3.245 \(\int \frac{(c+\frac{d}{x})^3}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (-2 a^2 d^2+9 a b c d+3 b^2 c^2\right )+\frac{b d (2 a d+3 b c)}{x}\right )}{3 a b^2}-\frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2}{a} \]

[Out]

-(d*Sqrt[a + b/x]*(2*(3*b^2*c^2 + 9*a*b*c*d - 2*a^2*d^2) + (b*d*(3*b*c + 2*a*d))/x))/(3*a*b^2) + (c*Sqrt[a + b
/x]*(c + d/x)^2*x)/a - (c^2*(b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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Rubi [A]  time = 0.0899013, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {375, 98, 147, 63, 208} \[ -\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (-2 a^2 d^2+9 a b c d+3 b^2 c^2\right )+\frac{b d (2 a d+3 b c)}{x}\right )}{3 a b^2}-\frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2}{a} \]

Antiderivative was successfully verified.

[In]

Int[(c + d/x)^3/Sqrt[a + b/x],x]

[Out]

-(d*Sqrt[a + b/x]*(2*(3*b^2*c^2 + 9*a*b*c*d - 2*a^2*d^2) + (b*d*(3*b*c + 2*a*d))/x))/(3*a*b^2) + (c*Sqrt[a + b
/x]*(c + d/x)^2*x)/a - (c^2*(b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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 98

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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 \frac{\left (c+\frac{d}{x}\right )^3}{\sqrt{a+\frac{b}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{(c+d x)^3}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 x}{a}+\frac{\operatorname{Subst}\left (\int \frac{(c+d x) \left (\frac{1}{2} c (b c-6 a d)-\frac{1}{2} d (3 b c+2 a d) x\right )}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (3 b^2 c^2+9 a b c d-2 a^2 d^2\right )+\frac{b d (3 b c+2 a d)}{x}\right )}{3 a b^2}+\frac{c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 x}{a}+\frac{\left (c^2 (b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (3 b^2 c^2+9 a b c d-2 a^2 d^2\right )+\frac{b d (3 b c+2 a d)}{x}\right )}{3 a b^2}+\frac{c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 x}{a}+\frac{\left (c^2 (b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{a b}\\ &=-\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (3 b^2 c^2+9 a b c d-2 a^2 d^2\right )+\frac{b d (3 b c+2 a d)}{x}\right )}{3 a b^2}+\frac{c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 x}{a}-\frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.109219, size = 95, normalized size = 0.75 \[ \frac{\sqrt{a+\frac{b}{x}} \left (4 a^2 d^3 x-2 a b d^2 (9 c x+d)+3 b^2 c^3 x^2\right )}{3 a b^2 x}+\frac{c^2 (6 a d-b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b/x]*(4*a^2*d^3*x + 3*b^2*c^3*x^2 - 2*a*b*d^2*(d + 9*c*x)))/(3*a*b^2*x) + (c^2*(-(b*c) + 6*a*d)*ArcT
anh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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Maple [B]  time = 0.014, size = 535, normalized size = 4.3 \begin{align*} -{\frac{1}{6\,{b}^{3}{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 6\,{a}^{7/2}\sqrt{a{x}^{2}+bx}{x}^{3}{d}^{3}+6\,{a}^{7/2}\sqrt{ \left ( ax+b \right ) x}{x}^{3}{d}^{3}-18\,{a}^{5/2}\sqrt{a{x}^{2}+bx}{x}^{3}bc{d}^{2}-18\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{3}bc{d}^{2}-12\,{a}^{5/2} \left ( a{x}^{2}+bx \right ) ^{3/2}x{d}^{3}-18\,{a}^{3/2}\sqrt{a{x}^{2}+bx}{x}^{3}{b}^{2}{c}^{2}d+18\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}{x}^{3}{b}^{2}{c}^{2}d+36\,{a}^{3/2} \left ( a{x}^{2}+bx \right ) ^{3/2}xbc{d}^{2}-6\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{x}^{3}{b}^{3}{c}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{3}b{d}^{3}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{2}{b}^{2}c{d}^{2}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}a{b}^{3}{c}^{2}d-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{3}b{d}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{2}{b}^{2}c{d}^{2}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}a{b}^{3}{c}^{2}d+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{b}^{4}{c}^{3}+4\,{d}^{3} \left ( a{x}^{2}+bx \right ) ^{3/2}b{a}^{3/2} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d/x)^3/(a+b/x)^(1/2),x)

[Out]

-1/6*((a*x+b)/x)^(1/2)*(6*a^(7/2)*(a*x^2+b*x)^(1/2)*x^3*d^3+6*a^(7/2)*((a*x+b)*x)^(1/2)*x^3*d^3-18*a^(5/2)*(a*
x^2+b*x)^(1/2)*x^3*b*c*d^2-18*a^(5/2)*((a*x+b)*x)^(1/2)*x^3*b*c*d^2-12*a^(5/2)*(a*x^2+b*x)^(3/2)*x*d^3-18*a^(3
/2)*(a*x^2+b*x)^(1/2)*x^3*b^2*c^2*d+18*a^(3/2)*((a*x+b)*x)^(1/2)*x^3*b^2*c^2*d+36*a^(3/2)*(a*x^2+b*x)^(3/2)*x*
b*c*d^2-6*a^(1/2)*((a*x+b)*x)^(1/2)*x^3*b^3*c^3+3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^
3*b*d^3-9*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^2*b^2*c*d^2-9*ln(1/2*(2*(a*x^2+b*x)^(1/2
)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a*b^3*c^2*d-3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^3*b*
d^3+9*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^2*b^2*c*d^2-9*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^
(1/2)+2*a*x+b)/a^(1/2))*x^3*a*b^3*c^2*d+3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*b^4*c^3+4*
d^3*(a*x^2+b*x)^(3/2)*b*a^(3/2))/x^2/((a*x+b)*x)^(1/2)/b^3/a^(3/2)

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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((c+d/x)^3/(a+b/x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37605, size = 510, normalized size = 4.05 \begin{align*} \left [-\frac{3 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d\right )} \sqrt{a} x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (3 \, a b^{2} c^{3} x^{2} - 2 \, a^{2} b d^{3} - 2 \,{\left (9 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{6 \, a^{2} b^{2} x}, \frac{3 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (3 \, a b^{2} c^{3} x^{2} - 2 \, a^{2} b d^{3} - 2 \,{\left (9 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{3 \, a^{2} b^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d/x)^3/(a+b/x)^(1/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(b^3*c^3 - 6*a*b^2*c^2*d)*sqrt(a)*x*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(3*a*b^2*c^3*x
^2 - 2*a^2*b*d^3 - 2*(9*a^2*b*c*d^2 - 2*a^3*d^3)*x)*sqrt((a*x + b)/x))/(a^2*b^2*x), 1/3*(3*(b^3*c^3 - 6*a*b^2*
c^2*d)*sqrt(-a)*x*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (3*a*b^2*c^3*x^2 - 2*a^2*b*d^3 - 2*(9*a^2*b*c*d^2 - 2
*a^3*d^3)*x)*sqrt((a*x + b)/x))/(a^2*b^2*x)]

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Sympy [A]  time = 40.8405, size = 386, normalized size = 3.06 \begin{align*} \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} d^{3} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} d^{3} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} d^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{4} b d^{3} x^{\frac{5}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{3} b^{2} d^{3} x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + 3 c d^{2} \left (\begin{cases} - \frac{1}{\sqrt{a} x} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{\sqrt{b} c^{3} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{6 c^{2} d \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + \frac{b}{x}}} \right )}}{a \sqrt{- \frac{1}{a}}} - \frac{b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d/x)**3/(a+b/x)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.2012, size = 207, normalized size = 1.64 \begin{align*} -\frac{1}{3} \,{\left (\frac{3 \, c^{3} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} - \frac{3 \,{\left (b c^{3} - 6 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b} + \frac{2 \,{\left (9 \, b^{7} c d^{2} \sqrt{\frac{a x + b}{x}} - 3 \, a b^{6} d^{3} \sqrt{\frac{a x + b}{x}} + \frac{{\left (a x + b\right )} b^{6} d^{3} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{b^{9}}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d/x)^3/(a+b/x)^(1/2),x, algorithm="giac")

[Out]

-1/3*(3*c^3*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a) - 3*(b*c^3 - 6*a*c^2*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))
/(sqrt(-a)*a*b) + 2*(9*b^7*c*d^2*sqrt((a*x + b)/x) - 3*a*b^6*d^3*sqrt((a*x + b)/x) + (a*x + b)*b^6*d^3*sqrt((a
*x + b)/x)/x)/b^9)*b