3.799 \(\int \frac{(a^2-b^2 x^2)^{3/2}}{(a+b x)^9} \, dx\)

Optimal. Leaf size=166 \[ -\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(13*a*b*(a + b*x)^9) - (4*(a^2 - b^2*x^2)^(5/2))/(143*a^2*b*(a + b*x)^8) - (4*(a^2 - b^
2*x^2)^(5/2))/(429*a^3*b*(a + b*x)^7) - (8*(a^2 - b^2*x^2)^(5/2))/(3003*a^4*b*(a + b*x)^6) - (8*(a^2 - b^2*x^2
)^(5/2))/(15015*a^5*b*(a + b*x)^5)

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Rubi [A]  time = 0.0750511, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(13*a*b*(a + b*x)^9) - (4*(a^2 - b^2*x^2)^(5/2))/(143*a^2*b*(a + b*x)^8) - (4*(a^2 - b^
2*x^2)^(5/2))/(429*a^3*b*(a + b*x)^7) - (8*(a^2 - b^2*x^2)^(5/2))/(3003*a^4*b*(a + b*x)^6) - (8*(a^2 - b^2*x^2
)^(5/2))/(15015*a^5*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)^9} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}+\frac{4 \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx}{13 a}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}+\frac{12 \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx}{143 a^2}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}+\frac{8 \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx}{429 a^3}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}+\frac{8 \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{3003 a^4}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0591876, size = 82, normalized size = 0.49 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left (308 a^2 b^2 x^2+852 a^3 b x+1763 a^4+72 a b^3 x^3+8 b^4 x^4\right )}{15015 a^5 b (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]

[Out]

-((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(1763*a^4 + 852*a^3*b*x + 308*a^2*b^2*x^2 + 72*a*b^3*x^3 + 8*b^4*x^4))/(1501
5*a^5*b*(a + b*x)^7)

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Maple [A]  time = 0.046, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 8\,{b}^{4}{x}^{4}+72\,a{b}^{3}{x}^{3}+308\,{b}^{2}{x}^{2}{a}^{2}+852\,x{a}^{3}b+1763\,{a}^{4} \right ) \left ( -bx+a \right ) }{15015\, \left ( bx+a \right ) ^{8}{a}^{5}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)^9,x)

[Out]

-1/15015*(-b*x+a)*(8*b^4*x^4+72*a*b^3*x^3+308*a^2*b^2*x^2+852*a^3*b*x+1763*a^4)*(-b^2*x^2+a^2)^(3/2)/(b*x+a)^8
/a^5/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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.91788, size = 544, normalized size = 3.28 \begin{align*} -\frac{1763 \, b^{7} x^{7} + 12341 \, a b^{6} x^{6} + 37023 \, a^{2} b^{5} x^{5} + 61705 \, a^{3} b^{4} x^{4} + 61705 \, a^{4} b^{3} x^{3} + 37023 \, a^{5} b^{2} x^{2} + 12341 \, a^{6} b x + 1763 \, a^{7} +{\left (8 \, b^{6} x^{6} + 56 \, a b^{5} x^{5} + 172 \, a^{2} b^{4} x^{4} + 308 \, a^{3} b^{3} x^{3} + 367 \, a^{4} b^{2} x^{2} - 2674 \, a^{5} b x + 1763 \, a^{6}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{15015 \,{\left (a^{5} b^{8} x^{7} + 7 \, a^{6} b^{7} x^{6} + 21 \, a^{7} b^{6} x^{5} + 35 \, a^{8} b^{5} x^{4} + 35 \, a^{9} b^{4} x^{3} + 21 \, a^{10} b^{3} x^{2} + 7 \, a^{11} b^{2} x + a^{12} 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)^9,x, algorithm="fricas")

[Out]

-1/15015*(1763*b^7*x^7 + 12341*a*b^6*x^6 + 37023*a^2*b^5*x^5 + 61705*a^3*b^4*x^4 + 61705*a^4*b^3*x^3 + 37023*a
^5*b^2*x^2 + 12341*a^6*b*x + 1763*a^7 + (8*b^6*x^6 + 56*a*b^5*x^5 + 172*a^2*b^4*x^4 + 308*a^3*b^3*x^3 + 367*a^
4*b^2*x^2 - 2674*a^5*b*x + 1763*a^6)*sqrt(-b^2*x^2 + a^2))/(a^5*b^8*x^7 + 7*a^6*b^7*x^6 + 21*a^7*b^6*x^5 + 35*
a^8*b^5*x^4 + 35*a^9*b^4*x^3 + 21*a^10*b^3*x^2 + 7*a^11*b^2*x + a^12*b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**9,x)

[Out]

Timed out

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Giac [B]  time = 1.25809, size = 558, normalized size = 3.36 \begin{align*} \frac{2 \,{\left (\frac{7904 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{77454 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{233948 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{659945 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{1094808 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{1559844 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{1465464 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{1174173 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac{600600 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac{270270 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + \frac{60060 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{11}}{b^{22} x^{11}} + \frac{15015 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{12}}{b^{24} x^{12}} + 1763\right )}}{15015 \, a^{5}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{13}{\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)^9,x, algorithm="giac")

[Out]

2/15015*(7904*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 77454*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x
^2) + 233948*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 659945*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b
^8*x^4) + 1094808*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 1559844*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b
))^6/(b^12*x^6) + 1465464*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 1174173*(a*b + sqrt(-b^2*x^2 + a^
2)*abs(b))^8/(b^16*x^8) + 600600*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^9/(b^18*x^9) + 270270*(a*b + sqrt(-b^2*x^
2 + a^2)*abs(b))^10/(b^20*x^10) + 60060*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^11/(b^22*x^11) + 15015*(a*b + sqrt
(-b^2*x^2 + a^2)*abs(b))^12/(b^24*x^12) + 1763)/(a^5*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^13*abs(
b))