Integrand size = 26, antiderivative size = 96 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=a^2 c \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right ) \] Output:
a^2*c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)+1/3*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2 )-a^3*c^(3/2)*arctanh((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/a/c^(1/2))
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=-\frac {(c (a-b x))^{3/2} \left (\sqrt {a-b x} \sqrt {a+b x} \left (-4 a^2+b^2 x^2\right )+6 a^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{3 (a-b x)^{3/2}} \] Input:
Integrate[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x,x]
Output:
-1/3*((c*(a - b*x))^(3/2)*(Sqrt[a - b*x]*Sqrt[a + b*x]*(-4*a^2 + b^2*x^2) + 6*a^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a - b*x]]))/(a - b*x)^(3/2)
Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {112, 27, 112, 25, 27, 103, 221}
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.
\(\displaystyle \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 112 |
\(\displaystyle \frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}-\frac {1}{3} \int -\frac {3 a^2 c \sqrt {a+b x} \sqrt {a c-b c x}}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a^2 c \int \frac {\sqrt {a+b x} \sqrt {a c-b c x}}{x}dx+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}\) |
\(\Big \downarrow \) 112 |
\(\displaystyle a^2 c \left (\sqrt {a+b x} \sqrt {a c-b c x}-\int -\frac {a^2 c}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx\right )+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a^2 c \left (\int \frac {a^2 c}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx+\sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a^2 c \left (a^2 c \int \frac {1}{x \sqrt {a+b x} \sqrt {a c-b c x}}dx+\sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle a^2 c \left (a^2 b c \int \frac {1}{b (a+b x) (a c-b c x)-a^2 b c}d\left (\sqrt {a+b x} \sqrt {a c-b c x}\right )+\sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle a^2 c \left (\sqrt {a+b x} \sqrt {a c-b c x}-a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {a c-b c x}}{a \sqrt {c}}\right )\right )+\frac {1}{3} (a+b x)^{3/2} (a c-b c x)^{3/2}\) |
Input:
Int[((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/x,x]
Output:
((a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/3 + a^2*c*(Sqrt[a + b*x]*Sqrt[a*c - b*c*x] - a*Sqrt[c]*ArcTanh[(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(a*Sqrt[c])])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.51
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (3 \ln \left (\frac {2 a^{2} c +2 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}{x}\right ) a^{4} c +b^{2} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}-4 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {a^{2} c}\, a^{2}\right )}{3 \sqrt {a^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\) | \(145\) |
Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/3*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*c*(3*ln(2*(a^2*c+(a^2*c)^(1/2)*(c*(- b^2*x^2+a^2))^(1/2))/x)*a^4*c+b^2*x^2*(c*(-b^2*x^2+a^2))^(1/2)*(a^2*c)^(1/ 2)-4*(c*(-b^2*x^2+a^2))^(1/2)*(a^2*c)^(1/2)*a^2)/(a^2*c)^(1/2)/(c*(-b^2*x^ 2+a^2))^(1/2)
Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=\left [\frac {1}{2} \, a^{3} c^{\frac {3}{2}} \log \left (-\frac {b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {c}}{x^{2}}\right ) - \frac {1}{3} \, {\left (b^{2} c x^{2} - 4 \, a^{2} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}, -a^{3} \sqrt {-c} c \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} a \sqrt {-c}}{b^{2} c x^{2} - a^{2} c}\right ) - \frac {1}{3} \, {\left (b^{2} c x^{2} - 4 \, a^{2} c\right )} \sqrt {-b c x + a c} \sqrt {b x + a}\right ] \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x,x, algorithm="fricas")
Output:
[1/2*a^3*c^(3/2)*log(-(b^2*c*x^2 - 2*a^2*c + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqrt(c))/x^2) - 1/3*(b^2*c*x^2 - 4*a^2*c)*sqrt(-b*c*x + a*c)*sqrt( b*x + a), -a^3*sqrt(-c)*c*arctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqrt(- c)/(b^2*c*x^2 - a^2*c)) - 1/3*(b^2*c*x^2 - 4*a^2*c)*sqrt(-b*c*x + a*c)*sqr t(b*x + a)]
\[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=\int \frac {\left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}}{x}\, dx \] Input:
integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2)/x,x)
Output:
Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2)/x, x)
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=-a^{3} c^{\frac {3}{2}} \log \left (\frac {2 \, a^{2} c}{{\left | x \right |}} + \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a \sqrt {c}}{{\left | x \right |}}\right ) + \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c + \frac {1}{3} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x,x, algorithm="maxima")
Output:
-a^3*c^(3/2)*log(2*a^2*c/abs(x) + 2*sqrt(-b^2*c*x^2 + a^2*c)*a*sqrt(c)/abs (x)) + sqrt(-b^2*c*x^2 + a^2*c)*a^2*c + 1/3*(-b^2*c*x^2 + a^2*c)^(3/2)
Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=-\frac {6 \, a^{3} b \sqrt {-c} c \arctan \left (\frac {{\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}}{2 \, a c}\right ) - {\left (3 \, a^{2} b c - {\left ({\left (b x + a\right )} b c - 2 \, a b c\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{3 \, b} \] Input:
integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x,x, algorithm="giac")
Output:
-1/3*(6*a^3*b*sqrt(-c)*c*arctan(1/2*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2/(a*c)) - (3*a^2*b*c - ((b*x + a)*b*c - 2*a*b*c)*(b*x + a ))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))/b
Timed out. \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=\int \frac {{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2}}{x} \,d x \] Input:
int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x,x)
Output:
int(((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2))/x, x)
Time = 0.17 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^{3/2} (a c-b c x)^{3/2}}{x} \, dx=\frac {\sqrt {c}\, c \left (4 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2}-\sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} x^{2}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) a^{3}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) a^{3}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )-1\right ) a^{3}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right )}{2}\right )+1\right ) a^{3}\right )}{3} \] Input:
int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/x,x)
Output:
(sqrt(c)*c*(4*sqrt(a + b*x)*sqrt(a - b*x)*a**2 - sqrt(a + b*x)*sqrt(a - b* x)*b**2*x**2 - 3*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)) )/2) - 1)*a**3 + 3*log( - sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2 )))/2) + 1)*a**3 - 3*log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2) ))/2) - 1)*a**3 + 3*log(sqrt(2) + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)) )/2) + 1)*a**3))/3