\(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^5} \, dx\) [173]

Optimal result

Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]

[Out]

-1/4*a^5*((b*x+a)^2)^(1/2)/x^4/(b*x+a)-5/3*a^4*b*((b*x+a)^2)^(1/2)/x^3/(b*x+a)-5*a^3*b^2*((b*x+a)^2)^(1/2)/x^2
/(b*x+a)-10*a^2*b^3*((b*x+a)^2)^(1/2)/x/(b*x+a)+b^5*x*((b*x+a)^2)^(1/2)/(b*x+a)+5*a*b^4*ln(x)*((b*x+a)^2)^(1/2
)/(b*x+a)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 219, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^5,x]

[Out]

-1/4*(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^4*(a + b*x)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a +
b*x)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^2*(a + b*x)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(x*(a + b*x)) + (b^5*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a*b^4*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 )^5}{x^5} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^{10}+\frac {a^5 b^5}{x^5}+\frac {5 a^4 b^6}{x^4}+\frac {10 a^3 b^7}{x^3}+\frac {10 a^2 b^8}{x^2}+\frac {5 a b^9}{x}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \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. \(634\) vs. \(2(219)=438\).

Time = 1.35 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {48 a^6 \sqrt {a^2}+368 a^5 \sqrt {a^2} b x+1280 a^4 \sqrt {a^2} b^2 x^2+2880 a^3 \sqrt {a^2} b^3 x^3+2677 \left (a^2\right )^{3/2} b^4 x^4+565 a \sqrt {a^2} b^5 x^5-192 \sqrt {a^2} b^6 x^6-48 a^6 \sqrt {(a+b x)^2}-320 a^5 b x \sqrt {(a+b x)^2}-960 a^4 b^2 x^2 \sqrt {(a+b x)^2}-1920 a^3 b^3 x^3 \sqrt {(a+b x)^2}-757 a^2 b^4 x^4 \sqrt {(a+b x)^2}+192 a b^5 x^5 \sqrt {(a+b x)^2}-960 a b^4 x^4 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right ) \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )-960 b^4 x^4 \left (\left (a^2\right )^{3/2}+a \sqrt {a^2} b x-a^2 \sqrt {(a+b x)^2}\right ) \log (x)+480 \left (a^2\right )^{3/2} b^4 x^4 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+480 a \sqrt {a^2} b^5 x^5 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )-480 a^2 b^4 x^4 \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+480 \left (a^2\right )^{3/2} b^4 x^4 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )+480 a \sqrt {a^2} b^5 x^5 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )-480 a^2 b^4 x^4 \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{192 x^4 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^5,x]

[Out]

(48*a^6*Sqrt[a^2] + 368*a^5*Sqrt[a^2]*b*x + 1280*a^4*Sqrt[a^2]*b^2*x^2 + 2880*a^3*Sqrt[a^2]*b^3*x^3 + 2677*(a^
2)^(3/2)*b^4*x^4 + 565*a*Sqrt[a^2]*b^5*x^5 - 192*Sqrt[a^2]*b^6*x^6 - 48*a^6*Sqrt[(a + b*x)^2] - 320*a^5*b*x*Sq
rt[(a + b*x)^2] - 960*a^4*b^2*x^2*Sqrt[(a + b*x)^2] - 1920*a^3*b^3*x^3*Sqrt[(a + b*x)^2] - 757*a^2*b^4*x^4*Sqr
t[(a + b*x)^2] + 192*a*b^5*x^5*Sqrt[(a + b*x)^2] - 960*a*b^4*x^4*(a^2 + a*b*x - Sqrt[a^2]*Sqrt[(a + b*x)^2])*A
rcTanh[(b*x)/(Sqrt[a^2] - Sqrt[(a + b*x)^2])] - 960*b^4*x^4*((a^2)^(3/2) + a*Sqrt[a^2]*b*x - a^2*Sqrt[(a + b*x
)^2])*Log[x] + 480*(a^2)^(3/2)*b^4*x^4*Log[Sqrt[a^2] - b*x - Sqrt[(a + b*x)^2]] + 480*a*Sqrt[a^2]*b^5*x^5*Log[
Sqrt[a^2] - b*x - Sqrt[(a + b*x)^2]] - 480*a^2*b^4*x^4*Sqrt[(a + b*x)^2]*Log[Sqrt[a^2] - b*x - Sqrt[(a + b*x)^
2]] + 480*(a^2)^(3/2)*b^4*x^4*Log[Sqrt[a^2] + b*x - Sqrt[(a + b*x)^2]] + 480*a*Sqrt[a^2]*b^5*x^5*Log[Sqrt[a^2]
+ b*x - Sqrt[(a + b*x)^2]] - 480*a^2*b^4*x^4*Sqrt[(a + b*x)^2]*Log[Sqrt[a^2] + b*x - Sqrt[(a + b*x)^2]])/(192
*x^4*(a^2 + a*b*x - Sqrt[a^2]*Sqrt[(a + b*x)^2]))

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35

method	result	size
default	\(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 a \,b^{4} \ln \left (x \right ) x^{4}+12 b^{5} x^{5}-120 a^{2} b^{3} x^{3}-60 a^{3} b^{2} x^{2}-20 a^{4} b x -3 a^{5}\right )}{12 x^{4} \left (b x +a \right )^{5}}\)	\(76\)
risch	\(\frac {b^{5} x \sqrt {\left (b x +a \right )^{2}}}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-10 a^{2} b^{3} x^{3}-5 a^{3} b^{2} x^{2}-\frac {5}{3} a^{4} b x -\frac {1}{4} a^{5}\right )}{\left (b x +a \right ) x^{4}}+\frac {5 a \,b^{4} \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(102\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/12*((b*x+a)^2)^(5/2)*(60*a*b^4*ln(x)*x^4+12*b^5*x^5-120*a^2*b^3*x^3-60*a^3*b^2*x^2-20*a^4*b*x-3*a^5)/x^4/(b*
x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.27 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \]

[In]

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

[Out]

1/12*(12*b^5*x^5 + 60*a*b^4*x^4*log(x) - 120*a^2*b^3*x^3 - 60*a^3*b^2*x^2 - 20*a^4*b*x - 3*a^5)/x^4

Sympy [F]

\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{5}}\, dx \]

[In]

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

[Out]

Integral(((a + b*x)**2)**(5/2)/x**5, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (149) = 298\).

Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a b^{4} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5} x}{2 \, a} + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5} x}{4 \, a^{3}} + \frac {35 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{12 \, a^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{4}}{3 \, a^{4}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{3 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{3 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{4 \, a^{2} x^{4}} \]

[In]

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

[Out]

5*(-1)^(2*b^2*x + 2*a*b)*a*b^4*log(2*b^2*x + 2*a*b) - 5*(-1)^(2*a*b*x + 2*a^2)*a*b^4*log(2*a*b*x/abs(x) + 2*a^
2/abs(x)) + 5/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5*x/a + 15/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4 + 5/4*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*b^5*x/a^3 + 35/12*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^4/a^2 + 1/3*(b^2*x^2 + 2*a*b*x + a^
2)^(5/2)*b^4/a^4 - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^3/(a^3*x) - 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^2/(
a^4*x^2) + 1/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b/(a^3*x^3) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/(a^2*x^4)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=b^{5} x \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {120 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 60 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{12 \, x^{4}} \]

[In]

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

[Out]

b^5*x*sgn(b*x + a) + 5*a*b^4*log(abs(x))*sgn(b*x + a) - 1/12*(120*a^2*b^3*x^3*sgn(b*x + a) + 60*a^3*b^2*x^2*sg
n(b*x + a) + 20*a^4*b*x*sgn(b*x + a) + 3*a^5*sgn(b*x + a))/x^4

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^5} \,d x \]

[In]

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

[Out]

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