\(\int \frac {(a^2-b^2 x^2)^{3/2}}{(a+b x)^6} \, dx\) [796]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6} \]

[In]

Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^6,x]

[Out]

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

Rule 665

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 673

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*} \text {integral}& = -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}+\frac {\int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{7 a} \\ & = -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=-\frac {(a-b x)^2 (6 a+b x) \sqrt {a^2-b^2 x^2}}{35 a^2 b (a+b x)^4} \]

[In]

Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^6,x]

[Out]

-1/35*((a - b*x)^2*(6*a + b*x)*Sqrt[a^2 - b^2*x^2])/(a^2*b*(a + b*x)^4)

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {\left (-b x +a \right ) \left (b x +6 a \right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{35 \left (b x +a \right )^{5} a^{2} b}\)	\(43\)
trager	\(-\frac {\left (b^{3} x^{3}+4 a \,b^{2} x^{2}-11 a^{2} b x +6 a^{3}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{35 a^{2} \left (b x +a \right )^{4} b}\)	\(59\)
default	\(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{6}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{35 a^{2} \left (x +\frac {a}{b}\right )^{5}}}{b^{6}}\)	\(93\)

[In]

int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (59) = 118\).

Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=-\frac {6 \, b^{4} x^{4} + 24 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 6 \, a^{4} + {\left (b^{3} x^{3} + 4 \, a b^{2} x^{2} - 11 \, a^{2} b x + 6 \, a^{3}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{35 \, {\left (a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{3} + 6 \, a^{4} b^{3} x^{2} + 4 \, a^{5} b^{2} x + a^{6} b\right )}} \]

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/35*(6*b^4*x^4 + 24*a*b^3*x^3 + 36*a^2*b^2*x^2 + 24*a^3*b*x + 6*a^4 + (b^3*x^3 + 4*a*b^2*x^2 - 11*a^2*b*x +
6*a^3)*sqrt(-b^2*x^2 + a^2))/(a^2*b^5*x^4 + 4*a^3*b^4*x^3 + 6*a^4*b^3*x^2 + 4*a^5*b^2*x + a^6*b)

Sympy [F]

\[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{6}}\, dx \]

[In]

integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**6,x)

[Out]

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**6, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (59) = 118\).

Time = 0.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.81 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=-\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{2 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {3 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{7 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {3 \, \sqrt {-b^{2} x^{2} + a^{2}}}{70 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{35 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{35 \, {\left (a^{2} b^{2} x + a^{3} b\right )}} \]

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/2*(-b^2*x^2 + a^2)^(3/2)/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b) +
3/7*sqrt(-b^2*x^2 + a^2)*a/(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) - 3/70*sqrt(-b^2*x^2
+ a^2)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/35*sqrt(-b^2*x^2 + a^2)/(a*b^3*x^2 + 2*a^2*b^2*x + a^
3*b) - 1/35*sqrt(-b^2*x^2 + a^2)/(a^2*b^2*x + a^3*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (59) = 118\).

Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.39 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {91 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {70 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {140 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {35 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {35 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + 6\right )}}{35 \, a^{2} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{7} {\left | b \right |}} \]

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^6,x, algorithm="giac")

[Out]

2/35*(7*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 91*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 70*
(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 140*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 35*(a*
b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 35*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6) + 6)/(a^2*
((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^7*abs(b))

Mupad [B] (verification not implemented)

Time = 10.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx=\frac {16\,\sqrt {a^2-b^2\,x^2}}{35\,b\,{\left (a+b\,x\right )}^3}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{7\,b\,{\left (a+b\,x\right )}^4}-\frac {\sqrt {a^2-b^2\,x^2}}{35\,a\,b\,{\left (a+b\,x\right )}^2}-\frac {\sqrt {a^2-b^2\,x^2}}{35\,a^2\,b\,\left (a+b\,x\right )} \]

[In]

int((a^2 - b^2*x^2)^(3/2)/(a + b*x)^6,x)

[Out]

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