Integrand size = 24, antiderivative size = 78 \[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {3 x^{5/3} \sqrt {a+b x} \sqrt {c+d x} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {1}{2},-\frac {1}{2},\frac {8}{3},-\frac {b x}{a},-\frac {d x}{c}\right )}{5 \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}}} \] Output:
3/5*x^(5/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)*AppellF1(5/3,-1/2,-1/2,8/3,-b*x/a, -d*x/c)/(1+b*x/a)^(1/2)/(1+d*x/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(199\) vs. \(2(78)=156\).
Time = 3.73 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.55 \[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {3 x^{2/3} \left (10 (a+b x) (c+d x) (3 b c+3 a d+10 b d x)-30 a c (b c+a d) \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {b x}{a},-\frac {d x}{c}\right )-3 \left (7 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},\frac {1}{2},\frac {8}{3},-\frac {b x}{a},-\frac {d x}{c}\right )\right )}{800 b d \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[x^(2/3)*Sqrt[a + b*x]*Sqrt[c + d*x],x]
Output:
(3*x^(2/3)*(10*(a + b*x)*(c + d*x)*(3*b*c + 3*a*d + 10*b*d*x) - 30*a*c*(b* c + a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*x)/a), -((d*x)/c)] - 3*(7*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*AppellF1[5/3, 1/2, 1/2, 8/3, -((b*x)/a), -((d* x)/c)]))/(800*b*d*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {152, 152, 150}
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.
| \(\displaystyle \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx\) |
| \(\Big \downarrow \) 152 |
| \(\displaystyle \frac {\sqrt {a+b x} \int x^{2/3} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x}dx}{\sqrt {\frac {b x}{a}+1}}\) |
| \(\Big \downarrow \) 152 |
| \(\displaystyle \frac {\sqrt {a+b x} \sqrt {c+d x} \int x^{2/3} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}dx}{\sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {3 x^{5/3} \sqrt {a+b x} \sqrt {c+d x} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {1}{2},-\frac {1}{2},\frac {8}{3},-\frac {b x}{a},-\frac {d x}{c}\right )}{5 \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}\) |
Input:
Int[x^(2/3)*Sqrt[a + b*x]*Sqrt[c + d*x],x]
Output:
(3*x^(5/3)*Sqrt[a + b*x]*Sqrt[c + d*x]*AppellF1[5/3, -1/2, -1/2, 8/3, -((b *x)/a), -((d*x)/c)])/(5*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0]
\[\int x^{\frac {2}{3}} \sqrt {b x +a}\, \sqrt {x d +c}d x\]
Input:
int(x^(2/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2),x)
Output:
int(x^(2/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2),x)
\[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\int { \sqrt {b x + a} \sqrt {d x + c} x^{\frac {2}{3}} \,d x } \] Input:
integrate(x^(2/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x + a)*sqrt(d*x + c)*x^(2/3), x)
\[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\int x^{\frac {2}{3}} \sqrt {a + b x} \sqrt {c + d x}\, dx \] Input:
integrate(x**(2/3)*(b*x+a)**(1/2)*(d*x+c)**(1/2),x)
Output:
Integral(x**(2/3)*sqrt(a + b*x)*sqrt(c + d*x), x)
\[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\int { \sqrt {b x + a} \sqrt {d x + c} x^{\frac {2}{3}} \,d x } \] Input:
integrate(x^(2/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x + a)*sqrt(d*x + c)*x^(2/3), x)
\[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\int { \sqrt {b x + a} \sqrt {d x + c} x^{\frac {2}{3}} \,d x } \] Input:
integrate(x^(2/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x + a)*sqrt(d*x + c)*x^(2/3), x)
Timed out. \[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\int x^{2/3}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x} \,d x \] Input:
int(x^(2/3)*(a + b*x)^(1/2)*(c + d*x)^(1/2),x)
Output:
int(x^(2/3)*(a + b*x)^(1/2)*(c + d*x)^(1/2), x)
\[ \int x^{2/3} \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {-\frac {63 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} d^{2}}{320}+\frac {27 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c d}{160}+\frac {9 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{2} x}{80}-\frac {63 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{2}}{320}+\frac {9 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c d x}{80}+\frac {3 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} d^{2} x^{2}}{8}-\frac {21 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {4}{3}} a c +x^{\frac {7}{3}} a d +x^{\frac {7}{3}} b c +x^{\frac {10}{3}} b d}d x \right ) a^{3} c \,d^{2}}{320}+\frac {9 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {4}{3}} a c +x^{\frac {7}{3}} a d +x^{\frac {7}{3}} b c +x^{\frac {10}{3}} b d}d x \right ) a^{2} b \,c^{2} d}{160}-\frac {21 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {4}{3}} a c +x^{\frac {7}{3}} a d +x^{\frac {7}{3}} b c +x^{\frac {10}{3}} b d}d x \right ) a \,b^{2} c^{3}}{320}+\frac {21 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {1}{3}} a c +x^{\frac {4}{3}} a d +x^{\frac {4}{3}} b c +x^{\frac {7}{3}} b d}d x \right ) a^{3} d^{3}}{640}-\frac {9 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {1}{3}} a c +x^{\frac {4}{3}} a d +x^{\frac {4}{3}} b c +x^{\frac {7}{3}} b d}d x \right ) a^{2} b c \,d^{2}}{128}-\frac {9 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {1}{3}} a c +x^{\frac {4}{3}} a d +x^{\frac {4}{3}} b c +x^{\frac {7}{3}} b d}d x \right ) a \,b^{2} c^{2} d}{128}+\frac {21 x^{\frac {1}{3}} \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{x^{\frac {1}{3}} a c +x^{\frac {4}{3}} a d +x^{\frac {4}{3}} b c +x^{\frac {7}{3}} b d}d x \right ) b^{3} c^{3}}{640}}{x^{\frac {1}{3}} b^{2} d^{2}} \] Input:
int(x^(2/3)*(b*x+a)^(1/2)*(d*x+c)^(1/2),x)
Output:
(3*( - 42*sqrt(c + d*x)*sqrt(a + b*x)*a**2*d**2 + 36*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c*d + 24*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d**2*x - 42*sqrt(c + d *x)*sqrt(a + b*x)*b**2*c**2 + 24*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c*d*x + 80*sqrt(c + d*x)*sqrt(a + b*x)*b**2*d**2*x**2 - 14*x**(1/3)*int((sqrt(c + d*x)*sqrt(a + b*x))/(x**(1/3)*a*c*x + x**(1/3)*a*d*x**2 + x**(1/3)*b*c*x** 2 + x**(1/3)*b*d*x**3),x)*a**3*c*d**2 + 12*x**(1/3)*int((sqrt(c + d*x)*sqr t(a + b*x))/(x**(1/3)*a*c*x + x**(1/3)*a*d*x**2 + x**(1/3)*b*c*x**2 + x**( 1/3)*b*d*x**3),x)*a**2*b*c**2*d - 14*x**(1/3)*int((sqrt(c + d*x)*sqrt(a + b*x))/(x**(1/3)*a*c*x + x**(1/3)*a*d*x**2 + x**(1/3)*b*c*x**2 + x**(1/3)*b *d*x**3),x)*a*b**2*c**3 + 7*x**(1/3)*int((sqrt(c + d*x)*sqrt(a + b*x))/(x* *(1/3)*a*c + x**(1/3)*a*d*x + x**(1/3)*b*c*x + x**(1/3)*b*d*x**2),x)*a**3* d**3 - 15*x**(1/3)*int((sqrt(c + d*x)*sqrt(a + b*x))/(x**(1/3)*a*c + x**(1 /3)*a*d*x + x**(1/3)*b*c*x + x**(1/3)*b*d*x**2),x)*a**2*b*c*d**2 - 15*x**( 1/3)*int((sqrt(c + d*x)*sqrt(a + b*x))/(x**(1/3)*a*c + x**(1/3)*a*d*x + x* *(1/3)*b*c*x + x**(1/3)*b*d*x**2),x)*a*b**2*c**2*d + 7*x**(1/3)*int((sqrt( c + d*x)*sqrt(a + b*x))/(x**(1/3)*a*c + x**(1/3)*a*d*x + x**(1/3)*b*c*x + x**(1/3)*b*d*x**2),x)*b**3*c**3))/(640*x**(1/3)*b**2*d**2)