3.599 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{15}} \, dx\)

Optimal. Leaf size=72 \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}-\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}} \]

[Out]

-((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*a*x^14) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(7/2)/(84*a^2*x^14)

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Rubi [A]  time = 0.0168661, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1110} \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}-\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

-((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*a*x^14) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(7/2)/(84*a^2*x^14)

Rule 1110

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2
+ c*x^4)^(p + 1))/(4*a*d*(p + 1)*(2*p + 1)), x] - Simp[((d*x)^(m + 1)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^p)/(4*
a*d*(2*p + 1)), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && EqQ[m + 4*p + 5,
0] && NeQ[p, -2^(-1)]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx &=-\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}}+\frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}\\ \end{align*}

Mathematica [A]  time = 0.0197089, size = 83, normalized size = 1.15 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (105 a^2 b^3 x^6+84 a^3 b^2 x^4+35 a^4 b x^2+6 a^5+70 a b^4 x^8+21 b^5 x^{10}\right )}{84 x^{14} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(6*a^5 + 35*a^4*b*x^2 + 84*a^3*b^2*x^4 + 105*a^2*b^3*x^6 + 70*a*b^4*x^8 + 21*b^5*x^10))/
(84*x^14*(a + b*x^2))

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Maple [A]  time = 0.173, size = 80, normalized size = 1.1 \begin{align*} -{\frac{21\,{b}^{5}{x}^{10}+70\,a{b}^{4}{x}^{8}+105\,{a}^{2}{b}^{3}{x}^{6}+84\,{b}^{2}{a}^{3}{x}^{4}+35\,{a}^{4}b{x}^{2}+6\,{a}^{5}}{84\,{x}^{14} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x)

[Out]

-1/84*(21*b^5*x^10+70*a*b^4*x^8+105*a^2*b^3*x^6+84*a^3*b^2*x^4+35*a^4*b*x^2+6*a^5)*((b*x^2+a)^2)^(5/2)/x^14/(b
*x^2+a)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.47869, size = 134, normalized size = 1.86 \begin{align*} -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="fricas")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*x^2 + 6*a^5)/x^14

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{15}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**15,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**15, x)

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Giac [A]  time = 1.11714, size = 144, normalized size = 2. \begin{align*} -\frac{21 \, b^{5} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + 70 \, a b^{4} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + 105 \, a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 84 \, a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 35 \, a^{4} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 6 \, a^{5} \mathrm{sgn}\left (b x^{2} + a\right )}{84 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="giac")

[Out]

-1/84*(21*b^5*x^10*sgn(b*x^2 + a) + 70*a*b^4*x^8*sgn(b*x^2 + a) + 105*a^2*b^3*x^6*sgn(b*x^2 + a) + 84*a^3*b^2*
x^4*sgn(b*x^2 + a) + 35*a^4*b*x^2*sgn(b*x^2 + a) + 6*a^5*sgn(b*x^2 + a))/x^14