\(\int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx\) [533]

Optimal result

Integrand size = 19, antiderivative size = 206 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=-\frac {c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}-\frac {a c^2 \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}} \]

[Out]

-1/4*e*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)/(e*x+d)^4-5/12*c*d*e*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^2/(e*x+d)^3-1/8*a*c^2*
(-a*e^2+4*c*d^2)*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(7/2)-1/8*c*(-a*e^2+4
*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {759, 821, 735, 739, 212} \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=-\frac {a c^2 \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 (d+e x)^3 \left (a e^2+c d^2\right )^2}-\frac {c \sqrt {a+c x^2} \left (4 c d^2-a e^2\right ) (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

[In]

Int[Sqrt[a + c*x^2]/(d + e*x)^5,x]

[Out]

-1/8*(c*(4*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)^3*(d + e*x)^2) - (e*(a + c*x^2)^(3/2
))/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (5*c*d*e*(a + c*x^2)^(3/2))/(12*(c*d^2 + a*e^2)^2*(d + e*x)^3) - (a*c^2*(
4*c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2))

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 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 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 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(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
] && EqQ[Simplify[m + 2*p + 3], 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {c \int \frac {(-4 d+e x) \sqrt {a+c x^2}}{(d+e x)^4} \, dx}{4 \left (c d^2+a e^2\right )} \\ & = -\frac {e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {\left (c \left (4 c d^2-a e^2\right )\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{4 \left (c d^2+a e^2\right )^2} \\ & = -\frac {c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {\left (a c^2 \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^3} \\ & = -\frac {c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}-\frac {\left (a c^2 \left (4 c d^2-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 )}{8 \left (c d^2+a e^2\right )^3} \\ & = -\frac {c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}-\frac {a c^2 \left (4 c d^2-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 )}{8 \left (c d^2+a e^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.25 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=\frac {\sqrt {c d^2+a e^2} \sqrt {a+c x^2} \left (-6 \left (c d^2+a e^2\right )^3+2 c d \left (c d^2+a e^2\right )^2 (d+e x)+c \left (2 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^2+c^2 d \left (2 c d^2-13 a e^2\right ) (d+e x)^3\right )-3 a c^2 e \left (-4 c d^2+a e^2\right ) (d+e x)^4 \log (d+e x)+3 a c^2 e \left (-4 c d^2+a e^2\right ) (d+e x)^4 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{24 e \left (c d^2+a e^2\right )^{7/2} (d+e x)^4} \]

[In]

Integrate[Sqrt[a + c*x^2]/(d + e*x)^5,x]

[Out]

(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-6*(c*d^2 + a*e^2)^3 + 2*c*d*(c*d^2 + a*e^2)^2*(d + e*x) + c*(2*c*d^2 -
3*a*e^2)*(c*d^2 + a*e^2)*(d + e*x)^2 + c^2*d*(2*c*d^2 - 13*a*e^2)*(d + e*x)^3) - 3*a*c^2*e*(-4*c*d^2 + a*e^2)*
(d + e*x)^4*Log[d + e*x] + 3*a*c^2*e*(-4*c*d^2 + a*e^2)*(d + e*x)^4*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt
[a + c*x^2]])/(24*e*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1990\) vs. \(2(186)=372\).

Time = 2.31 (sec) , antiderivative size = 1991, normalized size of antiderivative = 9.67

method	result	size
default	\(\text {Expression too large to display}\)	\(1991\)

[In]

int((c*x^2+a)^(1/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

1/e^5*(-1/4/(a*e^2+c*d^2)*e^2/(x+d/e)^4*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+5/4*c*d*e/(a*e^2
+c*d^2)*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+c*d*e/(a*e^2+c
*d^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+1/2*c*d*e/(a*e^2
+c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d*e/(a*e^2+c*d^2
)*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^
2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^
2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))
+2*c/(a*e^2+c*d^2)*e^2*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*
(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^
2+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2+c*d^2)*e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*
d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a
*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/
e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))))-1/4*c/(a*e^2+c*d^2)*e^2*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c*(
x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+1/2*c*d*e/(a*e^2+c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+
d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)-c*d*e/(a*e^2+c*d^2)*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2
)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-
(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2
)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2+c*d^2)*e^2*(1/4*(2*c*(x+d/e)-2*c
*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2
)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2+c*d^2)*e
^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)
^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e
^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(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. 729 vs. \(2 (187) = 374\).

Time = 1.50 (sec) , antiderivative size = 1485, normalized size of antiderivative = 7.21 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

[-1/48*(3*(4*a*c^3*d^6 - a^2*c^2*d^4*e^2 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 4*(4*a*c^3*d^3*e^3 - a^2*c^2*
d*e^5)*x^3 + 6*(4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4)*x^2 + 4*(4*a*c^3*d^5*e - a^2*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*(28*a*c^3*d^6*e + 47*a^2*c^2*d^4*e^3 + 25*a^3*c*d^2*e^5 + 6*a
^4*e^7 - (2*c^4*d^5*e^2 - 11*a*c^3*d^3*e^4 - 13*a^2*c^2*d*e^6)*x^3 - (8*c^4*d^6*e - 32*a*c^3*d^4*e^3 - 43*a^2*
c^2*d^2*e^5 - 3*a^3*c*e^7)*x^2 - (12*c^4*d^7 - 25*a*c^3*d^5*e^2 - 41*a^2*c^2*d^3*e^4 - 4*a^3*c*d*e^6)*x)*sqrt(
c*x^2 + a))/(c^4*d^12 + 4*a*c^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 + 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 +
4*a*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 + 4*a*c^3*d^7*e^5 + 6*
a^2*c^2*d^5*e^7 + 4*a^3*c*d^3*e^9 + a^4*d*e^11)*x^3 + 6*(c^4*d^10*e^2 + 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 +
4*a^3*c*d^4*e^8 + a^4*d^2*e^10)*x^2 + 4*(c^4*d^11*e + 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 + 4*a^3*c*d^5*e^7 +
a^4*d^3*e^9)*x), -1/24*(3*(4*a*c^3*d^6 - a^2*c^2*d^4*e^2 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 4*(4*a*c^3*d^
3*e^3 - a^2*c^2*d*e^5)*x^3 + 6*(4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4)*x^2 + 4*(4*a*c^3*d^5*e - a^2*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)) + (28*a*c^3*d^6*e + 47*a^2*c^2*d^4*e^3 + 25*a^3*c*d^2*e^5 + 6*a^4*e^7 - (2*c^4*d^5*e^2 - 11*
a*c^3*d^3*e^4 - 13*a^2*c^2*d*e^6)*x^3 - (8*c^4*d^6*e - 32*a*c^3*d^4*e^3 - 43*a^2*c^2*d^2*e^5 - 3*a^3*c*e^7)*x^
2 - (12*c^4*d^7 - 25*a*c^3*d^5*e^2 - 41*a^2*c^2*d^3*e^4 - 4*a^3*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^12 + 4*a*c
^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 + 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 + 4*a*c^3*d^6*e^6 + 6*a^2*c^2*d
^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 + 4*a*c^3*d^7*e^5 + 6*a^2*c^2*d^5*e^7 + 4*a^3*c*d^3
*e^9 + a^4*d*e^11)*x^3 + 6*(c^4*d^10*e^2 + 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 + 4*a^3*c*d^4*e^8 + a^4*d^2*e^1
0)*x^2 + 4*(c^4*d^11*e + 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 + 4*a^3*c*d^5*e^7 + a^4*d^3*e^9)*x)]

Sympy [F]

\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=\int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \]

[In]

integrate((c*x**2+a)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**5, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^5,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]

\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{5}} \,d x } \]

[In]

integrate((c*x^2+a)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx=\int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^5} \,d x \]

[In]

int((a + c*x^2)^(1/2)/(d + e*x)^5,x)

[Out]

int((a + c*x^2)^(1/2)/(d + e*x)^5, x)