3.232 \(\int (a+\frac{b}{x})^{3/2} (c+\frac{d}{x})^2 \, dx\)

Optimal. Leaf size=126 \[ \frac{c^2 x \left (a+\frac{b}{x}\right )^{5/2}}{a}-\frac{c \left (a+\frac{b}{x}\right )^{3/2} (4 a d+3 b c)}{3 a}-c \sqrt{a+\frac{b}{x}} (4 a d+3 b c)+\sqrt{a} c (4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

[Out]

-(c*(3*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(3*b*c + 4*a*d)*(a + b/x)^(3/2))/(3*a) - (2*d^2*(a + b/x)^(5/2))/(5*b)
+ (c^2*(a + b/x)^(5/2)*x)/a + Sqrt[a]*c*(3*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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Rubi [A]  time = 0.0795285, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {375, 89, 80, 50, 63, 208} \[ \frac{c^2 x \left (a+\frac{b}{x}\right )^{5/2}}{a}-\frac{c \left (a+\frac{b}{x}\right )^{3/2} (4 a d+3 b c)}{3 a}-c \sqrt{a+\frac{b}{x}} (4 a d+3 b c)+\sqrt{a} c (4 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*(c + d/x)^2,x]

[Out]

-(c*(3*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(3*b*c + 4*a*d)*(a + b/x)^(3/2))/(3*a) - (2*d^2*(a + b/x)^(5/2))/(5*b)
+ (c^2*(a + b/x)^(5/2)*x)/a + Sqrt[a]*c*(3*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (c+d x)^2}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^2 \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} \left (\frac{1}{2} c (3 b c+4 a d)+a d^2 x\right )}{x} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b}+\frac{c^2 \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{(c (3 b c+4 a d)) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{c (3 b c+4 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b}+\frac{c^2 \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{1}{2} (c (3 b c+4 a d)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-c (3 b c+4 a d) \sqrt{a+\frac{b}{x}}-\frac{c (3 b c+4 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b}+\frac{c^2 \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{1}{2} (a c (3 b c+4 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-c (3 b c+4 a d) \sqrt{a+\frac{b}{x}}-\frac{c (3 b c+4 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b}+\frac{c^2 \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{(a c (3 b c+4 a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{b}\\ &=-c (3 b c+4 a d) \sqrt{a+\frac{b}{x}}-\frac{c (3 b c+4 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b}+\frac{c^2 \left (a+\frac{b}{x}\right )^{5/2} x}{a}+\sqrt{a} c (3 b c+4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [A]  time = 0.176633, size = 106, normalized size = 0.84 \[ -\frac{c (4 a d+3 b c) \left (\sqrt{a+\frac{b}{x}} (4 a x+b)-3 a^{3/2} x \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\right )}{3 a x}+\frac{c^2 x \left (a+\frac{b}{x}\right )^{5/2}}{a}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*(c + d/x)^2,x]

[Out]

(-2*d^2*(a + b/x)^(5/2))/(5*b) + (c^2*(a + b/x)^(5/2)*x)/a - (c*(3*b*c + 4*a*d)*(Sqrt[a + b/x]*(b + 4*a*x) - 3
*a^(3/2)*x*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]))/(3*a*x)

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Maple [B]  time = 0.011, size = 260, normalized size = 2.1 \begin{align*} -{\frac{1}{30\,b{x}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -120\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{4}cd-90\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{4}b{c}^{2}-60\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{4}{a}^{2}bcd-45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{4}a{b}^{2}{c}^{2}+120\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}{x}^{2}cd+60\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}{x}^{2}b{c}^{2}+12\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}x{d}^{2}+40\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}xbcd+12\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}b{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*((a*x+b)/x)^(1/2)/x^3/b*(-120*(a*x^2+b*x)^(1/2)*a^(5/2)*x^4*c*d-90*(a*x^2+b*x)^(1/2)*a^(3/2)*x^4*b*c^2-6
0*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^4*a^2*b*c*d-45*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2
*a*x+b)/a^(1/2))*x^4*a*b^2*c^2+120*(a*x^2+b*x)^(3/2)*a^(3/2)*x^2*c*d+60*(a*x^2+b*x)^(3/2)*a^(1/2)*x^2*b*c^2+12
*(a*x^2+b*x)^(3/2)*a^(3/2)*x*d^2+40*(a*x^2+b*x)^(3/2)*a^(1/2)*x*b*c*d+12*(a*x^2+b*x)^(3/2)*a^(1/2)*b*d^2)/((a*
x+b)*x)^(1/2)/a^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24766, size = 610, normalized size = 4.84 \begin{align*} \left [\frac{15 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt{a} x^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \,{\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{30 \, b x^{2}}, -\frac{15 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (15 \, a b c^{2} x^{3} - 6 \, b^{2} d^{2} - 2 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \,{\left (5 \, b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{15 \, b x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(a)*x^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(15*a*b*c^2*x
^3 - 6*b^2*d^2 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^2*d^2)*x^2 - 4*(5*b^2*c*d + 3*a*b*d^2)*x)*sqrt((a*x + b)/x))
/(b*x^2), -1/15*(15*(3*b^2*c^2 + 4*a*b*c*d)*sqrt(-a)*x^2*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (15*a*b*c^2*x^
3 - 6*b^2*d^2 - 2*(15*b^2*c^2 + 40*a*b*c*d + 3*a^2*d^2)*x^2 - 4*(5*b^2*c*d + 3*a*b*d^2)*x)*sqrt((a*x + b)/x))/
(b*x^2)]

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Sympy [A]  time = 44.1276, size = 534, normalized size = 4.24 \begin{align*} \frac{4 a^{\frac{11}{2}} b^{\frac{5}{2}} d^{2} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{7}{2}} d^{2} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{9}{2}} d^{2} x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} d^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \sqrt{a} b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{4 a^{6} b^{2} d^{2} x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{3} d^{2} x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{2} c d \operatorname{atan}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + a \sqrt{b} c^{2} \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{2 a b c^{2} \operatorname{atan}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - 4 a c d \sqrt{a + \frac{b}{x}} + a d^{2} \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) - 2 b c^{2} \sqrt{a + \frac{b}{x}} + 2 b c d \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a**(11/2)*b**(5/2)*d**2*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 2*a**
(9/2)*b**(7/2)*d**2*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**(7/2)*
b**(9/2)*d**2*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**(5/2)*b**(11/2)
*d**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + sqrt(a)*b*c**2*asinh(sqrt(a)*s
qrt(x)/sqrt(b)) - 4*a**6*b**2*d**2*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**5*b
**3*d**2*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**2*c*d*atan(sqrt(a + b/x)/sqrt
(-a))/sqrt(-a) + a*sqrt(b)*c**2*sqrt(x)*sqrt(a*x/b + 1) - 2*a*b*c**2*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) - 4
*a*c*d*sqrt(a + b/x) + a*d**2*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) - 2*b*c**2*
sqrt(a + b/x) + 2*b*c*d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError