\(\int \frac {(a^2+2 a b x+b^2 x^2)^{3/2}}{x^3} \, dx\) [156]

Optimal result

Integrand size = 24, antiderivative size = 141 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]

[Out]

-1/2*a^3*((b*x+a)^2)^(1/2)/x^2/(b*x+a)-3*a^2*b*((b*x+a)^2)^(1/2)/x/(b*x+a)+b^3*x*((b*x+a)^2)^(1/2)/(b*x+a)+3*a
*b^2*ln(x)*((b*x+a)^2)^(1/2)/(b*x+a)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b^2 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^3,x]

[Out]

-1/2*(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^2*(a + b*x)) - (3*a^2*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x
)) + (b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (3*a*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x
)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^3} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^6+\frac {a^3 b^3}{x^3}+\frac {3 a^2 b^4}{x^2}+\frac {3 a b^5}{x}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^3 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(554\) vs. \(2(141)=282\).

Time = 1.14 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.93 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {4 a^4 \sqrt {a^2}+28 a^3 \sqrt {a^2} b x+35 \left (a^2\right )^{3/2} b^2 x^2+3 a \sqrt {a^2} b^3 x^3-8 \sqrt {a^2} b^4 x^4-4 a^4 \sqrt {(a+b x)^2}-24 a^3 b x \sqrt {(a+b x)^2}-11 a^2 b^2 x^2 \sqrt {(a+b x)^2}+8 a b^3 x^3 \sqrt {(a+b x)^2}-24 a b^2 x^2 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right ) \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )-24 b^2 x^2 \left (\left (a^2\right )^{3/2}+a \sqrt {a^2} b x-a^2 \sqrt {(a+b x)^2}\right ) \log (x)+12 \left (a^2\right )^{3/2} b^2 x^2 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+12 a \sqrt {a^2} b^3 x^3 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )-12 a^2 b^2 x^2 \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+12 \left (a^2\right )^{3/2} b^2 x^2 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )+12 a \sqrt {a^2} b^3 x^3 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )-12 a^2 b^2 x^2 \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{8 x^2 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]

[In]

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

[Out]

(4*a^4*Sqrt[a^2] + 28*a^3*Sqrt[a^2]*b*x + 35*(a^2)^(3/2)*b^2*x^2 + 3*a*Sqrt[a^2]*b^3*x^3 - 8*Sqrt[a^2]*b^4*x^4
- 4*a^4*Sqrt[(a + b*x)^2] - 24*a^3*b*x*Sqrt[(a + b*x)^2] - 11*a^2*b^2*x^2*Sqrt[(a + b*x)^2] + 8*a*b^3*x^3*Sqr
t[(a + b*x)^2] - 24*a*b^2*x^2*(a^2 + a*b*x - Sqrt[a^2]*Sqrt[(a + b*x)^2])*ArcTanh[(b*x)/(Sqrt[a^2] - Sqrt[(a +
b*x)^2])] - 24*b^2*x^2*((a^2)^(3/2) + a*Sqrt[a^2]*b*x - a^2*Sqrt[(a + b*x)^2])*Log[x] + 12*(a^2)^(3/2)*b^2*x^
2*Log[Sqrt[a^2] - b*x - Sqrt[(a + b*x)^2]] + 12*a*Sqrt[a^2]*b^3*x^3*Log[Sqrt[a^2] - b*x - Sqrt[(a + b*x)^2]] -
12*a^2*b^2*x^2*Sqrt[(a + b*x)^2]*Log[Sqrt[a^2] - b*x - Sqrt[(a + b*x)^2]] + 12*(a^2)^(3/2)*b^2*x^2*Log[Sqrt[a
^2] + b*x - Sqrt[(a + b*x)^2]] + 12*a*Sqrt[a^2]*b^3*x^3*Log[Sqrt[a^2] + b*x - Sqrt[(a + b*x)^2]] - 12*a^2*b^2*
x^2*Sqrt[(a + b*x)^2]*Log[Sqrt[a^2] + b*x - Sqrt[(a + b*x)^2]])/(8*x^2*(a^2 + a*b*x - Sqrt[a^2]*Sqrt[(a + b*x)
^2]))

Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.38

method	result	size
default	\(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (6 a \,b^{2} \ln \left (x \right ) x^{2}+2 b^{3} x^{3}-6 a^{2} b x -a^{3}\right )}{2 x^{2} \left (b x +a \right )^{3}}\)	\(54\)
risch	\(\frac {b^{3} x \sqrt {\left (b x +a \right )^{2}}}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-3 a^{2} b x -\frac {1}{2} a^{3}\right )}{\left (b x +a \right ) x^{2}}+\frac {3 a \,b^{2} \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(80\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^(3/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/2*((b*x+a)^2)^(3/2)*(6*a*b^2*ln(x)*x^2+2*b^3*x^3-6*a^2*b*x-a^3)/x^2/(b*x+a)^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.26 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

1/2*(2*b^3*x^3 + 6*a*b^2*x^2*log(x) - 6*a^2*b*x - a^3)/x^2

Sympy [F]

\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(((a + b*x)**2)**(3/2)/x**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (95) = 190\).

Time = 0.20 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a b^{2} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3} x}{2 \, a} + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}}{2 \, a^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b}{2 \, a x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{2 \, a^{2} x^{2}} \]

[In]

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

[Out]

3*(-1)^(2*b^2*x + 2*a*b)*a*b^2*log(2*b^2*x + 2*a*b) - 3*(-1)^(2*a*b*x + 2*a^2)*a*b^2*log(2*a*b*x/abs(x) + 2*a^
2/abs(x)) + 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3*x/a + 9/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2 + 1/2*(b^2*x^2 +
2*a*b*x + a^2)^(3/2)*b^2/a^2 - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b/(a*x) - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(5
/2)/(a^2*x^2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=b^{3} x \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {6 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + a^{3} \mathrm {sgn}\left (b x + a\right )}{2 \, x^{2}} \]

[In]

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

[Out]

b^3*x*sgn(b*x + a) + 3*a*b^2*log(abs(x))*sgn(b*x + a) - 1/2*(6*a^2*b*x*sgn(b*x + a) + a^3*sgn(b*x + a))/x^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x^3} \,d x \]

[In]

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

[Out]

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