Integrand size = 19, antiderivative size = 314 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=-\frac {c^2 \left (d \left (8 c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+e \left (12 c^2 d^4+23 a c d^2 e^2+8 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{8 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c \left (d \left (4 c d^2+a e^2\right )+e \left (7 c d^2+4 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac {c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^6}+\frac {c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 e^6 \left (c d^2+a e^2\right )^{5/2}} \]
-1/12*c*(d*(a*e^2+4*c*d^2)+e*(4*a*e^2+7*c*d^2)*x)*(c*x^2+a)^(3/2)/e^3/(a*e ^2+c*d^2)/(e*x+d)^4-1/5*(c*x^2+a)^(5/2)/e/(e*x+d)^5+c^(5/2)*arctanh(x*c^(1 /2)/(c*x^2+a)^(1/2))/e^6+1/8*c^3*d*(15*a^2*e^4+20*a*c*d^2*e^2+8*c^2*d^4)*a rctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/e^6/(a*e^2+c*d^2) ^(5/2)-1/8*c^2*(d*(a^2*e^4+12*a*c*d^2*e^2+8*c^2*d^4)+e*(8*a^2*e^4+23*a*c*d ^2*e^2+12*c^2*d^4)*x)*(c*x^2+a)^(1/2)/e^5/(a*e^2+c*d^2)^2/(e*x+d)^2
Time = 10.53 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {-\frac {e \sqrt {a+c x^2} \left (24 \left (c d^2+a e^2\right )^4-126 c d \left (c d^2+a e^2\right )^3 (d+e x)+2 c \left (c d^2+a e^2\right )^2 \left (137 c d^2+44 a e^2\right ) (d+e x)^2-c^2 d \left (c d^2+a e^2\right ) \left (326 c d^2+311 a e^2\right ) (d+e x)^3+c^2 \left (274 c^2 d^4+503 a c d^2 e^2+184 a^2 e^4\right ) (d+e x)^4\right )}{\left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {15 c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{5/2}}+120 c^{5/2} \log \left (c x+\sqrt {c} \sqrt {a+c x^2}\right )+\frac {15 c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{5/2}}}{120 e^6} \]
(-((e*Sqrt[a + c*x^2]*(24*(c*d^2 + a*e^2)^4 - 126*c*d*(c*d^2 + a*e^2)^3*(d + e*x) + 2*c*(c*d^2 + a*e^2)^2*(137*c*d^2 + 44*a*e^2)*(d + e*x)^2 - c^2*d *(c*d^2 + a*e^2)*(326*c*d^2 + 311*a*e^2)*(d + e*x)^3 + c^2*(274*c^2*d^4 + 503*a*c*d^2*e^2 + 184*a^2*e^4)*(d + e*x)^4))/((c*d^2 + a*e^2)^2*(d + e*x)^ 5)) - (15*c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[d + e*x])/(c *d^2 + a*e^2)^(5/2) + 120*c^(5/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15 *c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[a*e - c*d*x + Sqrt[c* d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(5/2))/(120*e^6)
Time = 0.54 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {492, 589, 27, 680, 27, 719, 224, 219, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \frac {c \int \frac {x \left (c x^2+a\right )^{3/2}}{(d+e x)^5}dx}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 589 |
| \(\displaystyle \frac {c \left (\frac {c \int -\frac {2 \left (3 a d e-4 \left (c d^2+a e^2\right ) x\right ) \sqrt {c x^2+a}}{(d+e x)^3}dx}{8 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (-\frac {c \int \frac {\left (3 a d e-4 \left (c d^2+a e^2\right ) x\right ) \sqrt {c x^2+a}}{(d+e x)^3}dx}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {\int -\frac {2 c \left (a d e \left (4 c d^2+7 a e^2\right )-8 \left (c d^2+a e^2\right )^2 x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{4 e^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {c \int \frac {a d e \left (4 c d^2+7 a e^2\right )-8 \left (c d^2+a e^2\right )^2 x}{(d+e x) \sqrt {c x^2+a}}dx}{2 e^2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {c \left (\frac {d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {8 \left (a e^2+c d^2\right )^2 \int \frac {1}{\sqrt {c x^2+a}}dx}{e}\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {c \left (\frac {d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {8 \left (a e^2+c d^2\right )^2 \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {c \left (\frac {d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {8 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+c d^2\right )^2}{\sqrt {c} e}\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {c \left (-\frac {d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e}-\frac {8 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+c d^2\right )^2}{\sqrt {c} e}\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {c \left (-\frac {c \left (\frac {c \left (-\frac {d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \sqrt {a e^2+c d^2}}-\frac {8 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+c d^2\right )^2}{\sqrt {c} e}\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{2 e^2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 e^2 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^2 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{e}-\frac {\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}\) |
-1/5*(a + c*x^2)^(5/2)/(e*(d + e*x)^5) + (c*(-1/12*((d*(4*c*d^2 + a*e^2) + e*(7*c*d^2 + 4*a*e^2)*x)*(a + c*x^2)^(3/2))/(e^2*(c*d^2 + a*e^2)*(d + e*x )^4) - (c*(((d*(8*c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + e*(12*c^2*d^4 + 23 *a*c*d^2*e^2 + 8*a^2*e^4)*x)*Sqrt[a + c*x^2])/(2*e^2*(c*d^2 + a*e^2)*(d + e*x)^2) + (c*((-8*(c*d^2 + a*e^2)^2*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/ (Sqrt[c]*e) - (d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e*Sqrt[c*d^2 + a*e^2])))/( 2*e^2*(c*d^2 + a*e^2))))/(4*e^2*(c*d^2 + a*e^2))))/e
3.6.54.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 1))), x] - Simp[2*b*(p/(d*(n + 1)) ) Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] && !IL tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(a*d^2 + b*c^2*(2*p + 1)) - d*( a*d^2*(n + 1) + b*c^2*(n - 2*p + 1))*x)/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2 ))), x] + Simp[b*(p/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2))) Int[(c + d*x)^ (n + 2)*(a + b*x^2)^(p - 1)*Simp[2*a*c*d*(n + 2) - (2*a*d^2*(n + 1) - 2*b*c ^2*(2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n , -2] && LtQ[n + 2*p, 0] && !ILtQ[n + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 2)*(a + c*x^ 2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f *(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3 , 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(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] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(9761\) vs. \(2(288)=576\).
Time = 2.38 (sec) , antiderivative size = 9762, normalized size of antiderivative = 31.09
Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (289) = 578\).
Time = 60.38 (sec) , antiderivative size = 5169, normalized size of antiderivative = 16.46 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \]
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 1432 vs. \(2 (289) = 578\).
Time = 0.51 (sec) , antiderivative size = 1432, normalized size of antiderivative = 4.56 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
-1/4*(8*c^5*d^5 + 20*a*c^4*d^3*e^2 + 15*a^2*c^3*d*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^2*d^4*e^6 + 2 *a*c*d^2*e^8 + a^2*e^10)*sqrt(-c*d^2 - a*e^2)) - c^(5/2)*log(abs(-sqrt(c)* x + sqrt(c*x^2 + a)))/e^6 - 1/60*(600*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^5* d^5*e^4 + 1140*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^4*d^3*e^6 + 495*(sqrt(c )*x - sqrt(c*x^2 + a))^9*a^2*c^3*d*e^8 + 3600*(sqrt(c)*x - sqrt(c*x^2 + a) )^8*c^(11/2)*d^6*e^3 + 6300*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*d^4* e^5 + 1935*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d^2*e^7 - 360*(sqrt (c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e^9 + 8800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*d^7*e^2 + 12200*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*d^5*e^4 - 1250*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*d^3*e^6 - 3030*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*e^8 + 10000*(sqrt(c)*x - sqrt(c*x^2 + a))^6* c^(13/2)*d^8*e + 3800*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*d^6*e^3 - 18950*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*d^4*e^5 - 8250*(sqrt(c) *x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*d^2*e^7 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(5/2)*e^9 + 4384*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*d^9 - 1 3872*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^6*d^7*e^2 - 29076*(sqrt(c)*x - sq rt(c*x^2 + a))^5*a^2*c^5*d^5*e^4 + 370*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3 *c^4*d^3*e^6 + 5520*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*d*e^8 - 11920* (sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d^8*e - 3560*(sqrt(c)*x - sq...
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \]