3.1.63 \(\int \frac {1}{(a+c x^2)^{9/2}} \, dx\) [63]

Optimal. Leaf size=77 \[ \frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 x}{35 a^4 \sqrt {a+c x^2}} \]

[Out]

1/7*x/a/(c*x^2+a)^(7/2)+6/35*x/a^2/(c*x^2+a)^(5/2)+8/35*x/a^3/(c*x^2+a)^(3/2)+16/35*x/a^4/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197} \begin {gather*} \frac {16 x}{35 a^4 \sqrt {a+c x^2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {x}{7 a \left (a+c x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(-9/2),x]

[Out]

x/(7*a*(a + c*x^2)^(7/2)) + (6*x)/(35*a^2*(a + c*x^2)^(5/2)) + (8*x)/(35*a^3*(a + c*x^2)^(3/2)) + (16*x)/(35*a
^4*Sqrt[a + c*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+c x^2\right )^{9/2}} \, dx &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 \int \frac {1}{\left (a+c x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (a+c x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 x}{35 a^4 \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 51, normalized size = 0.66 \begin {gather*} \frac {35 a^3 x+70 a^2 c x^3+56 a c^2 x^5+16 c^3 x^7}{35 a^4 \left (a+c x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(-9/2),x]

[Out]

(35*a^3*x + 70*a^2*c*x^3 + 56*a*c^2*x^5 + 16*c^3*x^7)/(35*a^4*(a + c*x^2)^(7/2))

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Maple [A]
time = 0.41, size = 74, normalized size = 0.96

method	result	size
gosper	\(\frac {x \left (16 c^{3} x^{6}+56 a \,c^{2} x^{4}+70 a^{2} c \,x^{2}+35 a^{3}\right )}{35 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\)	\(48\)
trager	\(\frac {x \left (16 c^{3} x^{6}+56 a \,c^{2} x^{4}+70 a^{2} c \,x^{2}+35 a^{3}\right )}{35 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\)	\(48\)
default	\(\frac {x}{7 a \left (c \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {c \,x^{2}+a}}\right )}{7 a}}{a}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/7*x/a/(c*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(c*x^2+a)^(5/2)+4/5/a*(1/3*x/a/(c*x^2+a)^(3/2)+2/3*x/a^2/(c*x^2+a)^(1/2
)))

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Maxima [A]
time = 0.29, size = 61, normalized size = 0.79 \begin {gather*} \frac {16 \, x}{35 \, \sqrt {c x^{2} + a} a^{4}} + \frac {8 \, x}{35 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, x}{35 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {x}{7 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

16/35*x/(sqrt(c*x^2 + a)*a^4) + 8/35*x/((c*x^2 + a)^(3/2)*a^3) + 6/35*x/((c*x^2 + a)^(5/2)*a^2) + 1/7*x/((c*x^
2 + a)^(7/2)*a)

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Fricas [A]
time = 1.56, size = 91, normalized size = 1.18 \begin {gather*} \frac {{\left (16 \, c^{3} x^{7} + 56 \, a c^{2} x^{5} + 70 \, a^{2} c x^{3} + 35 \, a^{3} x\right )} \sqrt {c x^{2} + a}}{35 \, {\left (a^{4} c^{4} x^{8} + 4 \, a^{5} c^{3} x^{6} + 6 \, a^{6} c^{2} x^{4} + 4 \, a^{7} c x^{2} + a^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/35*(16*c^3*x^7 + 56*a*c^2*x^5 + 70*a^2*c*x^3 + 35*a^3*x)*sqrt(c*x^2 + a)/(a^4*c^4*x^8 + 4*a^5*c^3*x^6 + 6*a^
6*c^2*x^4 + 4*a^7*c*x^2 + a^8)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1265 vs. \(2 (70) = 140\).
time = 1.04, size = 1265, normalized size = 16.43 \begin {gather*} \frac {35 a^{14} x}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {175 a^{13} c x^{3}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {371 a^{12} c^{2} x^{5}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {429 a^{11} c^{3} x^{7}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {286 a^{10} c^{4} x^{9}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {104 a^{9} c^{5} x^{11}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {16 a^{8} c^{6} x^{13}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+a)**(9/2),x)

[Out]

35*a**14*x/(35*a**(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**
4*sqrt(1 + c*x**2/a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a)
+ 210*a**(27/2)*c**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 175*a**13*c*x**
3/(35*a**(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1
+ c*x**2/a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a*
*(27/2)*c**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 371*a**12*c**2*x**5/(35*
a**(37/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x*
*2/a) + 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2
)*c**5*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 429*a**11*c**3*x**7/(35*a**(37
/2)*sqrt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a)
+ 700*a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5
*x**10*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 286*a**10*c**4*x**9/(35*a**(37/2)*sq
rt(1 + c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 700*
a**(31/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5*x**10
*sqrt(1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 104*a**9*c**5*x**11/(35*a**(37/2)*sqrt(1 +
c*x**2/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 700*a**(31
/2)*c**3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5*x**10*sqrt(
1 + c*x**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a)) + 16*a**8*c**6*x**13/(35*a**(37/2)*sqrt(1 + c*x**2
/a) + 210*a**(35/2)*c*x**2*sqrt(1 + c*x**2/a) + 525*a**(33/2)*c**2*x**4*sqrt(1 + c*x**2/a) + 700*a**(31/2)*c**
3*x**6*sqrt(1 + c*x**2/a) + 525*a**(29/2)*c**4*x**8*sqrt(1 + c*x**2/a) + 210*a**(27/2)*c**5*x**10*sqrt(1 + c*x
**2/a) + 35*a**(25/2)*c**6*x**12*sqrt(1 + c*x**2/a))

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Giac [A]
time = 0.84, size = 55, normalized size = 0.71 \begin {gather*} \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, c^{3} x^{2}}{a^{4}} + \frac {7 \, c^{2}}{a^{3}}\right )} + \frac {35 \, c}{a^{2}}\right )} x^{2} + \frac {35}{a}\right )} x}{35 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/35*(2*(4*x^2*(2*c^3*x^2/a^4 + 7*c^2/a^3) + 35*c/a^2)*x^2 + 35/a)*x/(c*x^2 + a)^(7/2)

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Mupad [B]
time = 0.20, size = 61, normalized size = 0.79 \begin {gather*} \frac {16\,x}{35\,a^4\,\sqrt {c\,x^2+a}}+\frac {8\,x}{35\,a^3\,{\left (c\,x^2+a\right )}^{3/2}}+\frac {6\,x}{35\,a^2\,{\left (c\,x^2+a\right )}^{5/2}}+\frac {x}{7\,a\,{\left (c\,x^2+a\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + c*x^2)^(9/2),x)

[Out]

(16*x)/(35*a^4*(a + c*x^2)^(1/2)) + (8*x)/(35*a^3*(a + c*x^2)^(3/2)) + (6*x)/(35*a^2*(a + c*x^2)^(5/2)) + x/(7
*a*(a + c*x^2)^(7/2))

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