3.787 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^7} \, dx\)

Optimal. Leaf size=166 \[ -\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{3465 a^5 b (a+b x)^3}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7} \]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(11*a*b*(a + b*x)^7) - (4*(a^2 - b^2*x^2)^(3/2))/(99*a^2*b*(a + b*x)^6) - (4*(a^2 - b^2
*x^2)^(3/2))/(231*a^3*b*(a + b*x)^5) - (8*(a^2 - b^2*x^2)^(3/2))/(1155*a^4*b*(a + b*x)^4) - (8*(a^2 - b^2*x^2)
^(3/2))/(3465*a^5*b*(a + b*x)^3)

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Rubi [A]  time = 0.0708873, 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} \[ -\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{3465 a^5 b (a+b x)^3}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2 - b^2*x^2)^(3/2)/(11*a*b*(a + b*x)^7) - (4*(a^2 - b^2*x^2)^(3/2))/(99*a^2*b*(a + b*x)^6) - (4*(a^2 - b^2
*x^2)^(3/2))/(231*a^3*b*(a + b*x)^5) - (8*(a^2 - b^2*x^2)^(3/2))/(1155*a^4*b*(a + b*x)^4) - (8*(a^2 - b^2*x^2)
^(3/2))/(3465*a^5*b*(a + b*x)^3)

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{\sqrt{a^2-b^2 x^2}}{(a+b x)^7} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}+\frac{4 \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^6} \, dx}{11 a}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}+\frac{4 \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^5} \, dx}{33 a^2}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5}+\frac{8 \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^4} \, dx}{231 a^3}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}+\frac{8 \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^3} \, dx}{1155 a^4}\\ &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{3465 a^5 b (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0453151, size = 85, normalized size = 0.51 \[ \frac{\sqrt{a^2-b^2 x^2} \left (184 a^3 b^2 x^2+124 a^2 b^3 x^3+183 a^4 b x-547 a^5+48 a b^4 x^4+8 b^5 x^5\right )}{3465 a^5 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-547*a^5 + 183*a^4*b*x + 184*a^3*b^2*x^2 + 124*a^2*b^3*x^3 + 48*a*b^4*x^4 + 8*b^5*x^5))/
(3465*a^5*b*(a + b*x)^6)

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Maple [A]  time = 0.043, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 8\,{b}^{4}{x}^{4}+56\,a{b}^{3}{x}^{3}+180\,{b}^{2}{x}^{2}{a}^{2}+364\,x{a}^{3}b+547\,{a}^{4} \right ) \left ( -bx+a \right ) }{3465\, \left ( bx+a \right ) ^{6}{a}^{5}b}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^7,x)

[Out]

-1/3465*(-b*x+a)*(8*b^4*x^4+56*a*b^3*x^3+180*a^2*b^2*x^2+364*a^3*b*x+547*a^4)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^6/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)^(1/2)/(b*x+a)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.5877, size = 456, normalized size = 2.75 \begin{align*} -\frac{547 \, b^{6} x^{6} + 3282 \, a b^{5} x^{5} + 8205 \, a^{2} b^{4} x^{4} + 10940 \, a^{3} b^{3} x^{3} + 8205 \, a^{4} b^{2} x^{2} + 3282 \, a^{5} b x + 547 \, a^{6} -{\left (8 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} + 124 \, a^{2} b^{3} x^{3} + 184 \, a^{3} b^{2} x^{2} + 183 \, a^{4} b x - 547 \, a^{5}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{3465 \,{\left (a^{5} b^{7} x^{6} + 6 \, a^{6} b^{6} x^{5} + 15 \, a^{7} b^{5} x^{4} + 20 \, a^{8} b^{4} x^{3} + 15 \, a^{9} b^{3} x^{2} + 6 \, a^{10} b^{2} x + a^{11} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/3465*(547*b^6*x^6 + 3282*a*b^5*x^5 + 8205*a^2*b^4*x^4 + 10940*a^3*b^3*x^3 + 8205*a^4*b^2*x^2 + 3282*a^5*b*x
+ 547*a^6 - (8*b^5*x^5 + 48*a*b^4*x^4 + 124*a^2*b^3*x^3 + 184*a^3*b^2*x^2 + 183*a^4*b*x - 547*a^5)*sqrt(-b^2*
x^2 + a^2))/(a^5*b^7*x^6 + 6*a^6*b^6*x^5 + 15*a^7*b^5*x^4 + 20*a^8*b^4*x^3 + 15*a^9*b^3*x^2 + 6*a^10*b^2*x + a
^11*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\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)**(1/2)/(b*x+a)**7,x)

[Out]

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

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Giac [B]  time = 1.24197, size = 474, normalized size = 2.86 \begin{align*} \frac{2 \,{\left (\frac{2552 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{16225 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{42900 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{92730 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{122892 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{129822 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{87780 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{47355 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac{13860 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac{3465 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + 547\right )}}{3465 \, a^{5}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{11}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^7,x, algorithm="giac")

[Out]

2/3465*(2552*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 16225*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^
2) + 42900*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 92730*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*
x^4) + 122892*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 129822*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/
(b^12*x^6) + 87780*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 47355*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b)
)^8/(b^16*x^8) + 13860*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^9/(b^18*x^9) + 3465*(a*b + sqrt(-b^2*x^2 + a^2)*abs
(b))^10/(b^20*x^10) + 547)/(a^5*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^11*abs(b))