3.7.26 \(\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=100 \[ -\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^3}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(7*a*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)^(3/2))/(35*a^2*b*(a + b*x)^4) - (2*(a^2 - b^2*
x^2)^(3/2))/(105*a^3*b*(a + b*x)^3)

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]

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}+\frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^4} \, dx}{7 a}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}+\frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3} \, dx}{35 a^2}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 63, normalized size = 0.63 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-23 a^3+13 a^2 b x+8 a b^2 x^2+2 b^3 x^3\right )}{105 a^3 b (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-23*a^3 + 13*a^2*b*x + 8*a*b^2*x^2 + 2*b^3*x^3))/(105*a^3*b*(a + b*x)^4)

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IntegrateAlgebraic [A]  time = 0.51, size = 63, normalized size = 0.63 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-23 a^3+13 a^2 b x+8 a b^2 x^2+2 b^3 x^3\right )}{105 a^3 b (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-23*a^3 + 13*a^2*b*x + 8*a*b^2*x^2 + 2*b^3*x^3))/(105*a^3*b*(a + b*x)^4)

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fricas [A]  time = 0.42, size = 138, normalized size = 1.38 \begin {gather*} -\frac {23 \, b^{4} x^{4} + 92 \, a b^{3} x^{3} + 138 \, a^{2} b^{2} x^{2} + 92 \, a^{3} b x + 23 \, a^{4} - {\left (2 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + 13 \, a^{2} b x - 23 \, a^{3}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{3} + 6 \, a^{5} b^{3} x^{2} + 4 \, a^{6} b^{2} x + a^{7} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/105*(23*b^4*x^4 + 92*a*b^3*x^3 + 138*a^2*b^2*x^2 + 92*a^3*b*x + 23*a^4 - (2*b^3*x^3 + 8*a*b^2*x^2 + 13*a^2*
b*x - 23*a^3)*sqrt(-b^2*x^2 + a^2))/(a^3*b^5*x^4 + 4*a^4*b^4*x^3 + 6*a^5*b^3*x^2 + 4*a^6*b^2*x + a^7*b)

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giac [C]  time = 0.23, size = 178, normalized size = 1.78 \begin {gather*} -\frac {1}{420} \, {\left (\frac {\frac {3 \, {\left (5 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, a}{b x + a} - 1}\right )} \mathrm {sgn}\left (\frac {1}{b x + a}\right ) \mathrm {sgn}\relax (b)}{a^{2} b^{2}} - \frac {7 \, {\left (3 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, a}{b x + a} - 1}\right )} \mathrm {sgn}\left (\frac {1}{b x + a}\right ) \mathrm {sgn}\relax (b)}{a^{2} b^{2}}}{a} + \frac {8 i \, \mathrm {sgn}\left (\frac {1}{b x + a}\right ) \mathrm {sgn}\relax (b)}{a^{3} b^{2}}\right )} {\left | b \right |} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x, algorithm="giac")

[Out]

-1/420*((3*(5*(2*a/(b*x + a) - 1)^(7/2) + 21*(2*a/(b*x + a) - 1)^(5/2) + 35*(2*a/(b*x + a) - 1)^(3/2) + 35*sqr
t(2*a/(b*x + a) - 1))*sgn(1/(b*x + a))*sgn(b)/(a^2*b^2) - 7*(3*(2*a/(b*x + a) - 1)^(5/2) + 10*(2*a/(b*x + a) -
1)^(3/2) + 15*sqrt(2*a/(b*x + a) - 1))*sgn(1/(b*x + a))*sgn(b)/(a^2*b^2))/a + 8*I*sgn(1/(b*x + a))*sgn(b)/(a^
3*b^2))*abs(b)

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maple [A]  time = 0.04, size = 55, normalized size = 0.55 \begin {gather*} -\frac {\left (-b x +a \right ) \left (2 b^{2} x^{2}+10 a b x +23 a^{2}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{105 \left (b x +a \right )^{4} a^{3} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x)

[Out]

-1/105*(-b*x+a)*(2*b^2*x^2+10*a*b*x+23*a^2)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^4/a^3/b

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maxima [B]  time = 1.38, size = 188, normalized size = 1.88 \begin {gather*} -\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{7 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{35 \, {\left (a b^{4} x^{3} + 3 \, a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{3} b^{2} x + a^{4} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x, algorithm="maxima")

[Out]

-2/7*sqrt(-b^2*x^2 + a^2)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/35*sqrt(-b^2*x^2 +
a^2)/(a*b^4*x^3 + 3*a^2*b^3*x^2 + 3*a^3*b^2*x + a^4*b) + 2/105*sqrt(-b^2*x^2 + a^2)/(a^2*b^3*x^2 + 2*a^3*b^2*
x + a^4*b) + 2/105*sqrt(-b^2*x^2 + a^2)/(a^3*b^2*x + a^4*b)

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mupad [B]  time = 0.72, size = 114, normalized size = 1.14 \begin {gather*} \frac {\sqrt {a^2-b^2\,x^2}}{35\,a\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{7\,b\,{\left (a+b\,x\right )}^4}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{105\,a^2\,b\,{\left (a+b\,x\right )}^2}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{105\,a^3\,b\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 - b^2*x^2)^(1/2)/(a + b*x)^5,x)

[Out]

(a^2 - b^2*x^2)^(1/2)/(35*a*b*(a + b*x)^3) - (2*(a^2 - b^2*x^2)^(1/2))/(7*b*(a + b*x)^4) + (2*(a^2 - b^2*x^2)^
(1/2))/(105*a^2*b*(a + b*x)^2) + (2*(a^2 - b^2*x^2)^(1/2))/(105*a^3*b*(a + b*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**5,x)

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**5, x)

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