3.5.6 \(\int \frac {(a+b x^2)^{5/2}}{x^{14}} \, dx\) [406]

Optimal. Leaf size=92 \[ -\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac {16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7} \]

[Out]

-1/13*(b*x^2+a)^(7/2)/a/x^13+6/143*b*(b*x^2+a)^(7/2)/a^2/x^11-8/429*b^2*(b*x^2+a)^(7/2)/a^3/x^9+16/3003*b^3*(b
*x^2+a)^(7/2)/a^4/x^7

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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} \frac {16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^14,x]

[Out]

-1/13*(a + b*x^2)^(7/2)/(a*x^13) + (6*b*(a + b*x^2)^(7/2))/(143*a^2*x^11) - (8*b^2*(a + b*x^2)^(7/2))/(429*a^3
*x^9) + (16*b^3*(a + b*x^2)^(7/2))/(3003*a^4*x^7)

Rule 270

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

Rule 277

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^{14}} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}-\frac {(6 b) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}+\frac {\left (24 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{143 a^2}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}-\frac {\left (16 b^3\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{429 a^3}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac {6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac {16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 53, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-231 a^3+126 a^2 b x^2-56 a b^2 x^4+16 b^3 x^6\right )}{3003 a^4 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^14,x]

[Out]

((a + b*x^2)^(7/2)*(-231*a^3 + 126*a^2*b*x^2 - 56*a*b^2*x^4 + 16*b^3*x^6))/(3003*a^4*x^13)

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Maple [A]
time = 0.21, size = 85, normalized size = 0.92

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-16 b^{3} x^{6}+56 a \,b^{2} x^{4}-126 a^{2} b \,x^{2}+231 a^{3}\right )}{3003 x^{13} a^{4}}\)	\(50\)
trager	\(-\frac {\left (-16 b^{6} x^{12}+8 a \,b^{5} x^{10}-6 a^{2} b^{4} x^{8}+5 a^{3} x^{6} b^{3}+371 a^{4} b^{2} x^{4}+567 a^{5} b \,x^{2}+231 a^{6}\right ) \sqrt {b \,x^{2}+a}}{3003 x^{13} a^{4}}\)	\(83\)
risch	\(-\frac {\left (-16 b^{6} x^{12}+8 a \,b^{5} x^{10}-6 a^{2} b^{4} x^{8}+5 a^{3} x^{6} b^{3}+371 a^{4} b^{2} x^{4}+567 a^{5} b \,x^{2}+231 a^{6}\right ) \sqrt {b \,x^{2}+a}}{3003 x^{13} a^{4}}\)	\(83\)
default	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^14,x,method=_RETURNVERBOSE)

[Out]

-1/13*(b*x^2+a)^(7/2)/a/x^13-6/13*b/a*(-1/11*(b*x^2+a)^(7/2)/a/x^11-4/11*b/a*(-1/9*(b*x^2+a)^(7/2)/a/x^9+2/63*
b*(b*x^2+a)^(7/2)/a^2/x^7))

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Maxima [A]
time = 0.36, size = 76, normalized size = 0.83 \begin {gather*} \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{3003 \, a^{4} x^{7}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{429 \, a^{3} x^{9}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{143 \, a^{2} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{13 \, a x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^14,x, algorithm="maxima")

[Out]

16/3003*(b*x^2 + a)^(7/2)*b^3/(a^4*x^7) - 8/429*(b*x^2 + a)^(7/2)*b^2/(a^3*x^9) + 6/143*(b*x^2 + a)^(7/2)*b/(a
^2*x^11) - 1/13*(b*x^2 + a)^(7/2)/(a*x^13)

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Fricas [A]
time = 1.17, size = 82, normalized size = 0.89 \begin {gather*} \frac {{\left (16 \, b^{6} x^{12} - 8 \, a b^{5} x^{10} + 6 \, a^{2} b^{4} x^{8} - 5 \, a^{3} b^{3} x^{6} - 371 \, a^{4} b^{2} x^{4} - 567 \, a^{5} b x^{2} - 231 \, a^{6}\right )} \sqrt {b x^{2} + a}}{3003 \, a^{4} x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^14,x, algorithm="fricas")

[Out]

1/3003*(16*b^6*x^12 - 8*a*b^5*x^10 + 6*a^2*b^4*x^8 - 5*a^3*b^3*x^6 - 371*a^4*b^2*x^4 - 567*a^5*b*x^2 - 231*a^6
)*sqrt(b*x^2 + a)/(a^4*x^13)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (85) = 170\).
time = 1.42, size = 721, normalized size = 7.84 \begin {gather*} - \frac {231 a^{9} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {1260 a^{8} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {2765 a^{7} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {3050 a^{6} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {1689 a^{5} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac {376 a^{4} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {5 a^{3} b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {30 a^{2} b^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {40 a b^{\frac {35}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac {16 b^{\frac {37}{2}} x^{18} \sqrt {\frac {a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**14,x)

[Out]

-231*a**9*b**(19/2)*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16
+ 3003*a**4*b**12*x**18) - 1260*a**8*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b*
*10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) - 2765*a**7*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(30
03*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) - 3050*a**6*b**(25
/2)*x**6*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**
4*b**12*x**18) - 1689*a**5*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 +
9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) - 376*a**4*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**
9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) + 5*a**3*b**(31/2)*x**12*sqrt
(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18
) + 30*a**2*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**
11*x**16 + 3003*a**4*b**12*x**18) + 40*a*b**(35/2)*x**16*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**
6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) + 16*b**(37/2)*x**18*sqrt(a/(b*x**2) + 1)/(3003
*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (76) = 152\).
time = 0.73, size = 274, normalized size = 2.98 \begin {gather*} \frac {32 \, {\left (3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} b^{\frac {13}{2}} + 9009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a b^{\frac {13}{2}} + 18018 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{2} b^{\frac {13}{2}} + 16302 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{3} b^{\frac {13}{2}} + 10296 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{4} b^{\frac {13}{2}} + 2288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{5} b^{\frac {13}{2}} + 286 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{6} b^{\frac {13}{2}} - 78 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{7} b^{\frac {13}{2}} + 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{8} b^{\frac {13}{2}} - a^{9} b^{\frac {13}{2}}\right )}}{3003 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^14,x, algorithm="giac")

[Out]

32/3003*(3003*(sqrt(b)*x - sqrt(b*x^2 + a))^18*b^(13/2) + 9009*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a*b^(13/2) + 1
8018*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(13/2) + 16302*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(13/2) + 102
96*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(13/2) + 2288*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(13/2) + 286*(sq
rt(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(13/2) - 78*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(13/2) + 13*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^8*b^(13/2) - a^9*b^(13/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^13

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Mupad [B]
time = 5.98, size = 131, normalized size = 1.42 \begin {gather*} \frac {2\,b^4\,\sqrt {b\,x^2+a}}{1001\,a^2\,x^5}-\frac {53\,b^2\,\sqrt {b\,x^2+a}}{429\,x^9}-\frac {5\,b^3\,\sqrt {b\,x^2+a}}{3003\,a\,x^7}-\frac {a^2\,\sqrt {b\,x^2+a}}{13\,x^{13}}-\frac {8\,b^5\,\sqrt {b\,x^2+a}}{3003\,a^3\,x^3}+\frac {16\,b^6\,\sqrt {b\,x^2+a}}{3003\,a^4\,x}-\frac {27\,a\,b\,\sqrt {b\,x^2+a}}{143\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)/x^14,x)

[Out]

(2*b^4*(a + b*x^2)^(1/2))/(1001*a^2*x^5) - (53*b^2*(a + b*x^2)^(1/2))/(429*x^9) - (5*b^3*(a + b*x^2)^(1/2))/(3
003*a*x^7) - (a^2*(a + b*x^2)^(1/2))/(13*x^13) - (8*b^5*(a + b*x^2)^(1/2))/(3003*a^3*x^3) + (16*b^6*(a + b*x^2
)^(1/2))/(3003*a^4*x) - (27*a*b*(a + b*x^2)^(1/2))/(143*x^11)

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