3.5.27 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{15}} \, dx\)

Optimal. Leaf size=72 \[ \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}} \]

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1110} \begin {gather*} \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

-((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(12*a*x^14) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(7/2)/(84*a^2*x^14)

Rule 1110

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2
+ c*x^4)^(p + 1))/(4*a*d*(p + 1)*(2*p + 1)), x] - Simp[((d*x)^(m + 1)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^p)/(4*
a*d*(2*p + 1)), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && EqQ[m + 4*p + 5,
0] && NeQ[p, -2^(-1)]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx &=-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 1.15 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (6 a^5+35 a^4 b x^2+84 a^3 b^2 x^4+105 a^2 b^3 x^6+70 a b^4 x^8+21 b^5 x^{10}\right )}{84 x^{14} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

-1/84*(Sqrt[(a + b*x^2)^2]*(6*a^5 + 35*a^4*b*x^2 + 84*a^3*b^2*x^4 + 105*a^2*b^3*x^6 + 70*a*b^4*x^8 + 21*b^5*x^
10))/(x^14*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 1.34, size = 488, normalized size = 6.78 \begin {gather*} \frac {16 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-6 a^{11} b-71 a^{10} b^2 x^2-384 a^9 b^3 x^4-1254 a^8 b^4 x^6-2750 a^7 b^5 x^8-4257 a^6 b^6 x^{10}-4752 a^5 b^7 x^{12}-3829 a^4 b^8 x^{14}-2184 a^3 b^9 x^{16}-840 a^2 b^{10} x^{18}-196 a b^{11} x^{20}-21 b^{12} x^{22}\right )+16 \sqrt {b^2} b^6 \left (6 a^{12}+77 a^{11} b x^2+455 a^{10} b^2 x^4+1638 a^9 b^3 x^6+4004 a^8 b^4 x^8+7007 a^7 b^5 x^{10}+9009 a^6 b^6 x^{12}+8581 a^5 b^7 x^{14}+6013 a^4 b^8 x^{16}+3024 a^3 b^9 x^{18}+1036 a^2 b^{10} x^{20}+217 a b^{11} x^{22}+21 b^{12} x^{24}\right )}{21 \sqrt {b^2} x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-64 a^6 b^6-384 a^5 b^7 x^2-960 a^4 b^8 x^4-1280 a^3 b^9 x^6-960 a^2 b^{10} x^8-384 a b^{11} x^{10}-64 b^{12} x^{12}\right )+21 x^{14} \left (64 a^7 b^7+448 a^6 b^8 x^2+1344 a^5 b^9 x^4+2240 a^4 b^{10} x^6+2240 a^3 b^{11} x^8+1344 a^2 b^{12} x^{10}+448 a b^{13} x^{12}+64 b^{14} x^{14}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

(16*b^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-6*a^11*b - 71*a^10*b^2*x^2 - 384*a^9*b^3*x^4 - 1254*a^8*b^4*x^6 - 27
50*a^7*b^5*x^8 - 4257*a^6*b^6*x^10 - 4752*a^5*b^7*x^12 - 3829*a^4*b^8*x^14 - 2184*a^3*b^9*x^16 - 840*a^2*b^10*
x^18 - 196*a*b^11*x^20 - 21*b^12*x^22) + 16*b^6*Sqrt[b^2]*(6*a^12 + 77*a^11*b*x^2 + 455*a^10*b^2*x^4 + 1638*a^
9*b^3*x^6 + 4004*a^8*b^4*x^8 + 7007*a^7*b^5*x^10 + 9009*a^6*b^6*x^12 + 8581*a^5*b^7*x^14 + 6013*a^4*b^8*x^16 +
3024*a^3*b^9*x^18 + 1036*a^2*b^10*x^20 + 217*a*b^11*x^22 + 21*b^12*x^24))/(21*Sqrt[b^2]*x^14*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]*(-64*a^6*b^6 - 384*a^5*b^7*x^2 - 960*a^4*b^8*x^4 - 1280*a^3*b^9*x^6 - 960*a^2*b^10*x^8 - 384*a
*b^11*x^10 - 64*b^12*x^12) + 21*x^14*(64*a^7*b^7 + 448*a^6*b^8*x^2 + 1344*a^5*b^9*x^4 + 2240*a^4*b^10*x^6 + 22
40*a^3*b^11*x^8 + 1344*a^2*b^12*x^10 + 448*a*b^13*x^12 + 64*b^14*x^14))

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fricas [A]  time = 1.26, size = 59, normalized size = 0.82 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="fricas")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*x^2 + 6*a^5)/x^14

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giac [A]  time = 0.16, size = 107, normalized size = 1.49 \begin {gather*} -\frac {21 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 70 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 84 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 35 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="giac")

[Out]

-1/84*(21*b^5*x^10*sgn(b*x^2 + a) + 70*a*b^4*x^8*sgn(b*x^2 + a) + 105*a^2*b^3*x^6*sgn(b*x^2 + a) + 84*a^3*b^2*
x^4*sgn(b*x^2 + a) + 35*a^4*b*x^2*sgn(b*x^2 + a) + 6*a^5*sgn(b*x^2 + a))/x^14

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maple [A]  time = 0.01, size = 80, normalized size = 1.11 \begin {gather*} -\frac {\left (21 b^{5} x^{10}+70 a \,b^{4} x^{8}+105 a^{2} b^{3} x^{6}+84 a^{3} b^{2} x^{4}+35 a^{4} b \,x^{2}+6 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{84 \left (b \,x^{2}+a \right )^{5} x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x)

[Out]

-1/84*(21*b^5*x^10+70*a*b^4*x^8+105*a^2*b^3*x^6+84*a^3*b^2*x^4+35*a^4*b*x^2+6*a^5)*((b*x^2+a)^2)^(5/2)/x^14/(b
*x^2+a)^5

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maxima [A]  time = 1.35, size = 57, normalized size = 0.79 \begin {gather*} -\frac {b^{5}}{4 \, x^{4}} - \frac {5 \, a b^{4}}{6 \, x^{6}} - \frac {5 \, a^{2} b^{3}}{4 \, x^{8}} - \frac {a^{3} b^{2}}{x^{10}} - \frac {5 \, a^{4} b}{12 \, x^{12}} - \frac {a^{5}}{14 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="maxima")

[Out]

-1/4*b^5/x^4 - 5/6*a*b^4/x^6 - 5/4*a^2*b^3/x^8 - a^3*b^2/x^10 - 5/12*a^4*b/x^12 - 1/14*a^5/x^14

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mupad [B]  time = 4.22, size = 231, normalized size = 3.21 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^4\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{6\,x^6\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{12\,x^{12}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^8\,\left (b\,x^2+a\right )}-\frac {a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^{10}\,\left (b\,x^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)/x^15,x)

[Out]

- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(14*x^14*(a + b*x^2)) - (b^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4*x
^4*(a + b*x^2)) - (5*a*b^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(6*x^6*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^4 +
2*a*b*x^2)^(1/2))/(12*x^12*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4*x^8*(a + b*x^2)) -
(a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(x^10*(a + b*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{15}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**15,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**15, x)

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