Integrand size = 26, antiderivative size = 108 \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {3 x \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^4 c^2}-\frac {3 a^2 \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b^5 c^{3/2}} \] Output:
x^3/b^2/c/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)+3/2*x*(b*x+a)^(1/2)*(-b*c*x+a*c )^(1/2)/b^4/c^2-3*a^2*arctan(c^(1/2)*(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2))/b^5 /c^(3/2)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {-3 a^2 b x+b^3 x^3+6 a^2 \sqrt {a-b x} \sqrt {a+b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{2 b^5 c \sqrt {c (a-b x)} \sqrt {a+b x}} \] Input:
Integrate[x^4/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
-1/2*(-3*a^2*b*x + b^3*x^3 + 6*a^2*Sqrt[a - b*x]*Sqrt[a + b*x]*ArcTan[Sqrt [a + b*x]/Sqrt[a - b*x]])/(b^5*c*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {109, 27, 35, 101, 25, 27, 45, 218}
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.
\(\displaystyle \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {\int \frac {3 a c x^2 (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{a b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \int \frac {x^2 (a-b x)}{\sqrt {a+b x} (a c-b c x)^{3/2}}dx}{b^2}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \int \frac {x^2}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{b^2 c}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \left (-\frac {\int -\frac {a^2 c}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 b^2 c}-\frac {x \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^2 c}\right )}{b^2 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \left (\frac {\int \frac {a^2 c}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 b^2 c}-\frac {x \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^2 c}\right )}{b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \left (\frac {a^2 \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{2 b^2}-\frac {x \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^2 c}\right )}{b^2 c}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \left (\frac {a^2 \int \frac {1}{\frac {c (a+b x) b}{a c-b c x}+b}d\frac {\sqrt {a+b x}}{\sqrt {a c-b c x}}}{b^2}-\frac {x \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^2 c}\right )}{b^2 c}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {x^3}{b^2 c \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {3 \left (\frac {a^2 \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b^3 \sqrt {c}}-\frac {x \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^2 c}\right )}{b^2 c}\) |
Input:
Int[x^4/((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)),x]
Output:
x^3/(b^2*c*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (3*(-1/2*(x*Sqrt[a + b*x]*Sq rt[a*c - b*c*x])/(b^2*c) + (a^2*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*c - b*c*x]])/(b^3*Sqrt[c])))/(b^2*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(90)=180\).
Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.73
method | result | size |
default | \(-\frac {\sqrt {c \left (-b x +a \right )}\, \left (-3 \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{2} c \,x^{2}+b^{2} x^{3} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+3 \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{4} c -3 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} x \right )}{2 c^{2} \left (-b x +a \right ) \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} \sqrt {b x +a}}\) | \(187\) |
risch | \(\frac {x \sqrt {b x +a}\, \left (-b x +a \right )}{2 b^{4} \sqrt {-c \left (b x -a \right )}\, c}-\frac {a^{2} \left (\frac {\sqrt {-b^{2} c \left (x -\frac {a}{b}\right )^{2}-2 a b c \left (x -\frac {a}{b}\right )}}{b^{2} c \left (x -\frac {a}{b}\right )}+\frac {3 \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {b^{2} c}}+\frac {\sqrt {-b^{2} c \left (x +\frac {a}{b}\right )^{2}+2 a b c \left (x +\frac {a}{b}\right )}}{b^{2} c \left (x +\frac {a}{b}\right )}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{2 b^{4} \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}\, c}\) | \(215\) |
Input:
int(x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(c*(-b*x+a))^(1/2)/c^2*(-3*arctan((b^2*c)^(1/2)*x/(c*(-b^2*x^2+a^2))^ (1/2))*a^2*b^2*c*x^2+b^2*x^3*(b^2*c)^(1/2)*(c*(-b^2*x^2+a^2))^(1/2)+3*arct an((b^2*c)^(1/2)*x/(c*(-b^2*x^2+a^2))^(1/2))*a^4*c-3*(c*(-b^2*x^2+a^2))^(1 /2)*(b^2*c)^(1/2)*a^2*x)/(-b*x+a)/(b^2*c)^(1/2)/(c*(-b^2*x^2+a^2))^(1/2)/b ^4/(b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.35 \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\left [-\frac {3 \, {\left (a^{2} b^{2} x^{2} - a^{4}\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) - 2 \, {\left (b^{3} x^{3} - 3 \, a^{2} b x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{4 \, {\left (b^{7} c^{2} x^{2} - a^{2} b^{5} c^{2}\right )}}, \frac {3 \, {\left (a^{2} b^{2} x^{2} - a^{4}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (b^{3} x^{3} - 3 \, a^{2} b x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{2 \, {\left (b^{7} c^{2} x^{2} - a^{2} b^{5} c^{2}\right )}}\right ] \] Input:
integrate(x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
[-1/4*(3*(a^2*b^2*x^2 - a^4)*sqrt(-c)*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a* c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) - 2*(b^3*x^3 - 3*a^2*b*x)*sqrt(-b*c *x + a*c)*sqrt(b*x + a))/(b^7*c^2*x^2 - a^2*b^5*c^2), 1/2*(3*(a^2*b^2*x^2 - a^4)*sqrt(c)*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c* x^2 - a^2*c)) + (b^3*x^3 - 3*a^2*b*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b ^7*c^2*x^2 - a^2*b^5*c^2)]
Timed out. \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**4/(b*x+a)**(3/2)/(-b*c*x+a*c)**(3/2),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {x^{3}}{2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{2} c} + \frac {3 \, a^{2} x}{2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} b^{4} c} - \frac {3 \, a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{5} c^{\frac {3}{2}}} \] Input:
integrate(x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
-1/2*x^3/(sqrt(-b^2*c*x^2 + a^2*c)*b^2*c) + 3/2*a^2*x/(sqrt(-b^2*c*x^2 + a ^2*c)*b^4*c) - 3/2*a^2*arcsin(b*x/a)/(b^5*c^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (90) = 180\).
Time = 0.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.71 \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {b x + a}{b^{4} c} - \frac {3 \, a}{b^{4} c}\right )} + \frac {a^{2}}{b^{4} c}\right )}}{{\left (b x + a\right )} c - 2 \, a c} - \frac {4 \, a^{3}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )} b^{4} \sqrt {-c}} + \frac {3 \, a^{2} \log \left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}\right )}{b^{4} \sqrt {-c} c}}{2 \, b} \] Input:
integrate(x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
1/2*(sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*((b*x + a)*((b*x + a)/(b^4*c ) - 3*a/(b^4*c)) + a^2/(b^4*c))/((b*x + a)*c - 2*a*c) - 4*a^3/(((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2 - 2*a*c)*b^4*sqrt(-c)) + 3*a ^2*log((sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2)/(b^4*sqrt( -c)*c))/b
Timed out. \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\int \frac {x^4}{{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int(x^4/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)),x)
Output:
int(x^4/((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2)), x)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int \frac {x^4}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\sqrt {c}\, \left (6 \sqrt {b x +a}\, \sqrt {-b x +a}\, \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{2}+3 a^{2} b x -b^{3} x^{3}\right )}{2 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} c^{2}} \] Input:
int(x^4/(b*x+a)^(3/2)/(-b*c*x+a*c)^(3/2),x)
Output:
(sqrt(c)*(6*sqrt(a + b*x)*sqrt(a - b*x)*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2 )))*a**2 + 3*a**2*b*x - b**3*x**3))/(2*sqrt(a + b*x)*sqrt(a - b*x)*b**5*c* *2)