3.794 \(\int \frac{(a^2-b^2 x^2)^{3/2}}{(a+b x)^4} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}+\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

[Out]

(2*Sqrt[a^2 - b^2*x^2])/(b*(a + b*x)) - (2*(a^2 - b^2*x^2)^(3/2))/(3*b*(a + b*x)^3) + ArcTan[(b*x)/Sqrt[a^2 -
b^2*x^2]]/b

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Rubi [A]  time = 0.0229014, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {663, 217, 203} \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}+\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^4,x]

[Out]

(2*Sqrt[a^2 - b^2*x^2])/(b*(a + b*x)) - (2*(a^2 - b^2*x^2)^(3/2))/(3*b*(a + b*x)^3) + ArcTan[(b*x)/Sqrt[a^2 -
b^2*x^2]]/b

Rule 663

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^4} \, dx &=-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}-\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^2} \, dx\\ &=\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\operatorname{Subst}\left (\int \frac{1}{1+b^2 x^2} \, dx,x,\frac{x}{\sqrt{a^2-b^2 x^2}}\right )\\ &=\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{3 b (a+b x)^3}+\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0695871, size = 61, normalized size = 0.73 \[ \frac{\frac{4 \sqrt{a^2-b^2 x^2} (a+2 b x)}{(a+b x)^2}+3 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^4,x]

[Out]

((4*(a + 2*b*x)*Sqrt[a^2 - b^2*x^2])/(a + b*x)^2 + 3*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(3*b)

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Maple [B]  time = 0.053, size = 248, normalized size = 3. \begin{align*} -{\frac{1}{3\,{b}^{5}a} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-4}}+{\frac{1}{3\,{a}^{2}{b}^{4}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-3}}+{\frac{2}{3\,{b}^{3}{a}^{3}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-2}}+{\frac{2}{3\,b{a}^{3}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{{\frac{3}{2}}}}+{\frac{x}{{a}^{2}}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}+{\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}}} \right ){\frac{1}{\sqrt{{b}^{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)^4,x)

[Out]

-1/3/b^5/a/(x+1/b*a)^4*(-(x+1/b*a)^2*b^2+2*(x+1/b*a)*a*b)^(5/2)+1/3/b^4/a^2/(x+1/b*a)^3*(-(x+1/b*a)^2*b^2+2*(x
+1/b*a)*a*b)^(5/2)+2/3/b^3/a^3/(x+1/b*a)^2*(-(x+1/b*a)^2*b^2+2*(x+1/b*a)*a*b)^(5/2)+2/3/b/a^3*(-(x+1/b*a)^2*b^
2+2*(x+1/b*a)*a*b)^(3/2)+1/a^2*(-(x+1/b*a)^2*b^2+2*(x+1/b*a)*a*b)^(1/2)*x+1/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/(
-(x+1/b*a)^2*b^2+2*(x+1/b*a)*a*b)^(1/2))

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0793, size = 235, normalized size = 2.83 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} - 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + 2 \, \sqrt{-b^{2} x^{2} + a^{2}}{\left (2 \, b x + a\right )}\right )}}{3 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} 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)^4,x, algorithm="fricas")

[Out]

2/3*(2*b^2*x^2 + 4*a*b*x + 2*a^2 - 3*(b^2*x^2 + 2*a*b*x + a^2)*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) + 2*s
qrt(-b^2*x^2 + a^2)*(2*b*x + a))/(b^3*x^2 + 2*a*b^2*x + a^2*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 )^{4}}\, 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)**4,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**4, x)

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Giac [A]  time = 1.25805, size = 116, normalized size = 1.4 \begin{align*} \frac{\arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} - \frac{8 \,{\left (\frac{3 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + 1\right )}}{3 \,{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{3}{\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)^4,x, algorithm="giac")

[Out]

arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) - 8/3*(3*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)/(((a*b + sqrt(-b^
2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^3*abs(b))