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

Optimal. Leaf size=251 \[ \frac{b^5 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (5*a^4*b*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a +
b*x^3) + (5*a^3*b^2*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (10*a^2*b^3*x^7*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(7*(a + b*x^3)) + (a*b^4*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (b^5*x^13*Sqrt[
a^2 + 2*a*b*x^3 + b^2*x^6])/(13*(a + b*x^3))

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Rubi [A]  time = 0.0589937, 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} \[ \frac{b^5 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (5*a^4*b*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a +
b*x^3) + (5*a^3*b^2*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (10*a^2*b^3*x^7*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(7*(a + b*x^3)) + (a*b^4*x^10*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (b^5*x^13*Sqrt[
a^2 + 2*a*b*x^3 + b^2*x^6])/(13*(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^3} \, 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^3} \, 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 (5 a^4 b^6+\frac{a^5 b^5}{x^3}+10 a^3 b^7 x^3+10 a^2 b^8 x^6+5 a b^9 x^9+b^{10} x^{12}\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}}{2 x^2 \left (a+b x^3\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{10 a^2 b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b^5 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.0228377, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (260 a^2 b^3 x^9+455 a^3 b^2 x^6+910 a^4 b x^3-91 a^5+91 a b^4 x^{12}+14 b^5 x^{15}\right )}{182 x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-91*a^5 + 910*a^4*b*x^3 + 455*a^3*b^2*x^6 + 260*a^2*b^3*x^9 + 91*a*b^4*x^12 + 14*b^5*x^1
5))/(182*x^2*(a + b*x^3))

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Maple [A]  time = 0.006, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-14\,{b}^{5}{x}^{15}-91\,a{b}^{4}{x}^{12}-260\,{a}^{2}{b}^{3}{x}^{9}-455\,{a}^{3}{b}^{2}{x}^{6}-910\,{a}^{4}b{x}^{3}+91\,{a}^{5}}{182\,{x}^{2} \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^3,x)

[Out]

-1/182*(-14*b^5*x^15-91*a*b^4*x^12-260*a^2*b^3*x^9-455*a^3*b^2*x^6-910*a^4*b*x^3+91*a^5)*((b*x^3+a)^2)^(5/2)/x
^2/(b*x^3+a)^5

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Maxima [A]  time = 1.00467, size = 80, normalized size = 0.32 \begin{align*} \frac{14 \, b^{5} x^{15} + 91 \, a b^{4} x^{12} + 260 \, a^{2} b^{3} x^{9} + 455 \, a^{3} b^{2} x^{6} + 910 \, a^{4} b x^{3} - 91 \, a^{5}}{182 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

1/182*(14*b^5*x^15 + 91*a*b^4*x^12 + 260*a^2*b^3*x^9 + 455*a^3*b^2*x^6 + 910*a^4*b*x^3 - 91*a^5)/x^2

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Fricas [A]  time = 1.73851, size = 138, normalized size = 0.55 \begin{align*} \frac{14 \, b^{5} x^{15} + 91 \, a b^{4} x^{12} + 260 \, a^{2} b^{3} x^{9} + 455 \, a^{3} b^{2} x^{6} + 910 \, a^{4} b x^{3} - 91 \, a^{5}}{182 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/182*(14*b^5*x^15 + 91*a*b^4*x^12 + 260*a^2*b^3*x^9 + 455*a^3*b^2*x^6 + 910*a^4*b*x^3 - 91*a^5)/x^2

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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^{3}}\, 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**3,x)

[Out]

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

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Giac [A]  time = 1.12094, size = 139, normalized size = 0.55 \begin{align*} \frac{1}{13} \, b^{5} x^{13} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{1}{2} \, a b^{4} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{10}{7} \, a^{2} b^{3} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{2} \, a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a^{4} b x \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

1/13*b^5*x^13*sgn(b*x^3 + a) + 1/2*a*b^4*x^10*sgn(b*x^3 + a) + 10/7*a^2*b^3*x^7*sgn(b*x^3 + a) + 5/2*a^3*b^2*x
^4*sgn(b*x^3 + a) + 5*a^4*b*x*sgn(b*x^3 + a) - 1/2*a^5*sgn(b*x^3 + a)/x^2