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

Optimal. Leaf size=251 \[ \frac{b^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{2 a^3 b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]

[Out]

-((a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3))) + (5*a^4*b*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a
+ b*x^3)) + (2*a^3*b^2*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (5*a^2*b^3*x^8*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(4*(a + b*x^3)) + (5*a*b^4*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a + b*x^3)) + (b^5*x^14*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a + b*x^3))

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Rubi [A]  time = 0.0605986, 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} \[ \frac{b^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{2 a^3 b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3))) + (5*a^4*b*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a
+ b*x^3)) + (2*a^3*b^2*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(a + b*x^3) + (5*a^2*b^3*x^8*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(4*(a + b*x^3)) + (5*a*b^4*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a + b*x^3)) + (b^5*x^14*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(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^2} \, 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^2} \, 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 (\frac{a^5 b^5}{x^2}+5 a^4 b^6 x+10 a^3 b^7 x^4+10 a^2 b^8 x^7+5 a b^9 x^{10}+b^{10} x^{13}\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}}{x \left (a+b x^3\right )}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{2 a^3 b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^2 b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{b^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}\\ \end{align*}

Mathematica [A]  time = 0.020515, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (385 a^2 b^3 x^9+616 a^3 b^2 x^6+770 a^4 b x^3-308 a^5+140 a b^4 x^{12}+22 b^5 x^{15}\right )}{308 x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-308*a^5 + 770*a^4*b*x^3 + 616*a^3*b^2*x^6 + 385*a^2*b^3*x^9 + 140*a*b^4*x^12 + 22*b^5*x
^15))/(308*x*(a + b*x^3))

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Maple [A]  time = 0.006, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-22\,{b}^{5}{x}^{15}-140\,a{b}^{4}{x}^{12}-385\,{a}^{2}{b}^{3}{x}^{9}-616\,{a}^{3}{b}^{2}{x}^{6}-770\,{a}^{4}b{x}^{3}+308\,{a}^{5}}{308\,x \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^2,x)

[Out]

-1/308*(-22*b^5*x^15-140*a*b^4*x^12-385*a^2*b^3*x^9-616*a^3*b^2*x^6-770*a^4*b*x^3+308*a^5)*((b*x^3+a)^2)^(5/2)
/x/(b*x^3+a)^5

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Maxima [A]  time = 1.18157, size = 80, normalized size = 0.32 \begin{align*} \frac{22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \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^2,x, algorithm="maxima")

[Out]

1/308*(22*b^5*x^15 + 140*a*b^4*x^12 + 385*a^2*b^3*x^9 + 616*a^3*b^2*x^6 + 770*a^4*b*x^3 - 308*a^5)/x

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Fricas [A]  time = 1.71525, size = 138, normalized size = 0.55 \begin{align*} \frac{22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \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^2,x, algorithm="fricas")

[Out]

1/308*(22*b^5*x^15 + 140*a*b^4*x^12 + 385*a^2*b^3*x^9 + 616*a^3*b^2*x^6 + 770*a^4*b*x^3 - 308*a^5)/x

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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^{2}}\, 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**2,x)

[Out]

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

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Giac [A]  time = 1.12656, size = 142, normalized size = 0.57 \begin{align*} \frac{1}{14} \, b^{5} x^{14} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{11} \, a b^{4} x^{11} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{4} \, a^{2} b^{3} x^{8} \mathrm{sgn}\left (b x^{3} + a\right ) + 2 \, a^{3} b^{2} x^{5} \mathrm{sgn}\left (b x^{3} + a\right ) + \frac{5}{2} \, a^{4} b x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{x} \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^2,x, algorithm="giac")

[Out]

1/14*b^5*x^14*sgn(b*x^3 + a) + 5/11*a*b^4*x^11*sgn(b*x^3 + a) + 5/4*a^2*b^3*x^8*sgn(b*x^3 + a) + 2*a^3*b^2*x^5
*sgn(b*x^3 + a) + 5/2*a^4*b*x^2*sgn(b*x^3 + a) - a^5*sgn(b*x^3 + a)/x