\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^7} \, dx\) [555]

Optimal result

Integrand size = 19, antiderivative size = 203 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {5 a^2 c^2 (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {5 a^3 c^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{7/2}} \]

[Out]

-5/24*a*c*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^2/(e*x+d)^4-1/6*(-c*d*x+a*e)*(c*x^2+a)^(5/2)/(a*e^2+c*d^2
)/(e*x+d)^6-5/16*a^3*c^3*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)-5/16*a^
2*c^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 = 0.07 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {735, 739, 212} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=-\frac {5 a^3 c^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac {5 a^2 c^2 \sqrt {a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 10.50 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=\frac {1}{48} \left (\frac {\sqrt {a+c x^2} \left (-8 a^5 e^5+8 c^5 d^5 x^5-2 a^4 c e^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )+2 a c^4 d^3 x^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )-a^3 c^2 e \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )+a^2 c^3 d x \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^6}+\frac {15 a^3 c^3 \log (d+e x)}{\left (c d^2+a e^2\right )^{7/2}}-\frac {15 a^3 c^3 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{7/2}}\right ) \]

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

((Sqrt[a + c*x^2]*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 2*a^4*c*e^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) + 2*a*c^4*d^3*x^3*
(13*d^2 + 6*d*e*x + 13*e^2*x^2) - a^3*c^2*e*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4
) + a^2*c^3*d*x*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d +
e*x)^6) + (15*a^3*c^3*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^3*c^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*
Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/48

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(15867\) vs. \(2(183)=366\).

Time = 2.79 (sec) , antiderivative size = 15868, normalized size of antiderivative = 78.17

method	result	size
default	\(\text {Expression too large to display}\)	\(15868\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^7,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. 951 vs. \(2 (184) = 368\).

Time = 8.67 (sec) , antiderivative size = 1929, normalized size of antiderivative = 9.50 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

[1/96*(15*(a^3*c^3*e^6*x^6 + 6*a^3*c^3*d*e^5*x^5 + 15*a^3*c^3*d^2*e^4*x^4 + 20*a^3*c^3*d^3*e^3*x^3 + 15*a^3*c^
3*d^4*e^2*x^2 + 6*a^3*c^3*d^5*e*x + a^3*c^3*d^6)*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*(33*a^3*c^3*d^6*e + 59*a^4*c^2*d^4*e^3 + 34*a^5*c*d^2*e^5 + 8*a^6*e^7 - (8*c^6*d^7 + 34*a*c^5*d^5*e^2 + 59*a
^2*c^4*d^3*e^4 + 33*a^3*c^3*d*e^6)*x^5 - 3*(4*a*c^5*d^6*e + 22*a^2*c^4*d^4*e^3 + 7*a^3*c^3*d^2*e^5 - 11*a^4*c^
2*e^7)*x^4 - 2*(13*a*c^5*d^7 + 74*a^2*c^4*d^5*e^2 + 34*a^3*c^3*d^3*e^4 - 27*a^4*c^2*d*e^6)*x^3 - 2*(27*a^2*c^4
*d^6*e - 34*a^3*c^3*d^4*e^3 - 74*a^4*c^2*d^2*e^5 - 13*a^5*c*e^7)*x^2 - 3*(11*a^2*c^4*d^7 - 7*a^3*c^3*d^5*e^2 -
22*a^4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^14 + 4*a*c^3*d^12*e^2 + 6*a^2*c^2*d^10*e^4 + 4
*a^3*c*d^8*e^6 + a^4*d^6*e^8 + (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)*x^6 + 6*(c^4*d^9*e^5 + 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 + 4*a^3*c*d^3*e^11 + a^4*d*e^13)*x^5 + 15*(c^4*
d^10*e^4 + 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 + 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 + 4*a
*c^3*d^9*e^5 + 6*a^2*c^2*d^7*e^7 + 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^4*d^12*e^2 + 4*a*c^3*d^10*e^4 +
6*a^2*c^2*d^8*e^6 + 4*a^3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13*e + 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^
5 + 4*a^3*c*d^7*e^7 + a^4*d^5*e^9)*x), -1/48*(15*(a^3*c^3*e^6*x^6 + 6*a^3*c^3*d*e^5*x^5 + 15*a^3*c^3*d^2*e^4*x
^4 + 20*a^3*c^3*d^3*e^3*x^3 + 15*a^3*c^3*d^4*e^2*x^2 + 6*a^3*c^3*d^5*e*x + a^3*c^3*d^6)*sqrt(-c*d^2 - a*e^2)*a
rctan(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)) + (33*
a^3*c^3*d^6*e + 59*a^4*c^2*d^4*e^3 + 34*a^5*c*d^2*e^5 + 8*a^6*e^7 - (8*c^6*d^7 + 34*a*c^5*d^5*e^2 + 59*a^2*c^4
*d^3*e^4 + 33*a^3*c^3*d*e^6)*x^5 - 3*(4*a*c^5*d^6*e + 22*a^2*c^4*d^4*e^3 + 7*a^3*c^3*d^2*e^5 - 11*a^4*c^2*e^7)
*x^4 - 2*(13*a*c^5*d^7 + 74*a^2*c^4*d^5*e^2 + 34*a^3*c^3*d^3*e^4 - 27*a^4*c^2*d*e^6)*x^3 - 2*(27*a^2*c^4*d^6*e
- 34*a^3*c^3*d^4*e^3 - 74*a^4*c^2*d^2*e^5 - 13*a^5*c*e^7)*x^2 - 3*(11*a^2*c^4*d^7 - 7*a^3*c^3*d^5*e^2 - 22*a^
4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^14 + 4*a*c^3*d^12*e^2 + 6*a^2*c^2*d^10*e^4 + 4*a^3*c
*d^8*e^6 + a^4*d^6*e^8 + (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)*x^
6 + 6*(c^4*d^9*e^5 + 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 + 4*a^3*c*d^3*e^11 + a^4*d*e^13)*x^5 + 15*(c^4*d^10*e
^4 + 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 + 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 + 4*a*c^3*d
^9*e^5 + 6*a^2*c^2*d^7*e^7 + 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^4*d^12*e^2 + 4*a*c^3*d^10*e^4 + 6*a^2
*c^2*d^8*e^6 + 4*a^3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13*e + 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^5 + 4*
a^3*c*d^7*e^7 + a^4*d^5*e^9)*x)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{7}}\, dx \]

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**7,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**7, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,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. 1953 vs. \(2 (184) = 368\).

Time = 0.34 (sec) , antiderivative size = 1953, normalized size of antiderivative = 9.62 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

5/8*a^3*c^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^
4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + a))^11*c^6*d^6*e
^5 + 144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^5*d^4*e^7 + 144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d^2*e^9
+ 33*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*e^11 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(13/2)*d^7*e^4 +
720*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(11/2)*d^5*e^6 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*d^3
*e^8 + 75*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*e^10 + 640*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^
3 + 1840*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*e^7
- 340*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*d^2*e^9 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 + 960*(
sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 + 2160*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 72
0*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 - 2910*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*d^3*
e^8 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*d*e^10 + 768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*e +
576*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d^8*e^3 - 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 56
40*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^5*d^4*e^7 + 1800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d^2*e^9 + 90
*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11 + 256*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 - 1088*(sqrt
(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 - 3744*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 - 33
20*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d^5*e^6 + 5680*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(9/2)*d^3
*e^8 - 330*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*e^10 - 768*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^8*d^10
*e - 576*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^3 + 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^
5 + 7080*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*d^4*e^7 - 2160*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^
9 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*e^11 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2
+ 2160*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*d^7*e^4 + 1080*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/
2)*d^5*e^6 - 4620*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(9/2)*d^3*e^8 + 450*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^
6*c^(7/2)*d*e^10 - 640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d^8*e^3 - 1840*(sqrt(c)*x - sqrt(c*x^2 + a))^3*
a^4*c^6*d^6*e^5 - 2040*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^5*d^4*e^7 + 1640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*
a^6*c^4*d^2*e^9 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^3*e^11 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(
13/2)*d^7*e^4 + 792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*d^5*e^6 + 1104*(sqrt(c)*x - sqrt(c*x^2 + a))^
2*a^6*c^(9/2)*d^3*e^8 - 273*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(7/2)*d*e^10 - 48*(sqrt(c)*x - sqrt(c*x^2 +
a))*a^5*c^6*d^6*e^5 - 168*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d^4*e^7 - 252*(sqrt(c)*x - sqrt(c*x^2 + a))*a^
7*c^4*d^2*e^9 + 33*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^3*e^11 + 8*a^6*c^(11/2)*d^5*e^6 + 26*a^7*c^(9/2)*d^3*e^
8 + 33*a^8*c^(7/2)*d*e^10)/((c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12)*((sqrt(c)*x - sqrt(c
*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^7} \,d x \]

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^7,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^7, x)