3.1770 \(\int (a+\frac {b}{x})^{5/2} x^{7/2} \, dx\)

Optimal. Leaf size=48 \[ \frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{7/2}}{9 a}-\frac {4 b x^{7/2} \left (a+\frac {b}{x}\right )^{7/2}}{63 a^2} \]

[Out]

-4/63*b*(a+b/x)^(7/2)*x^(7/2)/a^2+2/9*(a+b/x)^(7/2)*x^(9/2)/a

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Rubi [A]  time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{7/2}}{9 a}-\frac {4 b x^{7/2} \left (a+\frac {b}{x}\right )^{7/2}}{63 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*x^(7/2),x]

[Out]

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

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Mathematica [A]  time = 0.03, size = 38, normalized size = 0.79 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} (a x+b)^3 (7 a x-2 b)}{63 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*x^(7/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^3*(-2*b + 7*a*x))/(63*a^2)

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fricas [A]  time = 0.80, size = 59, normalized size = 1.23 \[ \frac {2 \, {\left (7 \, a^{4} x^{4} + 19 \, a^{3} b x^{3} + 15 \, a^{2} b^{2} x^{2} + a b^{3} x - 2 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{63 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(7/2),x, algorithm="fricas")

[Out]

2/63*(7*a^4*x^4 + 19*a^3*b*x^3 + 15*a^2*b^2*x^2 + a*b^3*x - 2*b^4)*sqrt(x)*sqrt((a*x + b)/x)/a^2

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giac [B]  time = 0.20, size = 157, normalized size = 3.27 \[ \frac {2}{15} \, b^{2} {\left (\frac {2 \, b^{\frac {5}{2}}}{a^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} b}{a^{2}}\right )} \mathrm {sgn}\relax (x) - \frac {4}{105} \, a b {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\relax (x) + \frac {2}{315} \, a^{2} {\left (\frac {16 \, b^{\frac {9}{2}}}{a^{4}} + \frac {35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 135 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}}{a^{4}}\right )} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(7/2),x, algorithm="giac")

[Out]

2/15*b^2*(2*b^(5/2)/a^2 + (3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)/a^2)*sgn(x) - 4/105*a*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^2*(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)

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maple [A]  time = 0.00, size = 33, normalized size = 0.69 \[ \frac {2 \left (a x +b \right ) \left (7 a x -2 b \right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} x^{\frac {5}{2}}}{63 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)*x^(7/2),x)

[Out]

2/63*(a*x+b)*(7*a*x-2*b)*x^(5/2)*((a*x+b)/x)^(5/2)/a^2

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maxima [A]  time = 1.10, size = 35, normalized size = 0.73 \[ \frac {2 \, {\left (7 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} x^{\frac {9}{2}} - 9 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b x^{\frac {7}{2}}\right )}}{63 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(7/2),x, algorithm="maxima")

[Out]

2/63*(7*(a + b/x)^(9/2)*x^(9/2) - 9*(a + b/x)^(7/2)*b*x^(7/2))/a^2

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mupad [B]  time = 1.45, size = 56, normalized size = 1.17 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a^2\,x^{9/2}}{9}+\frac {10\,b^2\,x^{5/2}}{21}+\frac {2\,b^3\,x^{3/2}}{63\,a}-\frac {4\,b^4\,\sqrt {x}}{63\,a^2}+\frac {38\,a\,b\,x^{7/2}}{63}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + b/x)^(5/2),x)

[Out]

(a + b/x)^(1/2)*((2*a^2*x^(9/2))/9 + (10*b^2*x^(5/2))/21 + (2*b^3*x^(3/2))/(63*a) - (4*b^4*x^(1/2))/(63*a^2) +
(38*a*b*x^(7/2))/63)

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sympy [B]  time = 98.84, size = 114, normalized size = 2.38 \[ \frac {2 a^{2} \sqrt {b} x^{4} \sqrt {\frac {a x}{b} + 1}}{9} + \frac {38 a b^{\frac {3}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{63} + \frac {10 b^{\frac {5}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{21} + \frac {2 b^{\frac {7}{2}} x \sqrt {\frac {a x}{b} + 1}}{63 a} - \frac {4 b^{\frac {9}{2}} \sqrt {\frac {a x}{b} + 1}}{63 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(5/2)*x**(7/2),x)

[Out]

2*a**2*sqrt(b)*x**4*sqrt(a*x/b + 1)/9 + 38*a*b**(3/2)*x**3*sqrt(a*x/b + 1)/63 + 10*b**(5/2)*x**2*sqrt(a*x/b +
1)/21 + 2*b**(7/2)*x*sqrt(a*x/b + 1)/(63*a) - 4*b**(9/2)*sqrt(a*x/b + 1)/(63*a**2)

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