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

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

[Out]

-1/9*(-b^2*x^2+a^2)^(5/2)/a/b/(b*x+a)^7-2/63*(-b^2*x^2+a^2)^(5/2)/a^2/b/(b*x+a)^6-2/315*(-b^2*x^2+a^2)^(5/2)/a
^3/b/(b*x+a)^5

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

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 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 {\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*}

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Mathematica [A]  time = 0.05, size = 60, normalized size = 0.60 \[ -\frac {(a-b x)^2 \sqrt {a^2-b^2 x^2} \left (47 a^2+14 a b x+2 b^2 x^2\right )}{315 a^3 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(47*a^2 + 14*a*b*x + 2*b^2*x^2))/(a^3*b*(a + b*x)^5)

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fricas [A]  time = 0.95, size = 170, normalized size = 1.70 \[ -\frac {47 \, b^{5} x^{5} + 235 \, a b^{4} x^{4} + 470 \, a^{2} b^{3} x^{3} + 470 \, a^{3} b^{2} x^{2} + 235 \, a^{4} b x + 47 \, a^{5} + {\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} - 80 \, a^{3} b x + 47 \, a^{4}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{6} x^{5} + 5 \, a^{4} b^{5} x^{4} + 10 \, a^{5} b^{4} x^{3} + 10 \, a^{6} b^{3} x^{2} + 5 \, a^{7} b^{2} x + a^{8} b\right )}} \]

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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giac [B]  time = 0.22, size = 289, normalized size = 2.89 \[ \frac {2 \, {\left (\frac {108 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {1062 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {1638 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {3402 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {2520 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {2310 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {630 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {315 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 47\right )}}{315 \, a^{3} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{9} {\left | b \right |}} \]

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))

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maple [A]  time = 0.04, size = 55, normalized size = 0.55 \[ -\frac {\left (-b x +a \right ) \left (2 b^{2} x^{2}+14 a b x +47 a^{2}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{315 \left (b x +a \right )^{6} a^{3} b} \]

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 [B]  time = 1.44, size = 342, normalized size = 3.42 \[ -\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{9 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{63 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a b^{4} x^{3} + 3 \, a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{2} x + a^{4} b\right )}} \]

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]

-1/3*(-b^2*x^2 + a^2)^(3/2)/(b^7*x^6 + 6*a*b^6*x^5 + 15*a^2*b^5*x^4 + 20*a^3*b^4*x^3 + 15*a^4*b^3*x^2 + 6*a^5*
b^2*x + a^6*b) + 2/9*sqrt(-b^2*x^2 + a^2)*a/(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) - 1/63*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) - 1/10
5*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) - 2/315*sqrt(-b^2*x^2 + a^2)/(a^2*b^3
*x^2 + 2*a^3*b^2*x + a^4*b) - 2/315*sqrt(-b^2*x^2 + a^2)/(a^3*b^2*x + a^4*b)

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mupad [B]  time = 1.18, size = 141, normalized size = 1.41 \[ \frac {20\,\sqrt {a^2-b^2\,x^2}}{63\,b\,{\left (a+b\,x\right )}^4}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{9\,b\,{\left (a+b\,x\right )}^5}-\frac {\sqrt {a^2-b^2\,x^2}}{105\,a\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^2\,b\,{\left (a+b\,x\right )}^2}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^3\,b\,\left (a+b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*(a^2 - b^2*x^2)^(1/2))/(63*b*(a + b*x)^4) - (4*a*(a^2 - b^2*x^2)^(1/2))/(9*b*(a + b*x)^5) - (a^2 - b^2*x^2
)^(1/2)/(105*a*b*(a + b*x)^3) - (2*(a^2 - b^2*x^2)^(1/2))/(315*a^2*b*(a + b*x)^2) - (2*(a^2 - b^2*x^2)^(1/2))/
(315*a^3*b*(a + b*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{7}}\, dx \]

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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