Integrand size = 19, antiderivative size = 269 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {a^2 c^3 \left (6 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]
-1/24*c*(-a*e^2+6*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^3/(e*x +d)^4-1/6*e*(c*x^2+a)^(5/2)/(a*e^2+c*d^2)/(e*x+d)^6-7/30*c*d*e*(c*x^2+a)^( 5/2)/(a*e^2+c*d^2)^2/(e*x+d)^5-1/16*a^2*c^3*(-a*e^2+6*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)^(9/2)-1/16*a*c^ 2*(-a*e^2+6*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^4/(e*x+d)^2
Time = 10.65 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {1}{240} \left (-\frac {\sqrt {a+c x^2} \left (40 \left (c d^2+a e^2\right )^5-104 c d \left (c d^2+a e^2\right )^4 (d+e x)+2 c \left (c d^2+a e^2\right )^3 \left (38 c d^2+35 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right )^2 \left (2 c d^2+9 a e^2\right ) (d+e x)^3-c^2 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) (d+e x)^4-c^3 d \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) (d+e x)^5\right )}{e^3 \left (c d^2+a e^2\right )^4 (d+e x)^6}+\frac {15 a^2 c^3 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{9/2}}+\frac {15 a^2 c^3 \left (-6 c d^2+a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{9/2}}\right ) \]
(-((Sqrt[a + c*x^2]*(40*(c*d^2 + a*e^2)^5 - 104*c*d*(c*d^2 + a*e^2)^4*(d + e*x) + 2*c*(c*d^2 + a*e^2)^3*(38*c*d^2 + 35*a*e^2)*(d + e*x)^2 - 2*c^2*d* (c*d^2 + a*e^2)^2*(2*c*d^2 + 9*a*e^2)*(d + e*x)^3 - c^2*(c*d^2 + a*e^2)*(4 *c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*(d + e*x)^4 - c^3*d*(4*c^2*d^4 + 2 8*a*c*d^2*e^2 - 81*a^2*e^4)*(d + e*x)^5))/(e^3*(c*d^2 + a*e^2)^4*(d + e*x) ^6)) + (15*a^2*c^3*(6*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) + (15*a^2*c^3*(-6*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)^(9/2))/240
Time = 0.37 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {498, 25, 679, 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.
\(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx\) |
\(\Big \downarrow \) 498 |
\(\displaystyle -\frac {c \int -\frac {(6 d-e x) \left (c x^2+a\right )^{3/2}}{(d+e x)^6}dx}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {c \int \frac {(6 d-e x) \left (c x^2+a\right )^{3/2}}{(d+e x)^6}dx}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {c \left (\frac {\left (6 c d^2-a e^2\right ) \int \frac {\left (c x^2+a\right )^{3/2}}{(d+e x)^5}dx}{a e^2+c d^2}-\frac {7 d e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c \left (\frac {\left (6 c d^2-a e^2\right ) \left (\frac {3 a c \int \frac {\sqrt {c x^2+a}}{(d+e x)^3}dx}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {7 d e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 486 |
\(\displaystyle \frac {c \left (\frac {\left (6 c d^2-a e^2\right ) \left (\frac {3 a c \left (\frac {a c \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {7 d e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {c \left (\frac {\left (6 c d^2-a e^2\right ) \left (\frac {3 a c \left (-\frac {a c \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{2 \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {7 d e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {c \left (\frac {\left (6 c d^2-a e^2\right ) \left (\frac {3 a c \left (-\frac {a c \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{3/2}}-\frac {\sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )}\right )}{4 \left (a e^2+c d^2\right )}-\frac {\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {7 d e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )}\right )}{6 \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )}\) |
-1/6*(e*(a + c*x^2)^(5/2))/((c*d^2 + a*e^2)*(d + e*x)^6) + (c*((-7*d*e*(a + c*x^2)^(5/2))/(5*(c*d^2 + a*e^2)*(d + e*x)^5) + ((6*c*d^2 - a*e^2)*(-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))))/(c*d^2 + a*e^2)))/(6*(c*d^2 + a*e^2 ))
3.6.44.3.1 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/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
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] + Simp[(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] && EqQ[Simplify[m + 2*p + 3], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(8230\) vs. \(2(245)=490\).
Time = 2.20 (sec) , antiderivative size = 8231, normalized size of antiderivative = 30.60
Leaf count of result is larger than twice the leaf count of optimal. 1229 vs. \(2 (246) = 492\).
Time = 7.90 (sec) , antiderivative size = 2485, normalized size of antiderivative = 9.24 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]
[-1/480*(15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*d^2*e^6 - a^3*c^ 3*e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 - a^3*c^3*d*e^7)*x^5 + 15*(6*a^2*c^4*d^4 *e^4 - a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)*x^3 + 15*(6*a^2*c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3 *c^3*d^5*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*(246*a^2*c^4*d^8*e + 513*a^3*c^3 *d^6*e^3 + 433*a^4*c^2*d^4*e^5 + 206*a^5*c*d^2*e^7 + 40*a^6*e^9 - (4*c^6*d ^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x^5 - 3 *(8*c^6*d^8*e + 64*a*c^5*d^6*e^3 - 76*a^2*c^4*d^4*e^5 - 137*a^3*c^3*d^2*e^ 7 - 5*a^4*c^2*e^9)*x^4 - 2*(30*c^6*d^9 + 239*a*c^5*d^7*e^2 - 158*a^2*c^4*d ^5*e^4 - 388*a^3*c^3*d^3*e^6 - 21*a^4*c^2*d*e^8)*x^3 - 2*(114*a*c^5*d^8*e - 423*a^2*c^4*d^6*e^3 - 698*a^3*c^3*d^4*e^5 - 196*a^4*c^2*d^2*e^7 - 35*a^5 *c*e^9)*x^2 - 3*(50*a*c^5*d^9 - 117*a^2*c^4*d^7*e^2 - 221*a^3*c^3*d^5*e^4 - 66*a^4*c^2*d^3*e^6 - 12*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*d^16 + 5*a *c^4*d^14*e^2 + 10*a^2*c^3*d^12*e^4 + 10*a^3*c^2*d^10*e^6 + 5*a^4*c*d^8*e^ 8 + a^5*d^6*e^10 + (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)*x^6 + 6*(c^5*d^11*e^5 + 5*a*c^4*d^9*e^7 + 10*a^2*c^3*d^7*e^9 + 10*a^3*c^2*d^5*e^11 + 5*a^4*c*d^3 *e^13 + a^5*d*e^15)*x^5 + 15*(c^5*d^12*e^4 + 5*a*c^4*d^10*e^6 + 10*a^2*...
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]
Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
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. 1875 vs. \(2 (246) = 492\).
Time = 0.36 (sec) , antiderivative size = 1875, normalized size of antiderivative = 6.97 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]
-1/8*(6*a^2*c^4*d^2 - a^3*c^3*e^2)*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)) - 1/120*(90*( sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d^2*e^9 - 15*(sqrt(c)*x - sqrt(c*x ^2 + a))^11*a^3*c^3*e^11 + 990*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2 )*d^3*e^8 - 165*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*e^10 - 320* (sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 + 2520*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4* e^7 - 2530*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*d^2*e^9 - 235*(sqrt(c)* x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8* c^(15/2)*d^9*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 7380*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 - 8220*(sqrt( c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*d^3*e^8 + 285*(sqrt(c)*x - sqrt(c*x^ 2 + a))^8*a^4*c^(7/2)*d*e^10 - 384*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^1 0*e - 1728*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 9456*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 20760*(sqrt(c)*x - sqrt(c*x^2 + a) )^7*a^3*c^5*d^4*e^7 + 2700*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d^2*e^9 - 390*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11 - 128*(sqrt(c)*x - sqr t(c*x^2 + a))^6*c^(17/2)*d^11 + 64*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15 /2)*d^9*e^2 + 8592*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4...
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^7} \,d x \]