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

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

[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)

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Rubi [A]  time = 0.0441931, antiderivative size = 126, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0246925, size = 64, normalized size = 0.51 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (48 a^2 b^2 x^2-40 a^3 b x^3+35 a^4 x^4-64 a b^3 x+128 b^4\right )}{315 a^5} \]

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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Maple [A]  time = 0.006, size = 66, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 35\,{x}^{4}{a}^{4}-40\,b{x}^{3}{a}^{3}+48\,{b}^{2}{x}^{2}{a}^{2}-64\,{b}^{3}xa+128\,{b}^{4} \right ) }{315\,{a}^{5}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}

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 = 0.958884, size = 116, normalized size = 0.92 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.4332, size = 142, normalized size = 1.13 \begin{align*} \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}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18252, size = 95, normalized size = 0.75 \begin{align*} -\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} + 315 \, \sqrt{a x + b} b^{4}\right )}}{315 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 315*sqrt(a*x + b)*b^4)/a^5