*x
)*sqrt(c*x^2 + a))/(c^3*d^2*e^4 + a*c^2*e^6), -1/6*(6*sqrt(-c*d^2 - a*e^2)*c^2*d^4*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)) - 3*(2*c^2*d^5 + a*c*d^3*e^2 - a
^2*d*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (6*c^2*d^4*e + 2*a*c*d^2*e^3 - 4*a^2*e^5 + 2*(c^2*d^2*
e^3 + a*c*e^5)*x^2 - 3*(c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^4 + a*c^2*e^6)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^4/(e*x+d)/(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(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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