\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^8} \, dx\) [556]

Optimal result

Integrand size = 19, antiderivative size = 246 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {5 a^3 c^4 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]

[Out]

-5/24*a*c^2*d*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^3/(e*x+d)^4-1/6*c*d*(-c*d*x+a*e)*(c*x^2+a)^(5/2)/(a*e
^2+c*d^2)^2/(e*x+d)^6-1/7*e*(c*x^2+a)^(7/2)/(a*e^2+c*d^2)/(e*x+d)^7-5/16*a^3*c^4*d*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)^(9/2)-5/16*a^2*c^3*d*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^4
/(e*x+d)^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {745, 735, 739, 212} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {5 a^3 c^4 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac {5 a^2 c^3 d \sqrt {a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac {5 a c^2 d \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac {c d \left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )} \]

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

(-5*a^2*c^3*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(d + e*x)^2) - (5*a*c^2*d*(a*e - c*d*x)*(a
+ c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4) - (c*d*(a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)^
2*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(7*(c*d^2 + a*e^2)*(d + e*x)^7) - (5*a^3*c^4*d*ArcTanh[(a*e - c*d*x)/(S
qrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))

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 745

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*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {(c d) \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx}{c d^2+a e^2} \\ & = -\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {\left (5 a c^2 d\right ) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )^2} \\ & = -\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {\left (5 a^2 c^3 d\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^3} \\ & = -\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac {\left (5 a^3 c^4 d\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^4} \\ & = -\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {\left (5 a^3 c^4 d\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 )}{16 \left (c d^2+a e^2\right )^4} \\ & = -\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {5 a^3 c^4 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.53 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {\sqrt {a+c x^2} \left (48 \left (c d^2+a e^2\right )^6-232 c d \left (c d^2+a e^2\right )^5 (d+e x)+8 c \left (c d^2+a e^2\right )^4 \left (55 c d^2+18 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right )^3 \left (200 c d^2+197 a e^2\right ) (d+e x)^3+2 c^2 \left (c d^2+a e^2\right )^2 \left (80 c^2 d^4+159 a c d^2 e^2+72 a^2 e^4\right ) (d+e x)^4-c^3 d \left (c d^2+a e^2\right ) \left (8 c^2 d^4+30 a c d^2 e^2+57 a^2 e^4\right ) (d+e x)^5-c^3 \left (8 c^3 d^6+38 a c^2 d^4 e^2+87 a^2 c d^2 e^4-48 a^3 e^6\right ) (d+e x)^6\right )}{336 e^5 \left (c d^2+a e^2\right )^4 (d+e x)^7}+\frac {5 a^3 c^4 d \log (d+e x)}{16 \left (c d^2+a e^2\right )^{9/2}}-\frac {5 a^3 c^4 d \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

-1/336*(Sqrt[a + c*x^2]*(48*(c*d^2 + a*e^2)^6 - 232*c*d*(c*d^2 + a*e^2)^5*(d + e*x) + 8*c*(c*d^2 + a*e^2)^4*(5
5*c*d^2 + 18*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^3*(200*c*d^2 + 197*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2
+ a*e^2)^2*(80*c^2*d^4 + 159*a*c*d^2*e^2 + 72*a^2*e^4)*(d + e*x)^4 - c^3*d*(c*d^2 + a*e^2)*(8*c^2*d^4 + 30*a*
c*d^2*e^2 + 57*a^2*e^4)*(d + e*x)^5 - c^3*(8*c^3*d^6 + 38*a*c^2*d^4*e^2 + 87*a^2*c*d^2*e^4 - 48*a^3*e^6)*(d +
e*x)^6))/(e^5*(c*d^2 + a*e^2)^4*(d + e*x)^7) + (5*a^3*c^4*d*Log[d + e*x])/(16*(c*d^2 + a*e^2)^(9/2)) - (5*a^3*
c^4*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(16*(c*d^2 + a*e^2)^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(15955\) vs. \(2(222)=444\).

Time = 2.74 (sec) , antiderivative size = 15956, normalized size of antiderivative = 64.86

method	result	size
default	\(\text {Expression too large to display}\)	\(15956\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (223) = 446\).

Time = 18.31 (sec) , antiderivative size = 2593, normalized size of antiderivative = 10.54 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

[1/672*(105*(a^3*c^4*d*e^7*x^7 + 7*a^3*c^4*d^2*e^6*x^6 + 21*a^3*c^4*d^3*e^5*x^5 + 35*a^3*c^4*d^4*e^4*x^4 + 35*
a^3*c^4*d^5*e^3*x^3 + 21*a^3*c^4*d^6*e^2*x^2 + 7*a^3*c^4*d^7*e*x + a^3*c^4*d^8)*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*(279*a^3*c^4*d^8*e + 605*a^4*c^3*d^6*e^3 + 526*a^5*c^2*d^4*e^5 + 248*a^6*c*d^
2*e^7 + 48*a^7*e^9 - (8*c^7*d^8*e + 46*a*c^6*d^6*e^3 + 125*a^2*c^5*d^4*e^5 + 39*a^3*c^4*d^2*e^7 - 48*a^4*c^3*e
^9)*x^6 - 7*(8*c^7*d^9 + 46*a*c^6*d^7*e^2 + 125*a^2*c^5*d^5*e^4 + 54*a^3*c^4*d^3*e^6 - 33*a^4*c^3*d*e^8)*x^5 -
(122*a*c^6*d^8*e + 922*a^2*c^5*d^6*e^3 - 241*a^3*c^4*d^4*e^5 - 1185*a^4*c^3*d^2*e^7 - 144*a^5*c^2*e^9)*x^4 -
14*(13*a*c^6*d^9 + 101*a^2*c^5*d^7*e^2 - 101*a^4*c^3*d^3*e^6 - 13*a^5*c^2*d*e^8)*x^3 - (465*a^2*c^5*d^8*e - 11
99*a^3*c^4*d^6*e^3 - 2362*a^4*c^3*d^4*e^5 - 842*a^5*c^2*d^2*e^7 - 144*a^6*c*e^9)*x^2 - 7*(33*a^2*c^5*d^9 - 54*
a^3*c^4*d^7*e^2 - 125*a^4*c^3*d^5*e^4 - 46*a^5*c^2*d^3*e^6 - 8*a^6*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*d^17 + 5*
a*c^4*d^15*e^2 + 10*a^2*c^3*d^13*e^4 + 10*a^3*c^2*d^11*e^6 + 5*a^4*c*d^9*e^8 + a^5*d^7*e^10 + (c^5*d^10*e^7 +
5*a*c^4*d^8*e^9 + 10*a^2*c^3*d^6*e^11 + 10*a^3*c^2*d^4*e^13 + 5*a^4*c*d^2*e^15 + a^5*e^17)*x^7 + 7*(c^5*d^11*e
^6 + 5*a*c^4*d^9*e^8 + 10*a^2*c^3*d^7*e^10 + 10*a^3*c^2*d^5*e^12 + 5*a^4*c*d^3*e^14 + a^5*d*e^16)*x^6 + 21*(c^
5*d^12*e^5 + 5*a*c^4*d^10*e^7 + 10*a^2*c^3*d^8*e^9 + 10*a^3*c^2*d^6*e^11 + 5*a^4*c*d^4*e^13 + a^5*d^2*e^15)*x^
5 + 35*(c^5*d^13*e^4 + 5*a*c^4*d^11*e^6 + 10*a^2*c^3*d^9*e^8 + 10*a^3*c^2*d^7*e^10 + 5*a^4*c*d^5*e^12 + a^5*d^
3*e^14)*x^4 + 35*(c^5*d^14*e^3 + 5*a*c^4*d^12*e^5 + 10*a^2*c^3*d^10*e^7 + 10*a^3*c^2*d^8*e^9 + 5*a^4*c*d^6*e^1
1 + a^5*d^4*e^13)*x^3 + 21*(c^5*d^15*e^2 + 5*a*c^4*d^13*e^4 + 10*a^2*c^3*d^11*e^6 + 10*a^3*c^2*d^9*e^8 + 5*a^4
*c*d^7*e^10 + a^5*d^5*e^12)*x^2 + 7*(c^5*d^16*e + 5*a*c^4*d^14*e^3 + 10*a^2*c^3*d^12*e^5 + 10*a^3*c^2*d^10*e^7
+ 5*a^4*c*d^8*e^9 + a^5*d^6*e^11)*x), -1/336*(105*(a^3*c^4*d*e^7*x^7 + 7*a^3*c^4*d^2*e^6*x^6 + 21*a^3*c^4*d^3
*e^5*x^5 + 35*a^3*c^4*d^4*e^4*x^4 + 35*a^3*c^4*d^5*e^3*x^3 + 21*a^3*c^4*d^6*e^2*x^2 + 7*a^3*c^4*d^7*e*x + a^3*
c^4*d^8)*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)) + (279*a^3*c^4*d^8*e + 605*a^4*c^3*d^6*e^3 + 526*a^5*c^2*d^4*e^5 + 248*a^6*c*d^2*e^7
+ 48*a^7*e^9 - (8*c^7*d^8*e + 46*a*c^6*d^6*e^3 + 125*a^2*c^5*d^4*e^5 + 39*a^3*c^4*d^2*e^7 - 48*a^4*c^3*e^9)*x^
6 - 7*(8*c^7*d^9 + 46*a*c^6*d^7*e^2 + 125*a^2*c^5*d^5*e^4 + 54*a^3*c^4*d^3*e^6 - 33*a^4*c^3*d*e^8)*x^5 - (122*
a*c^6*d^8*e + 922*a^2*c^5*d^6*e^3 - 241*a^3*c^4*d^4*e^5 - 1185*a^4*c^3*d^2*e^7 - 144*a^5*c^2*e^9)*x^4 - 14*(13
*a*c^6*d^9 + 101*a^2*c^5*d^7*e^2 - 101*a^4*c^3*d^3*e^6 - 13*a^5*c^2*d*e^8)*x^3 - (465*a^2*c^5*d^8*e - 1199*a^3
*c^4*d^6*e^3 - 2362*a^4*c^3*d^4*e^5 - 842*a^5*c^2*d^2*e^7 - 144*a^6*c*e^9)*x^2 - 7*(33*a^2*c^5*d^9 - 54*a^3*c^
4*d^7*e^2 - 125*a^4*c^3*d^5*e^4 - 46*a^5*c^2*d^3*e^6 - 8*a^6*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*d^17 + 5*a*c^4*
d^15*e^2 + 10*a^2*c^3*d^13*e^4 + 10*a^3*c^2*d^11*e^6 + 5*a^4*c*d^9*e^8 + a^5*d^7*e^10 + (c^5*d^10*e^7 + 5*a*c^
4*d^8*e^9 + 10*a^2*c^3*d^6*e^11 + 10*a^3*c^2*d^4*e^13 + 5*a^4*c*d^2*e^15 + a^5*e^17)*x^7 + 7*(c^5*d^11*e^6 + 5
*a*c^4*d^9*e^8 + 10*a^2*c^3*d^7*e^10 + 10*a^3*c^2*d^5*e^12 + 5*a^4*c*d^3*e^14 + a^5*d*e^16)*x^6 + 21*(c^5*d^12
*e^5 + 5*a*c^4*d^10*e^7 + 10*a^2*c^3*d^8*e^9 + 10*a^3*c^2*d^6*e^11 + 5*a^4*c*d^4*e^13 + a^5*d^2*e^15)*x^5 + 35
*(c^5*d^13*e^4 + 5*a*c^4*d^11*e^6 + 10*a^2*c^3*d^9*e^8 + 10*a^3*c^2*d^7*e^10 + 5*a^4*c*d^5*e^12 + a^5*d^3*e^14
)*x^4 + 35*(c^5*d^14*e^3 + 5*a*c^4*d^12*e^5 + 10*a^2*c^3*d^10*e^7 + 10*a^3*c^2*d^8*e^9 + 5*a^4*c*d^6*e^11 + a^
5*d^4*e^13)*x^3 + 21*(c^5*d^15*e^2 + 5*a*c^4*d^13*e^4 + 10*a^2*c^3*d^11*e^6 + 10*a^3*c^2*d^9*e^8 + 5*a^4*c*d^7
*e^10 + a^5*d^5*e^12)*x^2 + 7*(c^5*d^16*e + 5*a*c^4*d^14*e^3 + 10*a^2*c^3*d^12*e^5 + 10*a^3*c^2*d^10*e^7 + 5*a
^4*c*d^8*e^9 + a^5*d^6*e^11)*x)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{8}}\, dx \]

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**8,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**8, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^8,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2428 vs. \(2 (223) = 446\).

Time = 0.36 (sec) , antiderivative size = 2428, normalized size of antiderivative = 9.87 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

-5/8*a^3*c^4*d*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^4*d^8 + 4*a*c^3*
d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d^2 - a*e^2)) - 1/168*(105*(sqrt(c)*x - sqrt(
c*x^2 + a))^13*a^3*c^4*d*e^12 - 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*c^(15/2)*d^8*e^5 - 1344*(sqrt(c)*x - sqrt
(c*x^2 + a))^12*a*c^(13/2)*d^6*e^7 - 2016*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^2*c^(11/2)*d^4*e^9 + 21*(sqrt(c)*
x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)*d^2*e^11 - 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*e^13 - 1120*(s
qrt(c)*x - sqrt(c*x^2 + a))^11*c^8*d^9*e^4 - 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*d^7*e^6 - 6720*(sqrt(
c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*d^5*e^8 + 3010*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*d^3*e^10 - 1820*(sq
rt(c)*x - sqrt(c*x^2 + a))^11*a^4*c^4*d*e^12 - 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*d^10*e^3 - 8960*
(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/2)*d^8*e^5 - 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*d^6*
e^7 + 13370*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(11/2)*d^4*e^9 - 9940*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*
c^(9/2)*d^2*e^11 - 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*d^11*e^2 - 8288*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*
c^8*d^9*e^4 - 6272*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*d^7*e^6 + 42588*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3
*c^6*d^5*e^8 - 27370*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^5*d^3*e^10 + 4445*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a
^5*c^4*d*e^12 - 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(19/2)*d^12*e - 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*
c^(17/2)*d^10*e^3 + 9072*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(15/2)*d^8*e^5 + 55832*(sqrt(c)*x - sqrt(c*x^2
+ a))^8*a^3*c^(13/2)*d^6*e^7 - 70210*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(11/2)*d^4*e^9 + 16485*(sqrt(c)*x -
sqrt(c*x^2 + a))^8*a^5*c^(9/2)*d^2*e^11 - 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^6*c^(7/2)*e^13 - 512*(sqrt(c
)*x - sqrt(c*x^2 + a))^7*c^10*d^13 + 2944*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^9*d^11*e^2 + 13248*(sqrt(c)*x -
sqrt(c*x^2 + a))^7*a^2*c^8*d^9*e^4 + 30736*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^7*d^7*e^6 - 100016*(sqrt(c)*x
- sqrt(c*x^2 + a))^7*a^4*c^6*d^5*e^8 + 52500*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^5*d^3*e^10 - 6720*(sqrt(c)
*x - sqrt(c*x^2 + a))^7*a^6*c^4*d*e^12 + 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(19/2)*d^12*e + 1792*(sqrt(c
)*x - sqrt(c*x^2 + a))^6*a^2*c^(17/2)*d^10*e^3 - 9072*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(15/2)*d^8*e^5 - 8
0192*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(13/2)*d^6*e^7 + 82180*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(11/2)
*d^4*e^9 - 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*c^(9/2)*d^2*e^11 - 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a
^2*c^9*d^11*e^2 - 8288*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^8*d^9*e^4 - 11312*(sqrt(c)*x - sqrt(c*x^2 + a))^5
*a^4*c^7*d^7*e^6 + 79128*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^6*d^5*e^8 - 44660*(sqrt(c)*x - sqrt(c*x^2 + a))
^5*a^6*c^5*d^3*e^10 + 5635*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^4*d*e^12 + 2240*(sqrt(c)*x - sqrt(c*x^2 + a))
^4*a^3*c^(17/2)*d^10*e^3 + 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(15/2)*d^8*e^5 + 19488*(sqrt(c)*x - sqrt
(c*x^2 + a))^4*a^5*c^(13/2)*d^6*e^7 - 49252*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(11/2)*d^4*e^9 + 12047*(sqrt
(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(9/2)*d^2*e^11 - 1008*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^8*c^(7/2)*e^13 - 1120
*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^8*d^9*e^4 - 5488*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^7*d^7*e^6 - 1444
8*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^6*d^5*e^8 + 17738*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^5*d^3*e^10 - 2
212*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^8*c^4*d*e^12 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(15/2)*d^8*e^5
+ 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(13/2)*d^6*e^7 + 5026*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(11/2
)*d^4*e^9 - 4620*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(9/2)*d^2*e^11 - 112*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*
c^7*d^7*e^6 - 532*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^6*d^5*e^8 - 1218*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^5*d
^3*e^10 + 567*(sqrt(c)*x - sqrt(c*x^2 + a))*a^9*c^4*d*e^12 + 8*a^7*c^(13/2)*d^6*e^7 + 38*a^8*c^(11/2)*d^4*e^9
+ 87*a^9*c^(9/2)*d^2*e^11 - 48*a^10*c^(7/2)*e^13)/((c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 + 4*a^3
*c*d^2*e^12 + a^4*e^14)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^
7)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^8} \,d x \]

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^8,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^8, x)