3.8.87 \(\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx\) [787]

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

[Out]

-1/11*(-b^2*x^2+a^2)^(3/2)/a/b/(b*x+a)^7-4/99*(-b^2*x^2+a^2)^(3/2)/a^2/b/(b*x+a)^6-4/231*(-b^2*x^2+a^2)^(3/2)/
a^3/b/(b*x+a)^5-8/1155*(-b^2*x^2+a^2)^(3/2)/a^4/b/(b*x+a)^4-8/3465*(-b^2*x^2+a^2)^(3/2)/a^5/b/(b*x+a)^3

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Rubi [A]
time = 0.05, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 = {673, 665} \begin {gather*} -\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}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/11*(a^2 - b^2*x^2)^(3/2)/(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 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*} \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*}

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Mathematica [A]
time = 0.54, size = 85, normalized size = 0.51 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-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\right )}{3465 a^5 b (a+b x)^6} \end {gather*}

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 = 0.47, size = 249, normalized size = 1.50

method	result	size
gosper	\(-\frac {\left (-b x +a \right ) \left (8 b^{4} x^{4}+56 a \,b^{3} x^{3}+180 a^{2} b^{2} x^{2}+364 a^{3} b x +547 a^{4}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3465 \left (b x +a \right )^{6} a^{5} b}\)	\(77\)
trager	\(-\frac {\left (-8 b^{5} x^{5}-48 a \,b^{4} x^{4}-124 a^{2} b^{3} x^{3}-184 a^{3} x^{2} b^{2}-183 a^{4} b x +547 a^{5}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3465 a^{5} \left (b x +a \right )^{6} b}\)	\(82\)
default	\(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{11 a b \left (x +\frac {a}{b}\right )^{7}}+\frac {4 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{6}}+\frac {b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{5}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{5 a b \left (x +\frac {a}{b}\right )^{4}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{15 a^{2} \left (x +\frac {a}{b}\right )^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}}{b^{7}}\)	\(249\)

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,method=_RETURNVERBOSE)

[Out]

1/b^7*(-1/11/a/b/(x+a/b)^7*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(3/2)+4/11*b/a*(-1/9/a/b/(x+a/b)^6*(-b^2*(x+a/b)^2+2
*a*b*(x+a/b))^(3/2)+1/3*b/a*(-1/7/a/b/(x+a/b)^5*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(3/2)+2/7*b/a*(-1/5/a/b/(x+a/b)
^4*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(3/2)-1/15/a^2/(x+a/b)^3*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(3/2)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (146) = 292\).
time = 0.31, size = 351, normalized size = 2.11 \begin {gather*} -\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{11 \, {\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 {\sqrt {-b^{2} x^{2} + a^{2}}}{99 \, {\left (a b^{6} x^{5} + 5 \, a^{2} b^{5} x^{4} + 10 \, a^{3} b^{4} x^{3} + 10 \, a^{4} b^{3} x^{2} + 5 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {4 \, \sqrt {-b^{2} x^{2} + a^{2}}}{693 \, {\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 )}} + \frac {4 \, \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {8 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3465 \, {\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {8 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3465 \, {\left (a^{5} b^{2} x + a^{6} 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)^7,x, algorithm="maxima")

[Out]

-2/11*sqrt(-b^2*x^2 + a^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) + 1/99*sqrt(-b^2*x^2 + a^2)/(a*b^6*x^5 + 5*a^2*b^5*x^4 + 10*a^3*b^4*x^3 + 10*a^4*b^3*x^2 + 5*a^5
*b^2*x + a^6*b) + 4/693*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) + 4/1155*sqrt(-b^2*x^2 + a^2)/(a^3*b^4*x^3 + 3*a^4*b^3*x^2 + 3*a^5*b^2*x + a^6*b) + 8/3465*sqrt(-b^2*x^2 +
a^2)/(a^4*b^3*x^2 + 2*a^5*b^2*x + a^6*b) + 8/3465*sqrt(-b^2*x^2 + a^2)/(a^5*b^2*x + a^6*b)

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Fricas [A]
time = 2.45, size = 204, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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 = 0.00, 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 )^{7}}\, 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)**7,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (146) = 292\).
time = 0.76, size = 351, normalized size = 2.11 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 1.04, size = 172, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a^2-b^2\,x^2}}{99\,a\,b\,{\left (a+b\,x\right )}^5}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{11\,b\,{\left (a+b\,x\right )}^6}+\frac {4\,\sqrt {a^2-b^2\,x^2}}{693\,a^2\,b\,{\left (a+b\,x\right )}^4}+\frac {4\,\sqrt {a^2-b^2\,x^2}}{1155\,a^3\,b\,{\left (a+b\,x\right )}^3}+\frac {8\,\sqrt {a^2-b^2\,x^2}}{3465\,a^4\,b\,{\left (a+b\,x\right )}^2}+\frac {8\,\sqrt {a^2-b^2\,x^2}}{3465\,a^5\,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)^7,x)

[Out]

(a^2 - b^2*x^2)^(1/2)/(99*a*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)^(1/2))/(11*b*(a + b*x)^6) + (4*(a^2 - b^2*x^2)
^(1/2))/(693*a^2*b*(a + b*x)^4) + (4*(a^2 - b^2*x^2)^(1/2))/(1155*a^3*b*(a + b*x)^3) + (8*(a^2 - b^2*x^2)^(1/2
))/(3465*a^4*b*(a + b*x)^2) + (8*(a^2 - b^2*x^2)^(1/2))/(3465*a^5*b*(a + b*x))

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