Integrand size = 28, antiderivative size = 217 \[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {4}{15} a^2 c x^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}+\frac {2}{9} x^{3/2} (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {8 a^{11/2} c^2 \sqrt {1-\frac {b^2 x^2}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |-1\right )}{15 b^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}}-\frac {8 a^{11/2} c^2 \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{15 b^{3/2} \sqrt {a+b x} \sqrt {a c-b c x}} \] Output:
4/15*a^2*c*x^(3/2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)+2/9*x^(3/2)*(b*x+a)^(3 /2)*(-b*c*x+a*c)^(3/2)+8/15*a^(11/2)*c^2*(1-b^2*x^2/a^2)^(1/2)*EllipticE(b ^(1/2)*x^(1/2)/a^(1/2),I)/b^(3/2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)-8/15*a^ (11/2)*c^2*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(3 /2)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.80 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.33 \[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {2 a^2 c x^{3/2} \sqrt {c (a-b x)} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {b^2 x^2}{a^2}\right )}{3 \sqrt {1-\frac {b^2 x^2}{a^2}}} \] Input:
Integrate[Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]
Output:
(2*a^2*c*x^(3/2)*Sqrt[c*(a - b*x)]*Sqrt[a + b*x]*Hypergeometric2F1[-3/2, 3 /4, 7/4, (b^2*x^2)/a^2])/(3*Sqrt[1 - (b^2*x^2)/a^2])
Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {112, 27, 171, 27, 171, 27, 171, 27, 124, 27, 123}
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.
| \(\displaystyle \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2 \int \frac {a c \sqrt {a+b x} (a+7 b x) (a c-b c x)^{3/2}}{2 \sqrt {x}}dx}{9 b c}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {\sqrt {a+b x} (a+7 b x) (a c-b c x)^{3/2}}{\sqrt {x}}dx}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (-\frac {2 \int -\frac {7 a b c (a+2 b x) (a c-b c x)^{3/2}}{\sqrt {x} \sqrt {a+b x}}dx}{7 b c}-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (2 a \int \frac {(a+2 b x) (a c-b c x)^{3/2}}{\sqrt {x} \sqrt {a+b x}}dx-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {2 \int \frac {3 a b c (a+3 b x) \sqrt {a c-b c x}}{2 \sqrt {x} \sqrt {a+b x}}dx}{5 b}+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {3}{5} a c \int \frac {(a+3 b x) \sqrt {a c-b c x}}{\sqrt {x} \sqrt {a+b x}}dx+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {3}{5} a c \left (\frac {2 \int \frac {3 a b^2 c \sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx}{3 b}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {3}{5} a c \left (2 a b c \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {a c-b c x}}dx+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {3}{5} a c \left (\frac {\sqrt {2} a b c \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {2} \sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {3}{5} a c \left (\frac {2 a b c \sqrt {x} \sqrt {\frac {a-b x}{a}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {1-\frac {b x}{a}}}dx}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {a \left (2 a \left (\frac {3}{5} a c \left (\frac {4 a^{3/2} c \sqrt {x} \sqrt {\frac {a-b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt {a+b x}}{\sqrt {2} \sqrt {a}}\right )\right |2\right )}{\sqrt {-\frac {b x}{a}} \sqrt {a c-b c x}}+2 \sqrt {x} \sqrt {a+b x} \sqrt {a c-b c x}\right )+\frac {4}{5} \sqrt {x} \sqrt {a+b x} (a c-b c x)^{3/2}\right )-\frac {2 \sqrt {x} \sqrt {a+b x} (a c-b c x)^{5/2}}{c}\right )}{9 b}-\frac {2 \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{5/2}}{9 b c}\) |
Input:
Int[Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]
Output:
(-2*Sqrt[x]*(a + b*x)^(3/2)*(a*c - b*c*x)^(5/2))/(9*b*c) + (a*((-2*Sqrt[x] *Sqrt[a + b*x]*(a*c - b*c*x)^(5/2))/c + 2*a*((4*Sqrt[x]*Sqrt[a + b*x]*(a*c - b*c*x)^(3/2))/5 + (3*a*c*(2*Sqrt[x]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x] + ( 4*a^(3/2)*c*Sqrt[x]*Sqrt[(a - b*x)/a]*EllipticE[ArcSin[Sqrt[a + b*x]/(Sqrt [2]*Sqrt[a])], 2])/(Sqrt[-((b*x)/a)]*Sqrt[a*c - b*c*x])))/5)))/(9*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Time = 0.67 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(-\frac {2 \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, c \left (-5 b^{6} x^{6}+12 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{6}-6 \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right ) a^{6}+16 a^{2} x^{4} b^{4}-11 a^{4} x^{2} b^{2}\right )}{45 \sqrt {x}\, b^{2} \left (-b^{2} x^{2}+a^{2}\right )}\) | \(186\) |
| risch | \(\frac {2 x^{\frac {3}{2}} \left (-5 b^{2} x^{2}+11 a^{2}\right ) \sqrt {b x +a}\, \left (-b x +a \right ) c^{2}}{45 \sqrt {-c \left (b x -a \right )}}+\frac {4 a^{5} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {-x \left (b x +a \right ) c \left (b x -a \right )}\, c^{2}}{15 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}\, \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\) | \(209\) |
| elliptic | \(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \sqrt {c x \left (-b^{2} x^{2}+a^{2}\right )}\, \left (-\frac {2 b^{2} c \,x^{3} \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{9}+\frac {22 a^{2} c x \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}{45}+\frac {4 a^{5} c^{2} \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \left (-\frac {2 a \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 b \sqrt {-b^{2} c \,x^{3}+a^{2} c x}}\right )}{\sqrt {x}\, c \left (-b^{2} x^{2}+a^{2}\right )}\) | \(229\) |
Input:
int(x^(1/2)*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/45/x^(1/2)*(b*x+a)^(1/2)*(c*(-b*x+a))^(1/2)*c*(-5*b^6*x^6+12*((b*x+a)/a )^(1/2)*2^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/a)^(1/2)*EllipticE(((b*x+a)/a)^(1 /2),1/2*2^(1/2))*a^6-6*((b*x+a)/a)^(1/2)*2^(1/2)*((-b*x+a)/a)^(1/2)*(-b*x/ a)^(1/2)*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))*a^6+16*a^2*x^4*b^4-11*a^ 4*x^2*b^2)/b^2/(-b^2*x^2+a^2)
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.39 \[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {2 \, {\left (12 \, \sqrt {-b^{2} c} a^{4} c {\rm weierstrassZeta}\left (\frac {4 \, a^{2}}{b^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )\right ) - {\left (5 \, b^{4} c x^{3} - 11 \, a^{2} b^{2} c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a} \sqrt {x}\right )}}{45 \, b^{2}} \] Input:
integrate(x^(1/2)*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="fricas")
Output:
2/45*(12*sqrt(-b^2*c)*a^4*c*weierstrassZeta(4*a^2/b^2, 0, weierstrassPInve rse(4*a^2/b^2, 0, x)) - (5*b^4*c*x^3 - 11*a^2*b^2*c*x)*sqrt(-b*c*x + a*c)* sqrt(b*x + a)*sqrt(x))/b^2
\[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\int \sqrt {x} \left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**(1/2)*(b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2),x)
Output:
Integral(sqrt(x)*(-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2), x)
\[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\int { {\left (-b c x + a c\right )}^{\frac {3}{2}} {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {x} \,d x } \] Input:
integrate(x^(1/2)*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="maxima")
Output:
integrate((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2)*sqrt(x), x)
Timed out. \[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\text {Timed out} \] Input:
integrate(x^(1/2)*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\int \sqrt {x}\,{\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2} \,d x \] Input:
int(x^(1/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2),x)
Output:
int(x^(1/2)*(a*c - b*c*x)^(3/2)*(a + b*x)^(3/2), x)
\[ \int \sqrt {x} (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx=\frac {2 \sqrt {c}\, c \left (11 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} x -5 \sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} x^{3}+6 \left (\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{2}+a^{2}}d x \right ) a^{4}\right )}{45} \] Input:
int(x^(1/2)*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x)
Output:
(2*sqrt(c)*c*(11*sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x)*a**2*x - 5*sqrt(x)*sq rt(a + b*x)*sqrt(a - b*x)*b**2*x**3 + 6*int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**2 - b**2*x**2),x)*a**4))/45