3.1.73 \(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=252 \[ \frac {b^5 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {10 a^2 b^3 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \begin {gather*} \frac {b^5 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}+\frac {10 a^2 b^3 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^10,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(9*x^9*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*x^6*
(a + b*x^3)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*x^3*(a + b*x^3)) + (5*a*b^4*x^3*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (b^5*x^6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(6*(a + b*x^3)) + (10*a^2*b^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

Rule 43

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{10}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{10}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^4} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \operatorname {Subst}\left (\int \left (5 a b^9+\frac {a^5 b^5}{x^4}+\frac {5 a^4 b^6}{x^3}+\frac {10 a^3 b^7}{x^2}+\frac {10 a^2 b^8}{x}+b^{10} x\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {b^5 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 85, normalized size = 0.34 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-2 a^5-15 a^4 b x^3-60 a^3 b^2 x^6+180 a^2 b^3 x^9 \log (x)+30 a b^4 x^{12}+3 b^5 x^{15}\right )}{18 x^9 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^10,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(-2*a^5 - 15*a^4*b*x^3 - 60*a^3*b^2*x^6 + 30*a*b^4*x^12 + 3*b^5*x^15 + 180*a^2*b^3*x^9*Lo
g[x]))/(18*x^9*(a + b*x^3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.82, size = 366, normalized size = 1.45 \begin {gather*} -\frac {5}{3} a^2 \left (b^2\right )^{3/2} \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}-a-\sqrt {b^2} x^3\right )-\frac {5}{3} a^2 \left (b^2\right )^{3/2} \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}+a-\sqrt {b^2} x^3\right )+\frac {10}{3} a^2 b^3 \tanh ^{-1}\left (\frac {\sqrt {b^2} x^3}{a}-\frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{a}\right )+\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (-8 a^5 b-60 a^4 b^2 x^3-240 a^3 b^3 x^6-391 a^2 b^4 x^9+120 a b^5 x^{12}+12 b^6 x^{15}\right )+\sqrt {b^2} \left (8 a^6+68 a^5 b x^3+300 a^4 b^2 x^6+631 a^3 b^3 x^9+271 a^2 b^4 x^{12}-132 a b^5 x^{15}-12 b^6 x^{18}\right )}{72 x^9 \left (a b+b^2 x^3\right )-72 \sqrt {b^2} x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^10,x]

[Out]

(Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-8*a^5*b - 60*a^4*b^2*x^3 - 240*a^3*b^3*x^6 - 391*a^2*b^4*x^9 + 120*a*b^5*x^
12 + 12*b^6*x^15) + Sqrt[b^2]*(8*a^6 + 68*a^5*b*x^3 + 300*a^4*b^2*x^6 + 631*a^3*b^3*x^9 + 271*a^2*b^4*x^12 - 1
32*a*b^5*x^15 - 12*b^6*x^18))/(72*x^9*(a*b + b^2*x^3) - 72*Sqrt[b^2]*x^9*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (1
0*a^2*b^3*ArcTanh[(Sqrt[b^2]*x^3)/a - Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/a])/3 - (5*a^2*(b^2)^(3/2)*Log[-a - Sqrt
[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/3 - (5*a^2*(b^2)^(3/2)*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*
x^3 + b^2*x^6]])/3

________________________________________________________________________________________

fricas [A]  time = 1.72, size = 61, normalized size = 0.24 \begin {gather*} \frac {3 \, b^{5} x^{15} + 30 \, a b^{4} x^{12} + 180 \, a^{2} b^{3} x^{9} \log \relax (x) - 60 \, a^{3} b^{2} x^{6} - 15 \, a^{4} b x^{3} - 2 \, a^{5}}{18 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^10,x, algorithm="fricas")

[Out]

1/18*(3*b^5*x^15 + 30*a*b^4*x^12 + 180*a^2*b^3*x^9*log(x) - 60*a^3*b^2*x^6 - 15*a^4*b*x^3 - 2*a^5)/x^9

________________________________________________________________________________________

giac [A]  time = 0.40, size = 127, normalized size = 0.50 \begin {gather*} \frac {1}{6} \, b^{5} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{3} \, a b^{4} x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {110 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 60 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 15 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{18 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^10,x, algorithm="giac")

[Out]

1/6*b^5*x^6*sgn(b*x^3 + a) + 5/3*a*b^4*x^3*sgn(b*x^3 + a) + 10*a^2*b^3*log(abs(x))*sgn(b*x^3 + a) - 1/18*(110*
a^2*b^3*x^9*sgn(b*x^3 + a) + 60*a^3*b^2*x^6*sgn(b*x^3 + a) + 15*a^4*b*x^3*sgn(b*x^3 + a) + 2*a^5*sgn(b*x^3 + a
))/x^9

________________________________________________________________________________________

maple [A]  time = 0.01, size = 82, normalized size = 0.33 \begin {gather*} \frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} \left (3 b^{5} x^{15}+30 a \,b^{4} x^{12}+180 a^{2} b^{3} x^{9} \ln \relax (x )-60 a^{3} b^{2} x^{6}-15 a^{4} b \,x^{3}-2 a^{5}\right )}{18 \left (b \,x^{3}+a \right )^{5} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^10,x)

[Out]

1/18*((b*x^3+a)^2)^(5/2)*(3*b^5*x^15+30*a*b^4*x^12+180*a^2*b^3*ln(x)*x^9-60*a^3*b^2*x^6-15*a^4*b*x^3-2*a^5)/(b
*x^3+a)^5/x^9

________________________________________________________________________________________

maxima [A]  time = 0.80, size = 313, normalized size = 1.24 \begin {gather*} \frac {5}{3} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{4} x^{3} + \frac {10}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a^{2} b^{3} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {10}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a^{2} b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{4} x^{3}}{6 \, a^{2}} + 5 \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a b^{3} + \frac {35 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{3}}{18 \, a} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{3}}{18 \, a^{3}} - \frac {11 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{2}}{18 \, a^{2} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{18 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{9 \, a^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^10,x, algorithm="maxima")

[Out]

5/3*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*b^4*x^3 + 10/3*(-1)^(2*b^2*x^3 + 2*a*b)*a^2*b^3*log(2*b^2*x^3 + 2*a*b) - 1
0/3*(-1)^(2*a*b*x^3 + 2*a^2)*a^2*b^3*log(2*a*b*x/abs(x) + 2*a^2/(x^2*abs(x))) + 5/6*(b^2*x^6 + 2*a*b*x^3 + a^2
)^(3/2)*b^4*x^3/a^2 + 5*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)*a*b^3 + 35/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*b^3/a
+ 1/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^3/a^3 - 11/18*(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*b^2/(a^2*x^3) - 1/1
8*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)*b/(a^3*x^6) - 1/9*(b^2*x^6 + 2*a*b*x^3 + a^2)^(7/2)/(a^2*x^9)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^10,x)

[Out]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^10, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{10}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**10,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**10, x)

________________________________________________________________________________________