\(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^9} \, dx\) [557]

Optimal result

Integrand size = 19, antiderivative size = 332 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=-\frac {5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac {5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac {c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}-\frac {5 a^3 c^4 \left (8 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{128 \left (c d^2+a e^2\right )^{11/2}} \]

[Out]

-5/192*a*c^2*(-a*e^2+8*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^4/(e*x+d)^4-1/48*c*(-a*e^2+8*c*d^2)*(
-c*d*x+a*e)*(c*x^2+a)^(5/2)/(a*e^2+c*d^2)^3/(e*x+d)^6-1/8*e*(c*x^2+a)^(7/2)/(a*e^2+c*d^2)/(e*x+d)^8-9/56*c*d*e
*(c*x^2+a)^(7/2)/(a*e^2+c*d^2)^2/(e*x+d)^7-5/128*a^3*c^4*(-a*e^2+8*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)^(11/2)-5/128*a^2*c^3*(-a*e^2+8*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+
c*d^2)^5/(e*x+d)^2

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {759, 821, 735, 739, 212} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=-\frac {5 a^3 c^4 \left (8 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{128 \left (a e^2+c d^2\right )^{11/2}}-\frac {5 a^2 c^3 \sqrt {a+c x^2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{128 (d+e x)^2 \left (a e^2+c d^2\right )^5}-\frac {5 a c^2 \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{192 (d+e x)^4 \left (a e^2+c d^2\right )^4}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 (d+e x)^7 \left (a e^2+c d^2\right )^2}-\frac {c \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{48 (d+e x)^6 \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{7/2}}{8 (d+e x)^8 \left (a e^2+c d^2\right )} \]

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(-5*a^2*c^3*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(128*(c*d^2 + a*e^2)^5*(d + e*x)^2) - (5*a*c^2*(8
*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(192*(c*d^2 + a*e^2)^4*(d + e*x)^4) - (c*(8*c*d^2 - a*e^2)*(a
*e - c*d*x)*(a + c*x^2)^(5/2))/(48*(c*d^2 + a*e^2)^3*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(8*(c*d^2 + a*e^2)*(
d + e*x)^8) - (9*c*d*e*(a + c*x^2)^(7/2))/(56*(c*d^2 + a*e^2)^2*(d + e*x)^7) - (5*a^3*c^4*(8*c*d^2 - a*e^2)*Ar
cTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(128*(c*d^2 + a*e^2)^(11/2))

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 )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {c \int \frac {(-8 d+e x) \left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx}{8 \left (c d^2+a e^2\right )} \\ & = -\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac {\left (c \left (8 c d^2-a e^2\right )\right ) \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx}{8 \left (c d^2+a e^2\right )^2} \\ & = -\frac {c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac {\left (5 a c^2 \left (8 c d^2-a e^2\right )\right ) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{48 \left (c d^2+a e^2\right )^3} \\ & = -\frac {5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac {c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac {\left (5 a^2 c^3 \left (8 c d^2-a e^2\right )\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{64 \left (c d^2+a e^2\right )^4} \\ & = -\frac {5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac {5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac {c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}+\frac {\left (5 a^3 c^4 \left (8 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{128 \left (c d^2+a e^2\right )^5} \\ & = -\frac {5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac {5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac {c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}-\frac {\left (5 a^3 c^4 \left (8 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 )}{128 \left (c d^2+a e^2\right )^5} \\ & = -\frac {5 a^2 c^3 \left (8 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{128 \left (c d^2+a e^2\right )^5 (d+e x)^2}-\frac {5 a c^2 \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{192 \left (c d^2+a e^2\right )^4 (d+e x)^4}-\frac {c \left (8 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{5/2}}{48 \left (c d^2+a e^2\right )^3 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{8 \left (c d^2+a e^2\right ) (d+e x)^8}-\frac {9 c d e \left (a+c x^2\right )^{7/2}}{56 \left (c d^2+a e^2\right )^2 (d+e x)^7}-\frac {5 a^3 c^4 \left (8 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 )}{128 \left (c d^2+a e^2\right )^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.95 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=\frac {-\frac {\sqrt {a+c x^2} \left (336 \left (c d^2+a e^2\right )^7-1584 c d \left (c d^2+a e^2\right )^6 (d+e x)+8 c \left (c d^2+a e^2\right )^5 \left (362 c d^2+119 a e^2\right ) (d+e x)^2-8 c^2 d \left (c d^2+a e^2\right )^4 \left (310 c d^2+307 a e^2\right ) (d+e x)^3+2 c^2 \left (c d^2+a e^2\right )^3 \left (440 c^2 d^4+880 a c d^2 e^2+413 a^2 e^4\right ) (d+e x)^4-2 c^3 d \left (c d^2+a e^2\right )^2 \left (8 c^2 d^4+32 a c d^2 e^2+87 a^2 e^4\right ) (d+e x)^5-c^3 \left (c d^2+a e^2\right ) \left (16 c^3 d^6+88 a c^2 d^4 e^2+282 a^2 c d^2 e^4-105 a^3 e^6\right ) (d+e x)^6-c^4 d \left (16 c^3 d^6+104 a c^2 d^4 e^2+370 a^2 c d^2 e^4-663 a^3 e^6\right ) (d+e x)^7\right )}{\left (c d^2 e+a e^3\right )^5 (d+e x)^8}+\frac {105 a^3 c^4 \left (8 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{11/2}}+\frac {105 a^3 c^4 \left (-8 c d^2+a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{11/2}}}{2688} \]

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(-((Sqrt[a + c*x^2]*(336*(c*d^2 + a*e^2)^7 - 1584*c*d*(c*d^2 + a*e^2)^6*(d + e*x) + 8*c*(c*d^2 + a*e^2)^5*(362
*c*d^2 + 119*a*e^2)*(d + e*x)^2 - 8*c^2*d*(c*d^2 + a*e^2)^4*(310*c*d^2 + 307*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2
+ a*e^2)^3*(440*c^2*d^4 + 880*a*c*d^2*e^2 + 413*a^2*e^4)*(d + e*x)^4 - 2*c^3*d*(c*d^2 + a*e^2)^2*(8*c^2*d^4 +
32*a*c*d^2*e^2 + 87*a^2*e^4)*(d + e*x)^5 - c^3*(c*d^2 + a*e^2)*(16*c^3*d^6 + 88*a*c^2*d^4*e^2 + 282*a^2*c*d^2
*e^4 - 105*a^3*e^6)*(d + e*x)^6 - c^4*d*(16*c^3*d^6 + 104*a*c^2*d^4*e^2 + 370*a^2*c*d^2*e^4 - 663*a^3*e^6)*(d
+ e*x)^7))/((c*d^2*e + a*e^3)^5*(d + e*x)^8)) + (105*a^3*c^4*(8*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(
11/2) + (105*a^3*c^4*(-8*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2
)^(11/2))/2688

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(31926\) vs. \(2(304)=608\).

Time = 3.10 (sec) , antiderivative size = 31927, normalized size of antiderivative = 96.17

method	result	size
default	\(\text {Expression too large to display}\)	\(31927\)

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^9,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. 1833 vs. \(2 (305) = 610\).

Time = 35.33 (sec) , antiderivative size = 3693, normalized size of antiderivative = 11.12 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

[1/5376*(105*(8*a^3*c^5*d^10 - a^4*c^4*d^8*e^2 + (8*a^3*c^5*d^2*e^8 - a^4*c^4*e^10)*x^8 + 8*(8*a^3*c^5*d^3*e^7
- a^4*c^4*d*e^9)*x^7 + 28*(8*a^3*c^5*d^4*e^6 - a^4*c^4*d^2*e^8)*x^6 + 56*(8*a^3*c^5*d^5*e^5 - a^4*c^4*d^3*e^7
)*x^5 + 70*(8*a^3*c^5*d^6*e^4 - a^4*c^4*d^4*e^6)*x^4 + 56*(8*a^3*c^5*d^7*e^3 - a^4*c^4*d^5*e^5)*x^3 + 28*(8*a^
3*c^5*d^8*e^2 - a^4*c^4*d^6*e^4)*x^2 + 8*(8*a^3*c^5*d^9*e - a^4*c^4*d^7*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*(2616*a^3*c^5*d^10*e + 6481*a^4*c^4*d^8*e^3 + 7443*a^5*c^3*d^6*e^5 + 5298*a^6
*c^2*d^4*e^7 + 2056*a^7*c*d^2*e^9 + 336*a^8*e^11 - (16*c^8*d^9*e^2 + 120*a*c^7*d^7*e^4 + 474*a^2*c^6*d^5*e^6 -
293*a^3*c^5*d^3*e^8 - 663*a^4*c^4*d*e^10)*x^7 - (128*c^8*d^10*e + 960*a*c^7*d^8*e^3 + 3792*a^2*c^6*d^6*e^5 -
1504*a^3*c^5*d^4*e^7 - 4569*a^4*c^4*d^2*e^9 - 105*a^5*c^3*e^11)*x^6 - (448*c^8*d^11 + 3352*a*c^7*d^9*e^2 + 132
12*a^2*c^6*d^7*e^4 - 2141*a^3*c^5*d^5*e^6 - 12905*a^4*c^4*d^3*e^8 - 456*a^5*c^3*d*e^10)*x^5 - (1280*a*c^7*d^10
*e + 12624*a^2*c^6*d^8*e^3 - 15784*a^3*c^5*d^6*e^5 - 32071*a^4*c^4*d^4*e^7 - 5769*a^5*c^3*d^2*e^9 - 826*a^6*c^
2*e^11)*x^4 - (1456*a*c^7*d^11 + 14706*a^2*c^6*d^9*e^2 - 12191*a^3*c^5*d^7*e^4 - 30449*a^4*c^4*d^5*e^6 - 5856*
a^5*c^3*d^3*e^8 - 848*a^6*c^2*d*e^10)*x^3 - (4416*a^2*c^6*d^10*e - 19072*a^3*c^5*d^8*e^3 - 35839*a^4*c^4*d^6*e
^5 - 17595*a^5*c^3*d^4*e^7 - 6196*a^6*c^2*d^2*e^9 - 952*a^7*c*e^11)*x^2 - (1848*a^2*c^6*d^11 - 5535*a^3*c^5*d^
9*e^2 - 11423*a^4*c^4*d^7*e^4 - 5784*a^5*c^3*d^5*e^6 - 2064*a^6*c^2*d^3*e^8 - 320*a^7*c*d*e^10)*x)*sqrt(c*x^2
+ a))/(c^6*d^20 + 6*a*c^5*d^18*e^2 + 15*a^2*c^4*d^16*e^4 + 20*a^3*c^3*d^14*e^6 + 15*a^4*c^2*d^12*e^8 + 6*a^5*c
*d^10*e^10 + a^6*d^8*e^12 + (c^6*d^12*e^8 + 6*a*c^5*d^10*e^10 + 15*a^2*c^4*d^8*e^12 + 20*a^3*c^3*d^6*e^14 + 15
*a^4*c^2*d^4*e^16 + 6*a^5*c*d^2*e^18 + a^6*e^20)*x^8 + 8*(c^6*d^13*e^7 + 6*a*c^5*d^11*e^9 + 15*a^2*c^4*d^9*e^1
1 + 20*a^3*c^3*d^7*e^13 + 15*a^4*c^2*d^5*e^15 + 6*a^5*c*d^3*e^17 + a^6*d*e^19)*x^7 + 28*(c^6*d^14*e^6 + 6*a*c^
5*d^12*e^8 + 15*a^2*c^4*d^10*e^10 + 20*a^3*c^3*d^8*e^12 + 15*a^4*c^2*d^6*e^14 + 6*a^5*c*d^4*e^16 + a^6*d^2*e^1
8)*x^6 + 56*(c^6*d^15*e^5 + 6*a*c^5*d^13*e^7 + 15*a^2*c^4*d^11*e^9 + 20*a^3*c^3*d^9*e^11 + 15*a^4*c^2*d^7*e^13
+ 6*a^5*c*d^5*e^15 + a^6*d^3*e^17)*x^5 + 70*(c^6*d^16*e^4 + 6*a*c^5*d^14*e^6 + 15*a^2*c^4*d^12*e^8 + 20*a^3*c
^3*d^10*e^10 + 15*a^4*c^2*d^8*e^12 + 6*a^5*c*d^6*e^14 + a^6*d^4*e^16)*x^4 + 56*(c^6*d^17*e^3 + 6*a*c^5*d^15*e^
5 + 15*a^2*c^4*d^13*e^7 + 20*a^3*c^3*d^11*e^9 + 15*a^4*c^2*d^9*e^11 + 6*a^5*c*d^7*e^13 + a^6*d^5*e^15)*x^3 + 2
8*(c^6*d^18*e^2 + 6*a*c^5*d^16*e^4 + 15*a^2*c^4*d^14*e^6 + 20*a^3*c^3*d^12*e^8 + 15*a^4*c^2*d^10*e^10 + 6*a^5*
c*d^8*e^12 + a^6*d^6*e^14)*x^2 + 8*(c^6*d^19*e + 6*a*c^5*d^17*e^3 + 15*a^2*c^4*d^15*e^5 + 20*a^3*c^3*d^13*e^7
+ 15*a^4*c^2*d^11*e^9 + 6*a^5*c*d^9*e^11 + a^6*d^7*e^13)*x), -1/2688*(105*(8*a^3*c^5*d^10 - a^4*c^4*d^8*e^2 +
(8*a^3*c^5*d^2*e^8 - a^4*c^4*e^10)*x^8 + 8*(8*a^3*c^5*d^3*e^7 - a^4*c^4*d*e^9)*x^7 + 28*(8*a^3*c^5*d^4*e^6 - a
^4*c^4*d^2*e^8)*x^6 + 56*(8*a^3*c^5*d^5*e^5 - a^4*c^4*d^3*e^7)*x^5 + 70*(8*a^3*c^5*d^6*e^4 - a^4*c^4*d^4*e^6)*
x^4 + 56*(8*a^3*c^5*d^7*e^3 - a^4*c^4*d^5*e^5)*x^3 + 28*(8*a^3*c^5*d^8*e^2 - a^4*c^4*d^6*e^4)*x^2 + 8*(8*a^3*c
^5*d^9*e - a^4*c^4*d^7*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)) + (2616*a^3*c^5*d^10*e + 6481*a^4*c^4*d^8*e^3 + 7443*a^5*c^3*d^
6*e^5 + 5298*a^6*c^2*d^4*e^7 + 2056*a^7*c*d^2*e^9 + 336*a^8*e^11 - (16*c^8*d^9*e^2 + 120*a*c^7*d^7*e^4 + 474*a
^2*c^6*d^5*e^6 - 293*a^3*c^5*d^3*e^8 - 663*a^4*c^4*d*e^10)*x^7 - (128*c^8*d^10*e + 960*a*c^7*d^8*e^3 + 3792*a^
2*c^6*d^6*e^5 - 1504*a^3*c^5*d^4*e^7 - 4569*a^4*c^4*d^2*e^9 - 105*a^5*c^3*e^11)*x^6 - (448*c^8*d^11 + 3352*a*c
^7*d^9*e^2 + 13212*a^2*c^6*d^7*e^4 - 2141*a^3*c^5*d^5*e^6 - 12905*a^4*c^4*d^3*e^8 - 456*a^5*c^3*d*e^10)*x^5 -
(1280*a*c^7*d^10*e + 12624*a^2*c^6*d^8*e^3 - 15784*a^3*c^5*d^6*e^5 - 32071*a^4*c^4*d^4*e^7 - 5769*a^5*c^3*d^2*
e^9 - 826*a^6*c^2*e^11)*x^4 - (1456*a*c^7*d^11 + 14706*a^2*c^6*d^9*e^2 - 12191*a^3*c^5*d^7*e^4 - 30449*a^4*c^4
*d^5*e^6 - 5856*a^5*c^3*d^3*e^8 - 848*a^6*c^2*d*e^10)*x^3 - (4416*a^2*c^6*d^10*e - 19072*a^3*c^5*d^8*e^3 - 358
39*a^4*c^4*d^6*e^5 - 17595*a^5*c^3*d^4*e^7 - 6196*a^6*c^2*d^2*e^9 - 952*a^7*c*e^11)*x^2 - (1848*a^2*c^6*d^11 -
5535*a^3*c^5*d^9*e^2 - 11423*a^4*c^4*d^7*e^4 - 5784*a^5*c^3*d^5*e^6 - 2064*a^6*c^2*d^3*e^8 - 320*a^7*c*d*e^10
)*x)*sqrt(c*x^2 + a))/(c^6*d^20 + 6*a*c^5*d^18*e^2 + 15*a^2*c^4*d^16*e^4 + 20*a^3*c^3*d^14*e^6 + 15*a^4*c^2*d^
12*e^8 + 6*a^5*c*d^10*e^10 + a^6*d^8*e^12 + (c^6*d^12*e^8 + 6*a*c^5*d^10*e^10 + 15*a^2*c^4*d^8*e^12 + 20*a^3*c
^3*d^6*e^14 + 15*a^4*c^2*d^4*e^16 + 6*a^5*c*d^2*e^18 + a^6*e^20)*x^8 + 8*(c^6*d^13*e^7 + 6*a*c^5*d^11*e^9 + 15
*a^2*c^4*d^9*e^11 + 20*a^3*c^3*d^7*e^13 + 15*a^4*c^2*d^5*e^15 + 6*a^5*c*d^3*e^17 + a^6*d*e^19)*x^7 + 28*(c^6*d
^14*e^6 + 6*a*c^5*d^12*e^8 + 15*a^2*c^4*d^10*e^10 + 20*a^3*c^3*d^8*e^12 + 15*a^4*c^2*d^6*e^14 + 6*a^5*c*d^4*e^
16 + a^6*d^2*e^18)*x^6 + 56*(c^6*d^15*e^5 + 6*a*c^5*d^13*e^7 + 15*a^2*c^4*d^11*e^9 + 20*a^3*c^3*d^9*e^11 + 15*
a^4*c^2*d^7*e^13 + 6*a^5*c*d^5*e^15 + a^6*d^3*e^17)*x^5 + 70*(c^6*d^16*e^4 + 6*a*c^5*d^14*e^6 + 15*a^2*c^4*d^1
2*e^8 + 20*a^3*c^3*d^10*e^10 + 15*a^4*c^2*d^8*e^12 + 6*a^5*c*d^6*e^14 + a^6*d^4*e^16)*x^4 + 56*(c^6*d^17*e^3 +
6*a*c^5*d^15*e^5 + 15*a^2*c^4*d^13*e^7 + 20*a^3*c^3*d^11*e^9 + 15*a^4*c^2*d^9*e^11 + 6*a^5*c*d^7*e^13 + a^6*d
^5*e^15)*x^3 + 28*(c^6*d^18*e^2 + 6*a*c^5*d^16*e^4 + 15*a^2*c^4*d^14*e^6 + 20*a^3*c^3*d^12*e^8 + 15*a^4*c^2*d^
10*e^10 + 6*a^5*c*d^8*e^12 + a^6*d^6*e^14)*x^2 + 8*(c^6*d^19*e + 6*a*c^5*d^17*e^3 + 15*a^2*c^4*d^15*e^5 + 20*a
^3*c^3*d^13*e^7 + 15*a^4*c^2*d^11*e^9 + 6*a^5*c*d^9*e^11 + a^6*d^7*e^13)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**9,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^9,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. 3216 vs. \(2 (305) = 610\).

Time = 0.39 (sec) , antiderivative size = 3216, normalized size of antiderivative = 9.69 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

5/64*(8*a^3*c^5*d^2 - a^4*c^4*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))
/((c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10)*sqrt(-c*
d^2 - a*e^2)) - 1/1344*(840*(sqrt(c)*x - sqrt(c*x^2 + a))^15*a^3*c^5*d^2*e^13 - 105*(sqrt(c)*x - sqrt(c*x^2 +
a))^15*a^4*c^4*e^15 + 12600*(sqrt(c)*x - sqrt(c*x^2 + a))^14*a^3*c^(11/2)*d^3*e^12 - 1575*(sqrt(c)*x - sqrt(c*
x^2 + a))^14*a^4*c^(9/2)*d*e^14 - 3584*(sqrt(c)*x - sqrt(c*x^2 + a))^13*c^9*d^10*e^5 - 17920*(sqrt(c)*x - sqrt
(c*x^2 + a))^13*a*c^8*d^8*e^7 - 35840*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^2*c^7*d^6*e^9 + 45920*(sqrt(c)*x - sq
rt(c*x^2 + a))^13*a^3*c^6*d^4*e^11 - 34580*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^4*c^5*d^2*e^13 - 2779*(sqrt(c)*x
- sqrt(c*x^2 + a))^13*a^5*c^4*e^15 - 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^12*c^(19/2)*d^11*e^4 - 44800*(sqrt(c)
*x - sqrt(c*x^2 + a))^12*a*c^(17/2)*d^9*e^6 - 89600*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^2*c^(15/2)*d^7*e^8 + 20
8880*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(13/2)*d^5*e^10 - 165830*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(1
1/2)*d^3*e^12 + 1505*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^5*c^(9/2)*d*e^14 - 14336*(sqrt(c)*x - sqrt(c*x^2 + a))
^11*c^10*d^12*e^3 - 75264*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^9*d^10*e^5 - 161280*(sqrt(c)*x - sqrt(c*x^2 + a
))^11*a^2*c^8*d^8*e^7 + 486528*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^7*d^6*e^9 - 650160*(sqrt(c)*x - sqrt(c*x
^2 + a))^11*a^4*c^6*d^4*e^11 + 46620*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^5*c^5*d^2*e^13 - 6265*(sqrt(c)*x - sqr
t(c*x^2 + a))^11*a^6*c^4*e^15 - 14336*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(21/2)*d^13*e^2 - 57344*(sqrt(c)*x -
sqrt(c*x^2 + a))^10*a*c^(19/2)*d^11*e^4 - 71680*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(17/2)*d^9*e^6 + 916608
*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(15/2)*d^7*e^8 - 1412880*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(13/2)
*d^5*e^10 + 464520*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(11/2)*d^3*e^12 - 15155*(sqrt(c)*x - sqrt(c*x^2 + a)
)^10*a^6*c^(9/2)*d*e^14 - 8192*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^11*d^14*e - 10240*(sqrt(c)*x - sqrt(c*x^2 + a
))^9*a*c^10*d^12*e^3 + 54272*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^9*d^10*e^5 + 877056*(sqrt(c)*x - sqrt(c*x^2
+ a))^9*a^3*c^8*d^8*e^7 - 2366784*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^7*d^6*e^9 + 1325520*(sqrt(c)*x - sqrt
(c*x^2 + a))^9*a^5*c^6*d^4*e^11 - 229040*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^6*c^5*d^2*e^13 - 12355*(sqrt(c)*x -
sqrt(c*x^2 + a))^9*a^7*c^4*e^15 - 2048*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(23/2)*d^15 + 15360*(sqrt(c)*x - sqr
t(c*x^2 + a))^8*a*c^(21/2)*d^13*e^2 + 85248*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(19/2)*d^11*e^4 + 416384*(sq
rt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(17/2)*d^9*e^6 - 2315376*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(15/2)*d^7*e
^8 + 2638440*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^5*c^(13/2)*d^5*e^10 - 755510*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^
6*c^(11/2)*d^3*e^12 + 23205*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^7*c^(9/2)*d*e^14 + 8192*(sqrt(c)*x - sqrt(c*x^2
+ a))^7*a*c^11*d^14*e + 10240*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^10*d^12*e^3 - 54272*(sqrt(c)*x - sqrt(c*x^
2 + a))^7*a^3*c^9*d^10*e^5 - 1334016*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^8*d^8*e^7 + 2796864*(sqrt(c)*x - sq
rt(c*x^2 + a))^7*a^5*c^7*d^6*e^9 - 1851920*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^6*c^6*d^4*e^11 + 142240*(sqrt(c)*
x - sqrt(c*x^2 + a))^7*a^7*c^5*d^2*e^13 - 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^8*c^4*e^15 - 14336*(sqrt(c)*
x - sqrt(c*x^2 + a))^6*a^2*c^(21/2)*d^13*e^2 - 57344*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(19/2)*d^11*e^4 - 1
52320*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(17/2)*d^9*e^6 + 1755264*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(15
/2)*d^7*e^8 - 2173136*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*c^(13/2)*d^5*e^10 + 670040*(sqrt(c)*x - sqrt(c*x^2 +
a))^6*a^7*c^(11/2)*d^3*e^12 - 21973*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^8*c^(9/2)*d*e^14 + 14336*(sqrt(c)*x - s
qrt(c*x^2 + a))^5*a^3*c^10*d^12*e^3 + 75264*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^9*d^10*e^5 + 274176*(sqrt(c)
*x - sqrt(c*x^2 + a))^5*a^5*c^8*d^8*e^7 - 1495424*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^7*d^6*e^9 + 959280*(sq
rt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^6*d^4*e^11 - 176148*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^8*c^5*d^2*e^13 - 6265
*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^9*c^4*e^15 - 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(19/2)*d^11*e^4 - 6
0928*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(17/2)*d^9*e^6 - 255360*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(15/2
)*d^7*e^8 + 779408*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(13/2)*d^5*e^10 - 290066*(sqrt(c)*x - sqrt(c*x^2 + a)
)^4*a^8*c^(11/2)*d^3*e^12 + 17059*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^9*c^(9/2)*d*e^14 + 3584*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*a^5*c^9*d^10*e^5 + 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^8*d^8*e^7 + 118272*(sqrt(c)*x - s
qrt(c*x^2 + a))^3*a^7*c^7*d^6*e^9 - 292544*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^8*c^6*d^4*e^11 + 39676*(sqrt(c)*x
- sqrt(c*x^2 + a))^3*a^9*c^5*d^2*e^13 - 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^10*c^4*e^15 - 1792*(sqrt(c)*x
- sqrt(c*x^2 + a))^2*a^6*c^(17/2)*d^9*e^6 - 11520*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(15/2)*d^7*e^8 - 40608
*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(13/2)*d^5*e^10 + 64616*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^9*c^(11/2)*d^
3*e^12 - 3729*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^10*c^(9/2)*d*e^14 + 256*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^8*
d^8*e^7 + 1664*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^7*d^6*e^9 + 5920*(sqrt(c)*x - sqrt(c*x^2 + a))*a^9*c^6*d^4*
e^11 - 9768*(sqrt(c)*x - sqrt(c*x^2 + a))*a^10*c^5*d^2*e^13 - 105*(sqrt(c)*x - sqrt(c*x^2 + a))*a^11*c^4*e^15
- 16*a^8*c^(15/2)*d^7*e^8 - 104*a^9*c^(13/2)*d^5*e^10 - 370*a^10*c^(11/2)*d^3*e^12 + 663*a^11*c^(9/2)*d*e^14)/
((c^5*d^10*e^6 + 5*a*c^4*d^8*e^8 + 10*a^2*c^3*d^6*e^10 + 10*a^3*c^2*d^4*e^12 + 5*a^4*c*d^2*e^14 + a^5*e^16)*((
sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^8)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^9} \,d x \]

[In]

int((a + c*x^2)^(5/2)/(d + e*x)^9,x)

[Out]

int((a + c*x^2)^(5/2)/(d + e*x)^9, x)