3.8.99 \(\int \frac {(a^2-b^2 x^2)^{3/2}}{(a+b x)^9} \, dx\) [799]

Optimal. Leaf size=166 \[ -\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5} \]

[Out]

-1/13*(-b^2*x^2+a^2)^(5/2)/a/b/(b*x+a)^9-4/143*(-b^2*x^2+a^2)^(5/2)/a^2/b/(b*x+a)^8-4/429*(-b^2*x^2+a^2)^(5/2)
/a^3/b/(b*x+a)^7-8/3003*(-b^2*x^2+a^2)^(5/2)/a^4/b/(b*x+a)^6-8/15015*(-b^2*x^2+a^2)^(5/2)/a^5/b/(b*x+a)^5

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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 )^{5/2}}{143 a^2 b (a+b x)^8}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]

[Out]

-1/13*(a^2 - b^2*x^2)^(5/2)/(a*b*(a + b*x)^9) - (4*(a^2 - b^2*x^2)^(5/2))/(143*a^2*b*(a + b*x)^8) - (4*(a^2 -
b^2*x^2)^(5/2))/(429*a^3*b*(a + b*x)^7) - (8*(a^2 - b^2*x^2)^(5/2))/(3003*a^4*b*(a + b*x)^6) - (8*(a^2 - b^2*x
^2)^(5/2))/(15015*a^5*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*} \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^9} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}+\frac {4 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx}{13 a}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}+\frac {12 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx}{143 a^2}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}+\frac {8 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6} \, dx}{429 a^3}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}+\frac {8 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx}{3003 a^4}\\ &=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac {4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac {8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 82, normalized size = 0.49 \begin {gather*} -\frac {(a-b x)^2 \sqrt {a^2-b^2 x^2} \left (1763 a^4+852 a^3 b x+308 a^2 b^2 x^2+72 a b^3 x^3+8 b^4 x^4\right )}{15015 a^5 b (a+b x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]

[Out]

-1/15015*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(1763*a^4 + 852*a^3*b*x + 308*a^2*b^2*x^2 + 72*a*b^3*x^3 + 8*b^4*x^4
))/(a^5*b*(a + b*x)^7)

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Maple [A]
time = 0.48, size = 249, normalized size = 1.50

method	result	size
gosper	\(-\frac {\left (-b x +a \right ) \left (8 b^{4} x^{4}+72 a \,b^{3} x^{3}+308 a^{2} b^{2} x^{2}+852 a^{3} b x +1763 a^{4}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{15015 \left (b x +a \right )^{8} a^{5} b}\)	\(77\)
trager	\(-\frac {\left (8 b^{6} x^{6}+56 a \,b^{5} x^{5}+172 a^{2} b^{4} x^{4}+308 a^{3} b^{3} x^{3}+367 b^{2} a^{4} x^{2}-2674 a^{5} b x +1763 a^{6}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{15015 a^{5} \left (b x +a \right )^{7} b}\)	\(93\)
default	\(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{13 a b \left (x +\frac {a}{b}\right )^{9}}+\frac {4 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{11 a b \left (x +\frac {a}{b}\right )^{8}}+\frac {3 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{7}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{6}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{35 a^{2} \left (x +\frac {a}{b}\right )^{5}}\right )}{9 a}\right )}{11 a}\right )}{13 a}}{b^{9}}\)	\(249\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^9,x,method=_RETURNVERBOSE)

[Out]

1/b^9*(-1/13/a/b/(x+a/b)^9*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(5/2)+4/13*b/a*(-1/11/a/b/(x+a/b)^8*(-b^2*(x+a/b)^2+
2*a*b*(x+a/b))^(5/2)+3/11*b/a*(-1/9/a/b/(x+a/b)^7*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(5/2)+2/9*b/a*(-1/7/a/b/(x+a/
b)^6*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(5/2)-1/35/a^2/(x+a/b)^5*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(5/2)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (146) = 292\).
time = 0.30, size = 549, normalized size = 3.31 \begin {gather*} -\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{5 \, {\left (b^{9} x^{8} + 8 \, a b^{8} x^{7} + 28 \, a^{2} b^{7} x^{6} + 56 \, a^{3} b^{6} x^{5} + 70 \, a^{4} b^{5} x^{4} + 56 \, a^{5} b^{4} x^{3} + 28 \, a^{6} b^{3} x^{2} + 8 \, a^{7} b^{2} x + a^{8} b\right )}} + \frac {6 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{65 \, {\left (b^{8} x^{7} + 7 \, a b^{7} x^{6} + 21 \, a^{2} b^{6} x^{5} + 35 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{3} + 21 \, a^{5} b^{3} x^{2} + 7 \, a^{6} b^{2} x + a^{7} b\right )}} - \frac {3 \, \sqrt {-b^{2} x^{2} + a^{2}}}{715 \, {\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}}}{429 \, {\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}}}{3003 \, {\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}}}{5005 \, {\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}}}{15015 \, {\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}}}{15015 \, {\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)^(3/2)/(b*x+a)^9,x, algorithm="maxima")

[Out]

-1/5*(-b^2*x^2 + a^2)^(3/2)/(b^9*x^8 + 8*a*b^8*x^7 + 28*a^2*b^7*x^6 + 56*a^3*b^6*x^5 + 70*a^4*b^5*x^4 + 56*a^5
*b^4*x^3 + 28*a^6*b^3*x^2 + 8*a^7*b^2*x + a^8*b) + 6/65*sqrt(-b^2*x^2 + a^2)*a/(b^8*x^7 + 7*a*b^7*x^6 + 21*a^2
*b^6*x^5 + 35*a^3*b^5*x^4 + 35*a^4*b^4*x^3 + 21*a^5*b^3*x^2 + 7*a^6*b^2*x + a^7*b) - 3/715*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) - 1/429*sqr
t(-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) - 4/3003
*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) - 4/5005*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) - 8/15015*sqrt(-b^2*x^2 + a^2)/(a^4*b^3*x^2 + 2*
a^5*b^2*x + a^6*b) - 8/15015*sqrt(-b^2*x^2 + a^2)/(a^5*b^2*x + a^6*b)

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Fricas [A]
time = 2.81, size = 236, normalized size = 1.42 \begin {gather*} -\frac {1763 \, b^{7} x^{7} + 12341 \, a b^{6} x^{6} + 37023 \, a^{2} b^{5} x^{5} + 61705 \, a^{3} b^{4} x^{4} + 61705 \, a^{4} b^{3} x^{3} + 37023 \, a^{5} b^{2} x^{2} + 12341 \, a^{6} b x + 1763 \, a^{7} + {\left (8 \, b^{6} x^{6} + 56 \, a b^{5} x^{5} + 172 \, a^{2} b^{4} x^{4} + 308 \, a^{3} b^{3} x^{3} + 367 \, a^{4} b^{2} x^{2} - 2674 \, a^{5} b x + 1763 \, a^{6}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{15015 \, {\left (a^{5} b^{8} x^{7} + 7 \, a^{6} b^{7} x^{6} + 21 \, a^{7} b^{6} x^{5} + 35 \, a^{8} b^{5} x^{4} + 35 \, a^{9} b^{4} x^{3} + 21 \, a^{10} b^{3} x^{2} + 7 \, a^{11} b^{2} x + a^{12} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^9,x, algorithm="fricas")

[Out]

-1/15015*(1763*b^7*x^7 + 12341*a*b^6*x^6 + 37023*a^2*b^5*x^5 + 61705*a^3*b^4*x^4 + 61705*a^4*b^3*x^3 + 37023*a
^5*b^2*x^2 + 12341*a^6*b*x + 1763*a^7 + (8*b^6*x^6 + 56*a*b^5*x^5 + 172*a^2*b^4*x^4 + 308*a^3*b^3*x^3 + 367*a^
4*b^2*x^2 - 2674*a^5*b*x + 1763*a^6)*sqrt(-b^2*x^2 + a^2))/(a^5*b^8*x^7 + 7*a^6*b^7*x^6 + 21*a^7*b^6*x^5 + 35*
a^8*b^5*x^4 + 35*a^9*b^4*x^3 + 21*a^10*b^3*x^2 + 7*a^11*b^2*x + a^12*b)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**9,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (146) = 292\).
time = 1.10, size = 413, normalized size = 2.49 \begin {gather*} \frac {2 \, {\left (\frac {7904 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {77454 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {233948 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {659945 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {1094808 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {1559844 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {1465464 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {1174173 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac {600600 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac {270270 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + \frac {60060 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{11}}{b^{22} x^{11}} + \frac {15015 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{12}}{b^{24} x^{12}} + 1763\right )}}{15015 \, a^{5} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{13} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^9,x, algorithm="giac")

[Out]

2/15015*(7904*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 77454*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x
^2) + 233948*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*x^3) + 659945*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b
^8*x^4) + 1094808*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 1559844*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b
))^6/(b^12*x^6) + 1465464*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 1174173*(a*b + sqrt(-b^2*x^2 + a^
2)*abs(b))^8/(b^16*x^8) + 600600*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^9/(b^18*x^9) + 270270*(a*b + sqrt(-b^2*x^
2 + a^2)*abs(b))^10/(b^20*x^10) + 60060*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^11/(b^22*x^11) + 15015*(a*b + sqrt
(-b^2*x^2 + a^2)*abs(b))^12/(b^24*x^12) + 1763)/(a^5*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^13*abs(
b))

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Mupad [B]
time = 1.70, size = 199, normalized size = 1.20 \begin {gather*} \frac {28\,\sqrt {a^2-b^2\,x^2}}{143\,b\,{\left (a+b\,x\right )}^6}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{13\,b\,{\left (a+b\,x\right )}^7}-\frac {\sqrt {a^2-b^2\,x^2}}{429\,a\,b\,{\left (a+b\,x\right )}^5}-\frac {4\,\sqrt {a^2-b^2\,x^2}}{3003\,a^2\,b\,{\left (a+b\,x\right )}^4}-\frac {4\,\sqrt {a^2-b^2\,x^2}}{5005\,a^3\,b\,{\left (a+b\,x\right )}^3}-\frac {8\,\sqrt {a^2-b^2\,x^2}}{15015\,a^4\,b\,{\left (a+b\,x\right )}^2}-\frac {8\,\sqrt {a^2-b^2\,x^2}}{15015\,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)^(3/2)/(a + b*x)^9,x)

[Out]

(28*(a^2 - b^2*x^2)^(1/2))/(143*b*(a + b*x)^6) - (4*a*(a^2 - b^2*x^2)^(1/2))/(13*b*(a + b*x)^7) - (a^2 - b^2*x
^2)^(1/2)/(429*a*b*(a + b*x)^5) - (4*(a^2 - b^2*x^2)^(1/2))/(3003*a^2*b*(a + b*x)^4) - (4*(a^2 - b^2*x^2)^(1/2
))/(5005*a^3*b*(a + b*x)^3) - (8*(a^2 - b^2*x^2)^(1/2))/(15015*a^4*b*(a + b*x)^2) - (8*(a^2 - b^2*x^2)^(1/2))/
(15015*a^5*b*(a + b*x))

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