3.1777 \(\int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx\)

Optimal. Leaf size=126 \[ \frac {256 b^4 \sqrt {x} \sqrt {a+\frac {b}{x}}}{315 a^5}-\frac {128 b^3 x^{3/2} \sqrt {a+\frac {b}{x}}}{315 a^4}+\frac {32 b^2 x^{5/2} \sqrt {a+\frac {b}{x}}}{105 a^3}-\frac {16 b x^{7/2} \sqrt {a+\frac {b}{x}}}{63 a^2}+\frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a} \]

[Out]

-128/315*b^3*x^(3/2)*(a+b/x)^(1/2)/a^4+32/105*b^2*x^(5/2)*(a+b/x)^(1/2)/a^3-16/63*b*x^(7/2)*(a+b/x)^(1/2)/a^2+
2/9*x^(9/2)*(a+b/x)^(1/2)/a+256/315*b^4*(a+b/x)^(1/2)*x^(1/2)/a^5

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Rubi [A]  time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {32 b^2 x^{5/2} \sqrt {a+\frac {b}{x}}}{105 a^3}-\frac {128 b^3 x^{3/2} \sqrt {a+\frac {b}{x}}}{315 a^4}+\frac {256 b^4 \sqrt {x} \sqrt {a+\frac {b}{x}}}{315 a^5}-\frac {16 b x^{7/2} \sqrt {a+\frac {b}{x}}}{63 a^2}+\frac {2 x^{9/2} \sqrt {a+\frac {b}{x}}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)/Sqrt[a + b/x],x]

[Out]

(256*b^4*Sqrt[a + b/x]*Sqrt[x])/(315*a^5) - (128*b^3*Sqrt[a + b/x]*x^(3/2))/(315*a^4) + (32*b^2*Sqrt[a + b/x]*
x^(5/2))/(105*a^3) - (16*b*Sqrt[a + b/x]*x^(7/2))/(63*a^2) + (2*Sqrt[a + b/x]*x^(9/2))/(9*a)

Rule 264

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]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(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]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^{7/2}}{\sqrt {a+\frac {b}{x}}} \, dx &=\frac {2 \sqrt {a+\frac {b}{x}} x^{9/2}}{9 a}-\frac {(8 b) \int \frac {x^{5/2}}{\sqrt {a+\frac {b}{x}}} \, dx}{9 a}\\ &=-\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}+\frac {\left (16 b^2\right ) \int \frac {x^{3/2}}{\sqrt {a+\frac {b}{x}}} \, dx}{21 a^2}\\ &=\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}-\frac {\left (64 b^3\right ) \int \frac {\sqrt {x}}{\sqrt {a+\frac {b}{x}}} \, dx}{105 a^3}\\ &=-\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}+\frac {\left (128 b^4\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx}{315 a^4}\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 64, normalized size = 0.51 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} \left (35 a^4 x^4-40 a^3 b x^3+48 a^2 b^2 x^2-64 a b^3 x+128 b^4\right )}{315 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)/Sqrt[a + b/x],x]

[Out]

(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)

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fricas [A]  time = 0.80, size = 60, normalized size = 0.48 \[ \frac {2 \, {\left (35 \, a^{4} x^{4} - 40 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x + 128 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(a+b/x)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 73, normalized size = 0.58 \[ \frac {2 \, \sqrt {a x + b} b^{4}}{a^{5}} - \frac {256 \, b^{\frac {9}{2}}}{315 \, a^{5}} + \frac {2 \, {\left (35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}\right )}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(a+b/x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(a*x + b)*b^4/a^5 - 256/315*b^(9/2)/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)/a^5

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maple [A]  time = 0.01, size = 66, normalized size = 0.52 \[ \frac {2 \left (a x +b \right ) \left (35 a^{4} x^{4}-40 a^{3} x^{3} b +48 a^{2} x^{2} b^{2}-64 a x \,b^{3}+128 b^{4}\right )}{315 \sqrt {\frac {a x +b}{x}}\, a^{5} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(a+b/x)^(1/2),x)

[Out]

2/315*(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)/a^5/x^(1/2)/((a*x+b)/x)^(1/2)

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maxima [A]  time = 1.01, size = 86, normalized size = 0.68 \[ \frac {2 \, {\left (35 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} x^{\frac {9}{2}} - 180 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b x^{\frac {7}{2}} + 378 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{2} x^{\frac {5}{2}} - 420 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{3} x^{\frac {3}{2}} + 315 \, \sqrt {a + \frac {b}{x}} b^{4} \sqrt {x}\right )}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(a+b/x)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.49, size = 61, normalized size = 0.48 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{9/2}}{9\,a}-\frac {16\,b\,x^{7/2}}{63\,a^2}+\frac {32\,b^2\,x^{5/2}}{105\,a^3}-\frac {128\,b^3\,x^{3/2}}{315\,a^4}+\frac {256\,b^4\,\sqrt {x}}{315\,a^5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(a + b/x)^(1/2),x)

[Out]

(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))

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sympy [B]  time = 31.83, size = 692, normalized size = 5.49 \[ \frac {70 a^{8} b^{\frac {33}{2}} x^{8} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {200 a^{7} b^{\frac {35}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {196 a^{6} b^{\frac {37}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {56 a^{5} b^{\frac {39}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {70 a^{4} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {560 a^{3} b^{\frac {43}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {1120 a^{2} b^{\frac {45}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {896 a b^{\frac {47}{2}} x \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} + \frac {256 b^{\frac {49}{2}} \sqrt {\frac {a x}{b} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{3} + 1890 a^{7} b^{18} x^{2} + 1260 a^{6} b^{19} x + 315 a^{5} b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)/(a+b/x)**(1/2),x)

[Out]

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 + 12
60*a**6*b**19*x + 315*a**5*b**20) + 200*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**17*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 + 12
60*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) + 1
120*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 + 1
260*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 + 1890*a**7*b**18*x**2 + 1260*a**6*b**19*x + 315*a**5*b**20) + 256*b**(49/2)*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)

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