3.77 \(\int \frac{(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^{14}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{b^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.0594934, antiderivative size = 253, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{b^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^14,x]

[Out]

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

Mathematica [A]  time = 0.0168434, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (455 a^2 b^3 x^9+260 a^3 b^2 x^6+91 a^4 b x^3+14 a^5+910 a b^4 x^{12}-91 b^5 x^{15}\right )}{182 x^{13} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^14,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(14*a^5 + 91*a^4*b*x^3 + 260*a^3*b^2*x^6 + 455*a^2*b^3*x^9 + 910*a*b^4*x^12 - 91*b^5*x^1
5))/(182*x^13*(a + b*x^3))

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Maple [A]  time = 0.005, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-91\,{b}^{5}{x}^{15}+910\,a{b}^{4}{x}^{12}+455\,{a}^{2}{b}^{3}{x}^{9}+260\,{a}^{3}{b}^{2}{x}^{6}+91\,{a}^{4}b{x}^{3}+14\,{a}^{5}}{182\,{x}^{13} \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^14,x)

[Out]

-1/182*(-91*b^5*x^15+910*a*b^4*x^12+455*a^2*b^3*x^9+260*a^3*b^2*x^6+91*a^4*b*x^3+14*a^5)*((b*x^3+a)^2)^(5/2)/x
^13/(b*x^3+a)^5

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Maxima [A]  time = 1.07088, size = 80, normalized size = 0.32 \begin{align*} \frac{91 \, b^{5} x^{15} - 910 \, a b^{4} x^{12} - 455 \, a^{2} b^{3} x^{9} - 260 \, a^{3} b^{2} x^{6} - 91 \, a^{4} b x^{3} - 14 \, a^{5}}{182 \, x^{13}} \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^14,x, algorithm="maxima")

[Out]

1/182*(91*b^5*x^15 - 910*a*b^4*x^12 - 455*a^2*b^3*x^9 - 260*a^3*b^2*x^6 - 91*a^4*b*x^3 - 14*a^5)/x^13

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Fricas [A]  time = 1.72346, size = 139, normalized size = 0.55 \begin{align*} \frac{91 \, b^{5} x^{15} - 910 \, a b^{4} x^{12} - 455 \, a^{2} b^{3} x^{9} - 260 \, a^{3} b^{2} x^{6} - 91 \, a^{4} b x^{3} - 14 \, a^{5}}{182 \, x^{13}} \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^14,x, algorithm="fricas")

[Out]

1/182*(91*b^5*x^15 - 910*a*b^4*x^12 - 455*a^2*b^3*x^9 - 260*a^3*b^2*x^6 - 91*a^4*b*x^3 - 14*a^5)/x^13

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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^{14}}\, 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**14,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**14, x)

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Giac [A]  time = 1.12583, size = 146, normalized size = 0.58 \begin{align*} \frac{1}{2} \, b^{5} x^{2} \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{910 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 455 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 260 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 91 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 14 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{182 \, x^{13}} \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^14,x, algorithm="giac")

[Out]

1/2*b^5*x^2*sgn(b*x^3 + a) - 1/182*(910*a*b^4*x^12*sgn(b*x^3 + a) + 455*a^2*b^3*x^9*sgn(b*x^3 + a) + 260*a^3*b
^2*x^6*sgn(b*x^3 + a) + 91*a^4*b*x^3*sgn(b*x^3 + a) + 14*a^5*sgn(b*x^3 + a))/x^13