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

Optimal. Leaf size=67 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(7*a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(5/2)/(35*a^2*b*(a + b*x)^5)

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Rubi [A]  time = 0.0211747, antiderivative size = 67, 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 = {659, 651} \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2 - b^2*x^2)^(5/2)/(7*a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(5/2)/(35*a^2*b*(a + b*x)^5)

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rule 651

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

Rubi steps

\begin{align*} \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}+\frac{\int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{7 a}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0505912, size = 48, normalized size = 0.72 \[ -\frac{(a-b x)^2 (6 a+b x) \sqrt{a^2-b^2 x^2}}{35 a^2 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a - b*x)^2*(6*a + b*x)*Sqrt[a^2 - b^2*x^2])/(35*a^2*b*(a + b*x)^4)

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Maple [A]  time = 0.045, size = 43, normalized size = 0.6 \begin{align*} -{\frac{ \left ( bx+6\,a \right ) \left ( -bx+a \right ) }{35\, \left ( bx+a \right ) ^{5}b{a}^{2}} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x)

[Out]

-1/35*(-b*x+a)*(b*x+6*a)*(-b^2*x^2+a^2)^(3/2)/(b*x+a)^5/b/a^2

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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+a^2)^(3/2)/(b*x+a)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.23308, size = 281, normalized size = 4.19 \begin{align*} -\frac{6 \, b^{4} x^{4} + 24 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 6 \, a^{4} +{\left (b^{3} x^{3} + 4 \, a b^{2} x^{2} - 11 \, a^{2} b x + 6 \, a^{3}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{35 \,{\left (a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{3} + 6 \, a^{4} b^{3} x^{2} + 4 \, a^{5} b^{2} x + a^{6} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/35*(6*b^4*x^4 + 24*a*b^3*x^3 + 36*a^2*b^2*x^2 + 24*a^3*b*x + 6*a^4 + (b^3*x^3 + 4*a*b^2*x^2 - 11*a^2*b*x +
6*a^3)*sqrt(-b^2*x^2 + a^2))/(a^2*b^5*x^4 + 4*a^3*b^4*x^3 + 6*a^4*b^3*x^2 + 4*a^5*b^2*x + a^6*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac{3}{2}}}{\left (a + b x\right )^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**6,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**6, x)

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Giac [B]  time = 1.28945, size = 306, normalized size = 4.57 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{91 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{70 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{140 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{35 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{35 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + 6\right )}}{35 \, a^{2}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{7}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x, algorithm="giac")

[Out]

2/35*(7*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 91*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 70*
(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 140*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 35*(a*
b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 35*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6) + 6)/(a^2*
((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^7*abs(b))