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

Optimal. Leaf size=251 \[ \frac{b^5 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*x^5*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*
(a + b*x^3)) + (10*a^3*b^2*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (5*a^2*b^3*x^4*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(2*(a + b*x^3)) + (5*a*b^4*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (b^5*x^10*Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3))

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Rubi [A]  time = 0.0575716, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{b^5 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*x^5*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*
(a + b*x^3)) + (10*a^3*b^2*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (5*a^2*b^3*x^4*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(2*(a + b*x^3)) + (5*a*b^4*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (b^5*x^10*Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3))

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]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0217986, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (175 a^2 b^3 x^9+700 a^3 b^2 x^6-175 a^4 b x^3-14 a^5+50 a b^4 x^{12}+7 b^5 x^{15}\right )}{70 x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-14*a^5 - 175*a^4*b*x^3 + 700*a^3*b^2*x^6 + 175*a^2*b^3*x^9 + 50*a*b^4*x^12 + 7*b^5*x^15
))/(70*x^5*(a + b*x^3))

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Maple [A]  time = 0.008, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-7\,{b}^{5}{x}^{15}-50\,a{b}^{4}{x}^{12}-175\,{a}^{2}{b}^{3}{x}^{9}-700\,{a}^{3}{b}^{2}{x}^{6}+175\,{a}^{4}b{x}^{3}+14\,{a}^{5}}{70\,{x}^{5} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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^6,x)

[Out]

-1/70*(-7*b^5*x^15-50*a*b^4*x^12-175*a^2*b^3*x^9-700*a^3*b^2*x^6+175*a^4*b*x^3+14*a^5)*((b*x^3+a)^2)^(5/2)/x^5
/(b*x^3+a)^5

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Maxima [A]  time = 1.12618, size = 80, normalized size = 0.32 \begin{align*} \frac{7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \end{align*}

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^6,x, algorithm="maxima")

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b*x^3 - 14*a^5)/x^5

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Fricas [A]  time = 1.65492, size = 135, normalized size = 0.54 \begin{align*} \frac{7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \end{align*}

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^6,x, algorithm="fricas")

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b*x^3 - 14*a^5)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \end{align*}

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**6,x)

[Out]

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

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Giac [A]  time = 1.11071, size = 143, normalized size = 0.57 \begin{align*} \frac{1}{10} \, b^{5} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{7} \, a b^{4} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{2} \, a^{2} b^{3} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 10 \, a^{3} b^{2} x \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{25 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 2 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{10 \, x^{5}} \end{align*}

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^6,x, algorithm="giac")

[Out]

1/10*b^5*x^10*sgn(b*x^3 + a) + 5/7*a*b^4*x^7*sgn(b*x^3 + a) + 5/2*a^2*b^3*x^4*sgn(b*x^3 + a) + 10*a^3*b^2*x*sg
n(b*x^3 + a) - 1/10*(25*a^4*b*x^3*sgn(b*x^3 + a) + 2*a^5*sgn(b*x^3 + a))/x^5