Integrand size = 19, antiderivative size = 195 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {3 a^2 c^3 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}} \]
-1/4*c*d*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^2/(e*x+d)^4-1/5*e*(c*x ^2+a)^(5/2)/(a*e^2+c*d^2)/(e*x+d)^5-3/8*a^2*c^3*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)^(7/2)-3/8*a*c^2*d*(-c*d*x+ a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)^2
Time = 3.80 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\frac {\sqrt {a+c x^2} \left (-8 a^4 e^5+2 c^4 d^4 x^3 (5 d+e x)-2 a^3 c e^3 \left (13 d^2+5 d e x+8 e^2 x^2\right )+a c^3 d^2 x \left (25 d^3+29 d^2 e x+45 d e^2 x^2+9 e^3 x^3\right )-a^2 c^2 e \left (33 d^4+45 d^3 e x+77 d^2 e^2 x^2+25 d e^3 x^3+8 e^4 x^4\right )\right )}{40 \left (c d^2+a e^2\right )^3 (d+e x)^5}+\frac {3 a^2 c^3 d \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{4 \left (-c d^2-a e^2\right )^{7/2}} \]
(Sqrt[a + c*x^2]*(-8*a^4*e^5 + 2*c^4*d^4*x^3*(5*d + e*x) - 2*a^3*c*e^3*(13 *d^2 + 5*d*e*x + 8*e^2*x^2) + a*c^3*d^2*x*(25*d^3 + 29*d^2*e*x + 45*d*e^2* x^2 + 9*e^3*x^3) - a^2*c^2*e*(33*d^4 + 45*d^3*e*x + 77*d^2*e^2*x^2 + 25*d* e^3*x^3 + 8*e^4*x^4)))/(40*(c*d^2 + a*e^2)^3*(d + e*x)^5) + (3*a^2*c^3*d*A rcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(4* (-(c*d^2) - a*e^2)^(7/2))
Time = 0.32 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {491, 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.
\(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 491 |
\(\displaystyle \frac {c d \int \frac {\left (c x^2+a\right )^{3/2}}{(d+e x)^5}dx}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c d \left (\frac {3 a c \int \frac {\sqrt {c x^2+a}}{(d+e x)^3}dx}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c d \left (\frac {3 a c \left (\frac {a c \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {c d \left (\frac {3 a c \left (-\frac {a c \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{2 \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {c d \left (\frac {3 a c \left (-\frac {a c \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{3/2}}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\) |
-1/5*(e*(a + c*x^2)^(5/2))/((c*d^2 + a*e^2)*(d + e*x)^5) + (c*d*(-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*ArcT anh[(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))))/(c*d^2 + a*e^2)
3.6.43.3.1 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. \(4107\) vs. \(2(175)=350\).
Time = 2.13 (sec) , antiderivative size = 4108, normalized size of antiderivative = 21.07
1/e^6*(-1/5/(a*e^2+c*d^2)*e^2/(x+d/e)^5*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^ 2+c*d^2)/e^2)^(5/2)+c*d*e/(a*e^2+c*d^2)*(-1/4/(a*e^2+c*d^2)*e^2/(x+d/e)^4* (c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)+3/4*c*d*e/(a*e^2+c*d ^2)*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+ c*d^2)/e^2)^(5/2)+1/3*c*d*e/(a*e^2+c*d^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^ 2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(5/2)-1/2*c*d*e/(a*e^2+c *d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c* d^2)/e^2)^(5/2)-3*c*d*e/(a*e^2+c*d^2)*(1/3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a *e^2+c*d^2)/e^2)^(3/2)-c*d/e*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c *d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2 /e^2)/c^(3/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2-2*c*d/e*(x+d/e)+( a*e^2+c*d^2)/e^2)^(1/2)))+(a*e^2+c*d^2)/e^2*((c*(x+d/e)^2-2*c*d/e*(x+d/e)+ (a*e^2+c*d^2)/e^2)^(1/2)-c^(1/2)*d/e*ln((-c*d/e+c*(x+d/e))/c^(1/2)+(c*(x+d /e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))-(a*e^2+c*d^2)/e^2/((a*e^2+ c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2) /e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e) )))+4*c/(a*e^2+c*d^2)*e^2*(1/8*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/ e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e ^2)/c*(1/4*(2*c*(x+d/e)-2*c*d/e)/c*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d ^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/c^(3/2)*ln((-c...
Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (176) = 352\).
Time = 3.34 (sec) , antiderivative size = 1647, normalized size of antiderivative = 8.45 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
[1/80*(15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e^3* x^3 + 10*a^2*c^3*d^4*e^2*x^2 + 5*a^2*c^3*d^5*e*x + a^2*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^2*c^3*d^6*e + 59*a^3*c^2*d^4*e^3 + 34*a^4*c*d^2*e^5 + 8*a^5*e^7 - (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3*c^2 *e^7)*x^4 - 5*(2*c^5*d^7 + 11*a*c^4*d^5*e^2 + 4*a^2*c^3*d^3*e^4 - 5*a^3*c^ 2*d*e^6)*x^3 - (29*a*c^4*d^6*e - 48*a^2*c^3*d^4*e^3 - 93*a^3*c^2*d^2*e^5 - 16*a^4*c*e^7)*x^2 - 5*(5*a*c^4*d^7 - 4*a^2*c^3*d^5*e^2 - 11*a^3*c^2*d^3*e ^4 - 2*a^4*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^13 + 4*a*c^3*d^11*e^2 + 6*a ^2*c^2*d^9*e^4 + 4*a^3*c*d^7*e^6 + a^4*d^5*e^8 + (c^4*d^8*e^5 + 4*a*c^3*d^ 6*e^7 + 6*a^2*c^2*d^4*e^9 + 4*a^3*c*d^2*e^11 + a^4*e^13)*x^5 + 5*(c^4*d^9* e^4 + 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 + 4*a^3*c*d^3*e^10 + a^4*d*e^12) *x^4 + 10*(c^4*d^10*e^3 + 4*a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 + 4*a^3*c*d^ 4*e^9 + a^4*d^2*e^11)*x^3 + 10*(c^4*d^11*e^2 + 4*a*c^3*d^9*e^4 + 6*a^2*c^2 *d^7*e^6 + 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x^2 + 5*(c^4*d^12*e + 4*a*c^3*d ^10*e^3 + 6*a^2*c^2*d^8*e^5 + 4*a^3*c*d^6*e^7 + a^4*d^4*e^9)*x), -1/40*(15 *(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e^3*x^3 + 10* a^2*c^3*d^4*e^2*x^2 + 5*a^2*c^3*d^5*e*x + a^2*c^3*d^6)*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 + ...
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \]
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \]
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. 1283 vs. \(2 (176) = 352\).
Time = 0.35 (sec) , antiderivative size = 1283, normalized size of antiderivative = 6.58 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \]
-3/4*a^2*c^3*d*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)*s qrt(-c*d^2 - a*e^2)) - 1/20*(15*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*d* e^8 - 40*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*d^6*e^3 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*d^4*e^5 + 15*(sqrt(c)*x - sqrt(c*x^2 + a)) ^8*a^2*c^(7/2)*d^2*e^7 - 40*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e^ 9 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*d^7*e^2 - 240*(sqrt(c)*x - sqrt (c*x^2 + a))^7*a*c^5*d^5*e^4 + 230*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4 *d^3*e^6 - 150*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*e^8 - 80*(sqrt(c) *x - sqrt(c*x^2 + a))^6*c^(13/2)*d^8*e - 240*(sqrt(c)*x - sqrt(c*x^2 + a)) ^6*a*c^(11/2)*d^6*e^3 + 530*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*d^ 4*e^5 - 570*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*d^2*e^7 - 32*(sqrt (c)*x - sqrt(c*x^2 + a))^5*c^7*d^9 + 16*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a* c^6*d^7*e^2 + 788*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*d^5*e^4 - 910*(s qrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*d^3*e^6 + 240*(sqrt(c)*x - sqrt(c*x^ 2 + a))^5*a^4*c^3*d*e^8 + 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d^ 8*e + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*d^6*e^3 - 1170*(sqr t(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*d^4*e^5 + 480*(sqrt(c)*x - sqrt(c* x^2 + a))^4*a^4*c^(7/2)*d^2*e^7 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c ^(5/2)*e^9 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*d^7*e^2 - 400*(...
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \]