3.797 \(\int \frac{(a^2-b^2 x^2)^{3/2}}{(a+b x)^7} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(9*a*b*(a + b*x)^7) - (2*(a^2 - b^2*x^2)^(5/2))/(63*a^2*b*(a + b*x)^6) - (2*(a^2 - b^2*
x^2)^(5/2))/(315*a^3*b*(a + b*x)^5)

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Rubi [A]  time = 0.0367267, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^7,x]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(9*a*b*(a + b*x)^7) - (2*(a^2 - b^2*x^2)^(5/2))/(63*a^2*b*(a + b*x)^6) - (2*(a^2 - b^2*
x^2)^(5/2))/(315*a^3*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)^7} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}+\frac{2 \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx}{9 a}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}+\frac{2 \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{63 a^2}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0519771, size = 60, normalized size = 0.6 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left (47 a^2+14 a b x+2 b^2 x^2\right )}{315 a^3 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^7,x]

[Out]

-((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(47*a^2 + 14*a*b*x + 2*b^2*x^2))/(315*a^3*b*(a + b*x)^5)

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Maple [A]  time = 0.045, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,{b}^{2}{x}^{2}+14\,abx+47\,{a}^{2} \right ) \left ( -bx+a \right ) }{315\, \left ( bx+a \right ) ^{6}{a}^{3}b} \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)^7,x)

[Out]

-1/315*(-b*x+a)*(2*b^2*x^2+14*a*b*x+47*a^2)*(-b^2*x^2+a^2)^(3/2)/(b*x+a)^6/a^3/b

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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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.52184, size = 366, normalized size = 3.66 \begin{align*} -\frac{47 \, b^{5} x^{5} + 235 \, a b^{4} x^{4} + 470 \, a^{2} b^{3} x^{3} + 470 \, a^{3} b^{2} x^{2} + 235 \, a^{4} b x + 47 \, a^{5} +{\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} - 80 \, a^{3} b x + 47 \, a^{4}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{315 \,{\left (a^{3} b^{6} x^{5} + 5 \, a^{4} b^{5} x^{4} + 10 \, a^{5} b^{4} x^{3} + 10 \, a^{6} b^{3} x^{2} + 5 \, a^{7} b^{2} x + a^{8} 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)^7,x, algorithm="fricas")

[Out]

-1/315*(47*b^5*x^5 + 235*a*b^4*x^4 + 470*a^2*b^3*x^3 + 470*a^3*b^2*x^2 + 235*a^4*b*x + 47*a^5 + (2*b^4*x^4 + 1
0*a*b^3*x^3 + 21*a^2*b^2*x^2 - 80*a^3*b*x + 47*a^4)*sqrt(-b^2*x^2 + a^2))/(a^3*b^6*x^5 + 5*a^4*b^5*x^4 + 10*a^
5*b^4*x^3 + 10*a^6*b^3*x^2 + 5*a^7*b^2*x + a^8*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 )^{7}}\, 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)**7,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**7, x)

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Giac [B]  time = 1.24946, size = 390, normalized size = 3.9 \begin{align*} \frac{2 \,{\left (\frac{108 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{1062 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{1638 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{3402 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{2520 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{2310 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{630 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{315 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 47\right )}}{315 \, a^{3}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{9}{\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)^7,x, algorithm="giac")

[Out]

2/315*(108*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1062*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2)
+ 1638*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 3402*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4)
+ 2520*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 2310*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6
) + 630*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 315*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8
) + 47)/(a^3*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^9*abs(b))