3.175 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=37 \[ -\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.0137379, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 37} \[ -\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

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

Rule 646

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

Rule 37

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

Rubi steps

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

Mathematica [B]  time = 0.0150135, size = 75, normalized size = 2.03 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^3 b^2 x^2+20 a^2 b^3 x^3+6 a^4 b x+a^5+15 a b^4 x^4+6 b^5 x^5\right )}{6 x^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^5 + 6*a^4*b*x + 15*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 15*a*b^4*x^4 + 6*b^5*x^5))/(6*x^6*(a
+ b*x))

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Maple [B]  time = 0.175, size = 72, normalized size = 2. \begin{align*} -{\frac{6\,{b}^{5}{x}^{5}+15\,a{b}^{4}{x}^{4}+20\,{a}^{2}{b}^{3}{x}^{3}+15\,{a}^{3}{b}^{2}{x}^{2}+6\,{a}^{4}bx+{a}^{5}}{6\,{x}^{6} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x)

[Out]

-1/6*(6*b^5*x^5+15*a*b^4*x^4+20*a^2*b^3*x^3+15*a^3*b^2*x^2+6*a^4*b*x+a^5)*((b*x+a)^2)^(5/2)/x^6/(b*x+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^2+2*a*b*x+a^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.73067, size = 120, normalized size = 3.24 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**7,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**7, x)

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Giac [B]  time = 1.25764, size = 144, normalized size = 3.89 \begin{align*} -\frac{b^{6} \mathrm{sgn}\left (b x + a\right )}{6 \, a} - \frac{6 \, b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{4} b x \mathrm{sgn}\left (b x + a\right ) + a^{5} \mathrm{sgn}\left (b x + a\right )}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x, algorithm="giac")

[Out]

-1/6*b^6*sgn(b*x + a)/a - 1/6*(6*b^5*x^5*sgn(b*x + a) + 15*a*b^4*x^4*sgn(b*x + a) + 20*a^2*b^3*x^3*sgn(b*x + a
) + 15*a^3*b^2*x^2*sgn(b*x + a) + 6*a^4*b*x*sgn(b*x + a) + a^5*sgn(b*x + a))/x^6