Integrand size = 19, antiderivative size = 246 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {5 a^2 c^3 d (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {5 a c^2 d (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac {e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac {5 a^3 c^4 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]
-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
Time = 10.53 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=-\frac {\sqrt {a+c x^2} \left (48 \left (c d^2+a e^2\right )^6-232 c d \left (c d^2+a e^2\right )^5 (d+e x)+8 c \left (c d^2+a e^2\right )^4 \left (55 c d^2+18 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right )^3 \left (200 c d^2+197 a e^2\right ) (d+e x)^3+2 c^2 \left (c d^2+a e^2\right )^2 \left (80 c^2 d^4+159 a c d^2 e^2+72 a^2 e^4\right ) (d+e x)^4-c^3 d \left (c d^2+a e^2\right ) \left (8 c^2 d^4+30 a c d^2 e^2+57 a^2 e^4\right ) (d+e x)^5-c^3 \left (8 c^3 d^6+38 a c^2 d^4 e^2+87 a^2 c d^2 e^4-48 a^3 e^6\right ) (d+e x)^6\right )}{336 e^5 \left (c d^2+a e^2\right )^4 (d+e x)^7}+\frac {5 a^3 c^4 d \log (d+e x)}{16 \left (c d^2+a e^2\right )^{9/2}}-\frac {5 a^3 c^4 d \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]
-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*(55*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))
Time = 0.36 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {491, 486, 486, 486, 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.
\(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx\) |
\(\Big \downarrow \) 491 |
\(\displaystyle \frac {c d \int \frac {\left (c x^2+a\right )^{5/2}}{(d+e x)^7}dx}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c d \left (\frac {5 a c \int \frac {\left (c x^2+a\right )^{3/2}}{(d+e x)^5}dx}{6 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c d \left (\frac {5 a c \left (\frac {3 a c \int \frac {\sqrt {c x^2+a}}{(d+e x)^3}dx}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c d \left (\frac {5 a c \left (\frac {3 a c \left (\frac {a c \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {c d \left (\frac {5 a c \left (\frac {3 a c \left (-\frac {a c \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{2 \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {c d \left (\frac {5 a c \left (\frac {3 a c \left (-\frac {a c \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{3/2}}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}\) |
-1/7*(e*(a + c*x^2)^(7/2))/((c*d^2 + a*e^2)*(d + e*x)^7) + (c*d*(-1/6*((a* e - c*d*x)*(a + c*x^2)^(5/2))/((c*d^2 + a*e^2)*(d + e*x)^6) + (5*a*c*(-1/4 *((a*e - c*d*x)*(a + c*x^2)^(3/2))/((c*d^2 + a*e^2)*(d + e*x)^4) + (3*a*c* (-1/2*((a*e - c*d*x)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)^2) - (a*c *ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(3/2))))/(4*(c*d^2 + a*e^2))))/(6*(c*d^2 + a*e^2))))/(c*d^2 + a*e^ 2)
3.6.56.3.1 Defintions of rubi rules used
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && GtQ[p, 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[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b*(c/(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 3, 0]
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
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 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]
[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 + 3 9*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*...
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{8}}\, dx \]
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \]
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. 2428 vs. \(2 (223) = 446\).
Time = 0.36 (sec) , antiderivative size = 2428, normalized size of antiderivative = 9.87 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\text {Too large to display} \]
-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 - 201 6*(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*(sqrt(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*(sqrt(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 + ...
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^8} \,d x \]