Integrand size = 22, antiderivative size = 71 \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=-\frac {2}{3 c^2 \left (a^2+2 a c \sqrt {x}+c^2 x\right )^{3/2}}+\frac {a}{2 c^2 \left (a+c \sqrt {x}\right )^3 \sqrt {a^2+2 a c \sqrt {x}+c^2 x}} \] Output:
-2/3/c^2/(a^2+2*a*c*x^(1/2)+c^2*x)^(3/2)+1/2*a/c^2/(a+c*x^(1/2))^3/(a^2+2* a*c*x^(1/2)+c^2*x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=\frac {\left (-a-4 c \sqrt {x}\right ) \left (a+c \sqrt {x}\right )}{6 c^2 \left (\left (a+c \sqrt {x}\right )^2\right )^{5/2}} \] Input:
Integrate[(a^2 + 2*a*c*Sqrt[x] + c^2*x)^(-5/2),x]
Output:
((-a - 4*c*Sqrt[x])*(a + c*Sqrt[x]))/(6*c^2*((a + c*Sqrt[x])^2)^(5/2))
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1680, 1100, 1078}
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.
\(\displaystyle \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1680 |
\(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a^2+2 c \sqrt {x} a+c^2 x\right )^{5/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 1100 |
\(\displaystyle 2 \left (-\frac {a \int \frac {1}{\left (a^2+2 c \sqrt {x} a+c^2 x\right )^{5/2}}d\sqrt {x}}{c}-\frac {1}{3 c^2 \left (a^2+2 a c \sqrt {x}+c^2 x\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 1078 |
\(\displaystyle 2 \left (\frac {a}{4 c^2 \left (a+c \sqrt {x}\right ) \left (a^2+2 a c \sqrt {x}+c^2 x\right )^{3/2}}-\frac {1}{3 c^2 \left (a^2+2 a c \sqrt {x}+c^2 x\right )^{3/2}}\right )\) |
Input:
Int[(a^2 + 2*a*c*Sqrt[x] + c^2*x)^(-5/2),x]
Output:
2*(-1/3*1/(c^2*(a^2 + 2*a*c*Sqrt[x] + c^2*x)^(3/2)) + a/(4*c^2*(a + c*Sqrt [x])*(a^2 + 2*a*c*Sqrt[x] + c^2*x)^(3/2)))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b* e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k* n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && Fr actionQ[n]
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.45
method | result | size |
derivativedivides | \(-\frac {\left (4 c \sqrt {x}+a \right ) \left (a +c \sqrt {x}\right )}{6 c^{2} \left (\left (a +c \sqrt {x}\right )^{2}\right )^{\frac {5}{2}}}\) | \(32\) |
default | \(-\frac {\left (-15 x^{6} a \,c^{12}+50 x^{5} a^{3} c^{10}+4 x^{\frac {11}{2}} a^{2} c^{11}-56 x^{\frac {9}{2}} a^{4} c^{9}+104 x^{\frac {7}{2}} a^{6} c^{7}+a^{13}-14 a^{11} c^{2} x +4 x^{\frac {13}{2}} c^{13}-4 a^{7} c^{6} x^{3}+31 a^{9} c^{4} x^{2}-76 x^{\frac {5}{2}} a^{8} c^{5}-49 a^{5} c^{8} x^{4}+20 x^{\frac {3}{2}} a^{10} c^{3}\right ) \left (a +c \sqrt {x}\right )}{6 c^{2} \left (c \sqrt {x}-a \right )^{4} \left (-c^{2} x +a^{2}\right )^{4} \left (a^{2}+2 a c \sqrt {x}+c^{2} x \right )^{\frac {5}{2}}}\) | \(183\) |
Input:
int(1/(a^2+2*a*c*x^(1/2)+c^2*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(4*c*x^(1/2)+a)*(a+c*x^(1/2))/c^2/((a+c*x^(1/2))^2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (57) = 114\).
Time = 0.38 (sec) , antiderivative size = 764, normalized size of antiderivative = 10.76 \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(a^2+2*a*c*x^(1/2)+c^2*x)^(5/2),x, algorithm="fricas")
Output:
-1/6*(38*a^14 + 133*a^12*c^2 + 21*a^10*c^4 - a^8*c^6 + a^6*c^8 - (3*a^8*c^ 6 - 49*a^6*c^8 - 119*a^4*c^10 - 27*a^2*c^12)*x^4 - 4*(38*a^8*c^6 + 133*a^6 *c^8 + 21*a^4*c^10 - a^2*c^12 + c^14)*x^3 - (123*a^12*c^2 - 679*a^10*c^4 - 224*a^8*c^6 - 372*a^6*c^8 - 35*a^4*c^10 + 35*a^2*c^12)*x^2 - 2*(9*a^14 + 43*a^12*c^2 + 308*a^10*c^4 + 24*a^8*c^6 - 5*a^6*c^8 + 5*a^4*c^10)*x - (38* a^13*c + 133*a^11*c^3 + 21*a^9*c^5 - a^7*c^7 + a^5*c^9 - (3*a^7*c^7 - 49*a ^5*c^9 - 119*a^3*c^11 - 27*a*c^13)*x^4 + 4*(3*a^9*c^5 - 49*a^7*c^7 - 119*a ^5*c^9 - 27*a^3*c^11)*x^3 - (75*a^11*c^3 - 503*a^9*c^5 - 448*a^7*c^7 - 276 *a^5*c^9 - 19*a^3*c^11 + 19*a*c^13)*x^2 - 2*(39*a^13*c - 67*a^11*c^3 + 448 *a^9*c^5 - 36*a^7*c^7 - 15*a^5*c^9 + 15*a^3*c^11)*x)*sqrt(x))*sqrt(c^2*x + 2*a*c*sqrt(x) + a^2)/(3*a^20 - 11*a^18*c^2 + 14*a^16*c^4 - 6*a^14*c^6 - a ^12*c^8 + a^10*c^10 - (3*a^10*c^10 - 11*a^8*c^12 + 14*a^6*c^14 - 6*a^4*c^1 6 - a^2*c^18 + c^20)*x^5 + 5*(3*a^12*c^8 - 11*a^10*c^10 + 14*a^8*c^12 - 6* a^6*c^14 - a^4*c^16 + a^2*c^18)*x^4 - 10*(3*a^14*c^6 - 11*a^12*c^8 + 14*a^ 10*c^10 - 6*a^8*c^12 - a^6*c^14 + a^4*c^16)*x^3 + 10*(3*a^16*c^4 - 11*a^14 *c^6 + 14*a^12*c^8 - 6*a^10*c^10 - a^8*c^12 + a^6*c^14)*x^2 - 5*(3*a^18*c^ 2 - 11*a^16*c^4 + 14*a^14*c^6 - 6*a^12*c^8 - a^10*c^10 + a^8*c^12)*x)
\[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a c \sqrt {x} + c^{2} x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a**2+2*a*c*x**(1/2)+c**2*x)**(5/2),x)
Output:
Integral((a**2 + 2*a*c*sqrt(x) + c**2*x)**(-5/2), x)
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=\frac {c^{2} x^{2} + 4 \, a c x^{\frac {3}{2}} + 6 \, a^{2} x}{6 \, {\left (a^{3} c^{4} x^{2} + 4 \, a^{4} c^{3} x^{\frac {3}{2}} + 6 \, a^{5} c^{2} x + 4 \, a^{6} c \sqrt {x} + a^{7}\right )}} \] Input:
integrate(1/(a^2+2*a*c*x^(1/2)+c^2*x)^(5/2),x, algorithm="maxima")
Output:
1/6*(c^2*x^2 + 4*a*c*x^(3/2) + 6*a^2*x)/(a^3*c^4*x^2 + 4*a^4*c^3*x^(3/2) + 6*a^5*c^2*x + 4*a^6*c*sqrt(x) + a^7)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (57) = 114\).
Time = 0.13 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=-\frac {c^{2} {\left | a \right |} {\left | c \right |} \log \left ({\left | \sqrt {x} {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) + {\left | a \right |} \right |}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{4 \, {\left (a^{4} c^{4} {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) - a^{3} c^{5} {\left | a \right |}\right )}} - \frac {c^{3} {\left | a \right |} \log \left ({\left | c \sqrt {x} + a \right |}\right )}{4 \, {\left (a^{3} c^{4} {\left | a \right |} {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) - a^{4} c^{5}\right )}} - \frac {2 \, a^{4} {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) - 2 \, a^{3} c {\left | a \right |} + 3 \, {\left (a^{2} c^{2} {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) - a c^{3} {\left | a \right |}\right )} x + 9 \, {\left (a^{3} c {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) - a^{2} c^{2} {\left | a \right |}\right )} \sqrt {x}}{12 \, {\left ({\left | a \right |} {\left | c \right |} \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right ) - a c\right )} {\left (c \sqrt {x} + a\right )}^{3} a^{3} c^{2}} \] Input:
integrate(1/(a^2+2*a*c*x^(1/2)+c^2*x)^(5/2),x, algorithm="giac")
Output:
-1/4*c^2*abs(a)*abs(c)*log(abs(sqrt(x)*abs(c)*sgn(a)*sgn(c) + abs(a)))*sgn (a)*sgn(c)/(a^4*c^4*abs(c)*sgn(a)*sgn(c) - a^3*c^5*abs(a)) - 1/4*c^3*abs(a )*log(abs(c*sqrt(x) + a))/(a^3*c^4*abs(a)*abs(c)*sgn(a)*sgn(c) - a^4*c^5) - 1/12*(2*a^4*abs(c)*sgn(a)*sgn(c) - 2*a^3*c*abs(a) + 3*(a^2*c^2*abs(c)*sg n(a)*sgn(c) - a*c^3*abs(a))*x + 9*(a^3*c*abs(c)*sgn(a)*sgn(c) - a^2*c^2*ab s(a))*sqrt(x))/((abs(a)*abs(c)*sgn(a)*sgn(c) - a*c)*(c*sqrt(x) + a)^3*a^3* c^2)
Timed out. \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=\int \frac {1}{{\left (c^2\,x+a^2+2\,a\,c\,\sqrt {x}\right )}^{5/2}} \,d x \] Input:
int(1/(c^2*x + a^2 + 2*a*c*x^(1/2))^(5/2),x)
Output:
int(1/(c^2*x + a^2 + 2*a*c*x^(1/2))^(5/2), x)
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a^2+2 a c \sqrt {x}+c^2 x\right )^{5/2}} \, dx=\frac {-4 \sqrt {x}\, c -a}{6 c^{2} \left (4 \sqrt {x}\, a^{3} c +4 \sqrt {x}\, a \,c^{3} x +a^{4}+6 a^{2} c^{2} x +c^{4} x^{2}\right )} \] Input:
int(1/(a^2+2*a*c*x^(1/2)+c^2*x)^(5/2),x)
Output:
( - 4*sqrt(x)*c - a)/(6*c**2*(4*sqrt(x)*a**3*c + 4*sqrt(x)*a*c**3*x + a**4 + 6*a**2*c**2*x + c**4*x**2))