3.1761 \(\int (a+\frac {b}{x})^{3/2} x^{7/2} \, dx\)

Optimal. Leaf size=74 \[ \frac {16 b^2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{315 a^3}-\frac {8 b x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{63 a^2}+\frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{5/2}}{9 a} \]

[Out]

16/315*b^2*(a+b/x)^(5/2)*x^(5/2)/a^3-8/63*b*(a+b/x)^(5/2)*x^(7/2)/a^2+2/9*(a+b/x)^(5/2)*x^(9/2)/a

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {16 b^2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{315 a^3}-\frac {8 b x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{63 a^2}+\frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{5/2}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*x^(7/2),x]

[Out]

(16*b^2*(a + b/x)^(5/2)*x^(5/2))/(315*a^3) - (8*b*(a + b/x)^(5/2)*x^(7/2))/(63*a^2) + (2*(a + b/x)^(5/2)*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 \left (a+\frac {b}{x}\right )^{3/2} x^{7/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{9/2}}{9 a}-\frac {(4 b) \int \left (a+\frac {b}{x}\right )^{3/2} x^{5/2} \, dx}{9 a}\\ &=-\frac {8 b \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{63 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{9/2}}{9 a}+\frac {\left (8 b^2\right ) \int \left (a+\frac {b}{x}\right )^{3/2} x^{3/2} \, dx}{63 a^2}\\ &=\frac {16 b^2 \left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}{315 a^3}-\frac {8 b \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{63 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{9/2}}{9 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 49, normalized size = 0.66 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} (a x+b)^2 \left (35 a^2 x^2-20 a b x+8 b^2\right )}{315 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*x^(7/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^2*(8*b^2 - 20*a*b*x + 35*a^2*x^2))/(315*a^3)

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 60, normalized size = 0.81 \[ \frac {2 \, {\left (35 \, a^{4} x^{4} + 50 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 8 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{315 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*a^4*x^4 + 50*a^3*b*x^3 + 3*a^2*b^2*x^2 - 4*a*b^3*x + 8*b^4)*sqrt(x)*sqrt((a*x + b)/x)/a^3

________________________________________________________________________________________

giac [B]  time = 0.23, size = 114, normalized size = 1.54 \[ -\frac {2}{105} \, b {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\relax (x) + \frac {2}{315} \, a {\left (\frac {16 \, b^{\frac {9}{2}}}{a^{4}} + \frac {35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 135 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}}{a^{4}}\right )} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(7/2),x, algorithm="giac")

[Out]

-2/105*b*(8*b^(7/2)/a^3 - (15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 35*(a*x + b)^(3/2)*b^2)/a^3)*sgn(x) + 2
/315*a*(16*b^(9/2)/a^4 + (35*(a*x + b)^(9/2) - 135*(a*x + b)^(7/2)*b + 189*(a*x + b)^(5/2)*b^2 - 105*(a*x + b)
^(3/2)*b^3)/a^4)*sgn(x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 44, normalized size = 0.59 \[ \frac {2 \left (a x +b \right ) \left (35 a^{2} x^{2}-20 a b x +8 b^{2}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{315 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(3/2)*x^(7/2),x)

[Out]

2/315*(a*x+b)*(35*a^2*x^2-20*a*b*x+8*b^2)*x^(3/2)*((a*x+b)/x)^(3/2)/a^3

________________________________________________________________________________________

maxima [A]  time = 1.08, size = 52, normalized size = 0.70 \[ \frac {2 \, {\left (35 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} x^{\frac {9}{2}} - 90 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b x^{\frac {7}{2}} + 63 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{2} x^{\frac {5}{2}}\right )}}{315 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(7/2),x, algorithm="maxima")

[Out]

2/315*(35*(a + b/x)^(9/2)*x^(9/2) - 90*(a + b/x)^(7/2)*b*x^(7/2) + 63*(a + b/x)^(5/2)*b^2*x^(5/2))/a^3

________________________________________________________________________________________

mupad [B]  time = 1.47, size = 56, normalized size = 0.76 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{9/2}}{9}+\frac {20\,b\,x^{7/2}}{63}+\frac {2\,b^2\,x^{5/2}}{105\,a}-\frac {8\,b^3\,x^{3/2}}{315\,a^2}+\frac {16\,b^4\,\sqrt {x}}{315\,a^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + b/x)^(3/2),x)

[Out]

(a + b/x)^(1/2)*((2*a*x^(9/2))/9 + (20*b*x^(7/2))/63 + (2*b^2*x^(5/2))/(105*a) - (8*b^3*x^(3/2))/(315*a^2) + (
16*b^4*x^(1/2))/(315*a^3))

________________________________________________________________________________________

sympy [B]  time = 64.43, size = 369, normalized size = 4.99 \[ \frac {70 a^{6} b^{\frac {9}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} + \frac {240 a^{5} b^{\frac {11}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} + \frac {276 a^{4} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} + \frac {104 a^{3} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} + \frac {6 a^{2} b^{\frac {17}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} + \frac {24 a b^{\frac {19}{2}} x \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} + \frac {16 b^{\frac {21}{2}} \sqrt {\frac {a x}{b} + 1}}{315 a^{5} b^{4} x^{2} + 630 a^{4} b^{5} x + 315 a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(3/2)*x**(7/2),x)

[Out]

70*a**6*b**(9/2)*x**6*sqrt(a*x/b + 1)/(315*a**5*b**4*x**2 + 630*a**4*b**5*x + 315*a**3*b**6) + 240*a**5*b**(11
/2)*x**5*sqrt(a*x/b + 1)/(315*a**5*b**4*x**2 + 630*a**4*b**5*x + 315*a**3*b**6) + 276*a**4*b**(13/2)*x**4*sqrt
(a*x/b + 1)/(315*a**5*b**4*x**2 + 630*a**4*b**5*x + 315*a**3*b**6) + 104*a**3*b**(15/2)*x**3*sqrt(a*x/b + 1)/(
315*a**5*b**4*x**2 + 630*a**4*b**5*x + 315*a**3*b**6) + 6*a**2*b**(17/2)*x**2*sqrt(a*x/b + 1)/(315*a**5*b**4*x
**2 + 630*a**4*b**5*x + 315*a**3*b**6) + 24*a*b**(19/2)*x*sqrt(a*x/b + 1)/(315*a**5*b**4*x**2 + 630*a**4*b**5*
x + 315*a**3*b**6) + 16*b**(21/2)*sqrt(a*x/b + 1)/(315*a**5*b**4*x**2 + 630*a**4*b**5*x + 315*a**3*b**6)

________________________________________________________________________________________