Integrand size = 17, antiderivative size = 126 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {256 b^4 \sqrt {a+\frac {b}{x}} \sqrt {x}}{315 a^5}-\frac {128 b^3 \sqrt {a+\frac {b}{x}} x^{3/2}}{315 a^4}+\frac {32 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}{105 a^3}-\frac {16 b \sqrt {a+\frac {b}{x}} x^{7/2}}{63 a^2}+\frac {2 \sqrt {a+\frac {b}{x}} x^{9/2}}{9 a} \] Output:
256/315*b^4*(a+b/x)^(1/2)*x^(1/2)/a^5-128/315*b^3*(a+b/x)^(1/2)*x^(3/2)/a^ 4+32/105*b^2*(a+b/x)^(1/2)*x^(5/2)/a^3-16/63*b*(a+b/x)^(1/2)*x^(7/2)/a^2+2 /9*(a+b/x)^(1/2)*x^(9/2)/a
Time = 3.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.51 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} \left (128 b^4-64 a b^3 x+48 a^2 b^2 x^2-40 a^3 b x^3+35 a^4 x^4\right )}{315 a^5} \] Input:
Integrate[x^(7/2)/Sqrt[a + b/x],x]
Output:
(2*Sqrt[a + b/x]*Sqrt[x]*(128*b^4 - 64*a*b^3*x + 48*a^2*b^2*x^2 - 40*a^3*b *x^3 + 35*a^4*x^4))/(315*a^5)
Time = 0.41 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {803, 803, 803, 803, 796}
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.
\(\displaystyle \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a}-\frac {8 b \int \frac {x^{5/2}}{\sqrt {a+\frac {b}{x}}}dx}{9 a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a}-\frac {8 b \left (\frac {2 x^{7/2} \sqrt {a+\frac {b}{x}}}{7 a}-\frac {6 b \int \frac {x^{3/2}}{\sqrt {a+\frac {b}{x}}}dx}{7 a}\right )}{9 a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a}-\frac {8 b \left (\frac {2 x^{7/2} \sqrt {a+\frac {b}{x}}}{7 a}-\frac {6 b \left (\frac {2 x^{5/2} \sqrt {a+\frac {b}{x}}}{5 a}-\frac {4 b \int \frac {\sqrt {x}}{\sqrt {a+\frac {b}{x}}}dx}{5 a}\right )}{7 a}\right )}{9 a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a}-\frac {8 b \left (\frac {2 x^{7/2} \sqrt {a+\frac {b}{x}}}{7 a}-\frac {6 b \left (\frac {2 x^{5/2} \sqrt {a+\frac {b}{x}}}{5 a}-\frac {4 b \left (\frac {2 x^{3/2} \sqrt {a+\frac {b}{x}}}{3 a}-\frac {2 b \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}dx}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a}-\frac {8 b \left (\frac {2 x^{7/2} \sqrt {a+\frac {b}{x}}}{7 a}-\frac {6 b \left (\frac {2 x^{5/2} \sqrt {a+\frac {b}{x}}}{5 a}-\frac {4 b \left (\frac {2 x^{3/2} \sqrt {a+\frac {b}{x}}}{3 a}-\frac {4 b \sqrt {x} \sqrt {a+\frac {b}{x}}}{3 a^2}\right )}{5 a}\right )}{7 a}\right )}{9 a}\) |
Input:
Int[x^(7/2)/Sqrt[a + b/x],x]
Output:
(2*Sqrt[a + b/x]*x^(9/2))/(9*a) - (8*b*((2*Sqrt[a + b/x]*x^(7/2))/(7*a) - (6*b*((2*Sqrt[a + b/x]*x^(5/2))/(5*a) - (4*b*((-4*b*Sqrt[a + b/x]*Sqrt[x]) /(3*a^2) + (2*Sqrt[a + b/x]*x^(3/2))/(3*a)))/(5*a)))/(7*a)))/(9*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (35 a^{4} x^{4}-40 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-64 a \,b^{3} x +128 b^{4}\right )}{315 a^{5}}\) | \(61\) |
orering | \(\frac {2 \left (35 a^{4} x^{4}-40 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-64 a \,b^{3} x +128 b^{4}\right ) \left (a x +b \right )}{315 a^{5} \sqrt {x}\, \sqrt {a +\frac {b}{x}}}\) | \(64\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (35 a^{4} x^{4}-40 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-64 a \,b^{3} x +128 b^{4}\right )}{315 a^{5} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(66\) |
risch | \(\frac {2 \left (a x +b \right ) \left (35 a^{4} x^{4}-40 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-64 a \,b^{3} x +128 b^{4}\right )}{315 a^{5} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(66\) |
Input:
int(x^(7/2)/(a+b/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/315*((a*x+b)/x)^(1/2)*x^(1/2)*(35*a^4*x^4-40*a^3*b*x^3+48*a^2*b^2*x^2-64 *a*b^3*x+128*b^4)/a^5
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.48 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {2 \, {\left (35 \, a^{4} x^{4} - 40 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x + 128 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{315 \, a^{5}} \] Input:
integrate(x^(7/2)/(a+b/x)^(1/2),x, algorithm="fricas")
Output:
2/315*(35*a^4*x^4 - 40*a^3*b*x^3 + 48*a^2*b^2*x^2 - 64*a*b^3*x + 128*b^4)* sqrt(x)*sqrt((a*x + b)/x)/a^5
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (110) = 220\).
Time = 15.95 (sec) , antiderivative size = 692, normalized size of antiderivative = 5.49 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {70 a^{8} b^{\frac {33}{2}} x^{8} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {200 a^{7} b^{\frac {35}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {196 a^{6} b^{\frac {37}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {56 a^{5} b^{\frac {39}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {70 a^{4} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {560 a^{3} b^{\frac {43}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {1120 a^{2} b^{\frac {45}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {896 a b^{\frac {47}{2}} x \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {256 b^{\frac {49}{2}} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} \] Input:
integrate(x**(7/2)/(a+b/x)**(1/2),x)
Output:
70*a**8*b**(33/2)*x**8*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b* *17*x**3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 20 0*a**7*b**(35/2)*x**7*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b** 17*x**3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 196 *a**6*b**(37/2)*x**6*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**1 7*x**3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 56*a **5*b**(39/2)*x**5*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17* x**3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 70*a** 4*b**(41/2)*x**4*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x* *3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 560*a**3 *b**(43/2)*x**3*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x** 3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 1120*a**2 *b**(45/2)*x**2*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x** 3 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 896*a*b** (47/2)*x*sqrt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**3 + 189 0*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 256*b**(49/2)*sq rt(a*x/b + 1)/(315*a**9*b**16*x**4 + 1260*a**8*b**17*x**3 + 1890*a**7*b**1 8*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20)
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.68 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {2 \, {\left (35 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} x^{\frac {9}{2}} - 180 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b x^{\frac {7}{2}} + 378 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{2} x^{\frac {5}{2}} - 420 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{3} x^{\frac {3}{2}} + 315 \, \sqrt {a + \frac {b}{x}} b^{4} \sqrt {x}\right )}}{315 \, a^{5}} \] Input:
integrate(x^(7/2)/(a+b/x)^(1/2),x, algorithm="maxima")
Output:
2/315*(35*(a + b/x)^(9/2)*x^(9/2) - 180*(a + b/x)^(7/2)*b*x^(7/2) + 378*(a + b/x)^(5/2)*b^2*x^(5/2) - 420*(a + b/x)^(3/2)*b^3*x^(3/2) + 315*sqrt(a + b/x)*b^4*sqrt(x))/a^5
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.60 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=-\frac {256 \, b^{\frac {9}{2}} \mathrm {sgn}\left (x\right )}{315 \, a^{5}} + \frac {2 \, {\left (35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {a x + b} b^{4}\right )}}{315 \, a^{5} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^(7/2)/(a+b/x)^(1/2),x, algorithm="giac")
Output:
-256/315*b^(9/2)*sgn(x)/a^5 + 2/315*(35*(a*x + b)^(9/2) - 180*(a*x + b)^(7 /2)*b + 378*(a*x + b)^(5/2)*b^2 - 420*(a*x + b)^(3/2)*b^3 + 315*sqrt(a*x + b)*b^4)/(a^5*sgn(x))
Time = 0.76 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{9/2}}{9\,a}-\frac {16\,b\,x^{7/2}}{63\,a^2}+\frac {32\,b^2\,x^{5/2}}{105\,a^3}-\frac {128\,b^3\,x^{3/2}}{315\,a^4}+\frac {256\,b^4\,\sqrt {x}}{315\,a^5}\right ) \] Input:
int(x^(7/2)/(a + b/x)^(1/2),x)
Output:
(a + b/x)^(1/2)*((2*x^(9/2))/(9*a) - (16*b*x^(7/2))/(63*a^2) + (32*b^2*x^( 5/2))/(105*a^3) - (128*b^3*x^(3/2))/(315*a^4) + (256*b^4*x^(1/2))/(315*a^5 ))
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.41 \[ \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {2 \sqrt {a x +b}\, \left (35 a^{4} x^{4}-40 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-64 a \,b^{3} x +128 b^{4}\right )}{315 a^{5}} \] Input:
int(x^(7/2)/(a+b/x)^(1/2),x)
Output:
(2*sqrt(a*x + b)*(35*a**4*x**4 - 40*a**3*b*x**3 + 48*a**2*b**2*x**2 - 64*a *b**3*x + 128*b**4))/(315*a**5)