3.4.75 \(\int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^7} \, dx\)

Optimal. Leaf size=72 \[ \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{12 a^2 x^6}-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^6} \]

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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 )^{3/2}}{12 a^2 x^6}-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^7,x]

[Out]

-((a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*a*x^6) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/(12*a^2*x^6)

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 {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^7} \, dx &=-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^6}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{12 a^2 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (2 a+3 b x^2\right )}{12 x^6 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^7,x]

[Out]

-1/12*(Sqrt[(a + b*x^2)^2]*(2*a + 3*b*x^2))/(x^6*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 3.25, size = 751, normalized size = 10.43 \begin {gather*} \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-2 a^{15} b^3-55 a^{14} b^4 x^2-704 a^{13} b^5 x^4-5563 a^{12} b^6 x^6-30344 a^{11} b^7 x^8-121000 a^{10} b^8 x^{10}-364320 a^9 b^9 x^{12}-843216 a^8 b^{10} x^{14}-1512192 a^7 b^{11} x^{16}-2100736 a^6 b^{12} x^{18}-2241536 a^5 b^{13} x^{20}-1803520 a^4 b^{14} x^{22}-1058816 a^3 b^{15} x^{24}-428032 a^2 b^{16} x^{26}-106496 a b^{17} x^{28}-12288 b^{18} x^{30}\right )+\sqrt {b^2} \left (2 a^{16} b^2+57 a^{15} b^3 x^2+759 a^{14} b^4 x^4+6267 a^{13} b^5 x^6+35907 a^{12} b^6 x^8+151344 a^{11} b^7 x^{10}+485320 a^{10} b^8 x^{12}+1207536 a^9 b^9 x^{14}+2355408 a^8 b^{10} x^{16}+3612928 a^7 b^{11} x^{18}+4342272 a^6 b^{12} x^{20}+4045056 a^5 b^{13} x^{22}+2862336 a^4 b^{14} x^{24}+1486848 a^3 b^{15} x^{26}+534528 a^2 b^{16} x^{28}+118784 a b^{17} x^{30}+12288 b^{18} x^{32}\right )}{3 \sqrt {b^2} x^6 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-4 a^{14} b^2-104 a^{13} b^3 x^2-1252 a^{12} b^4 x^4-9248 a^{11} b^5 x^6-46816 a^{10} b^6 x^8-171776 a^9 b^7 x^{10}-470976 a^8 b^8 x^{12}-979968 a^7 b^9 x^{14}-1554432 a^6 b^{10} x^{16}-1869824 a^5 b^{11} x^{18}-1678336 a^4 b^{12} x^{20}-1089536 a^3 b^{13} x^{22}-483328 a^2 b^{14} x^{24}-131072 a b^{15} x^{26}-16384 b^{16} x^{28}\right )+3 x^6 \left (4 a^{15} b^3+108 a^{14} b^4 x^2+1356 a^{13} b^5 x^4+10500 a^{12} b^6 x^6+56064 a^{11} b^7 x^8+218592 a^{10} b^8 x^{10}+642752 a^9 b^9 x^{12}+1450944 a^8 b^{10} x^{14}+2534400 a^7 b^{11} x^{16}+3424256 a^6 b^{12} x^{18}+3548160 a^5 b^{13} x^{20}+2767872 a^4 b^{14} x^{22}+1572864 a^3 b^{15} x^{24}+614400 a^2 b^{16} x^{26}+147456 a b^{17} x^{28}+16384 b^{18} x^{30}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^7,x]

[Out]

(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-2*a^15*b^3 - 55*a^14*b^4*x^2 - 704*a^13*b^5*x^4 - 5563*a^12*b^6*x^6 - 30344
*a^11*b^7*x^8 - 121000*a^10*b^8*x^10 - 364320*a^9*b^9*x^12 - 843216*a^8*b^10*x^14 - 1512192*a^7*b^11*x^16 - 21
00736*a^6*b^12*x^18 - 2241536*a^5*b^13*x^20 - 1803520*a^4*b^14*x^22 - 1058816*a^3*b^15*x^24 - 428032*a^2*b^16*
x^26 - 106496*a*b^17*x^28 - 12288*b^18*x^30) + Sqrt[b^2]*(2*a^16*b^2 + 57*a^15*b^3*x^2 + 759*a^14*b^4*x^4 + 62
67*a^13*b^5*x^6 + 35907*a^12*b^6*x^8 + 151344*a^11*b^7*x^10 + 485320*a^10*b^8*x^12 + 1207536*a^9*b^9*x^14 + 23
55408*a^8*b^10*x^16 + 3612928*a^7*b^11*x^18 + 4342272*a^6*b^12*x^20 + 4045056*a^5*b^13*x^22 + 2862336*a^4*b^14
*x^24 + 1486848*a^3*b^15*x^26 + 534528*a^2*b^16*x^28 + 118784*a*b^17*x^30 + 12288*b^18*x^32))/(3*Sqrt[b^2]*x^6
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-4*a^14*b^2 - 104*a^13*b^3*x^2 - 1252*a^12*b^4*x^4 - 9248*a^11*b^5*x^6 - 468
16*a^10*b^6*x^8 - 171776*a^9*b^7*x^10 - 470976*a^8*b^8*x^12 - 979968*a^7*b^9*x^14 - 1554432*a^6*b^10*x^16 - 18
69824*a^5*b^11*x^18 - 1678336*a^4*b^12*x^20 - 1089536*a^3*b^13*x^22 - 483328*a^2*b^14*x^24 - 131072*a*b^15*x^2
6 - 16384*b^16*x^28) + 3*x^6*(4*a^15*b^3 + 108*a^14*b^4*x^2 + 1356*a^13*b^5*x^4 + 10500*a^12*b^6*x^6 + 56064*a
^11*b^7*x^8 + 218592*a^10*b^8*x^10 + 642752*a^9*b^9*x^12 + 1450944*a^8*b^10*x^14 + 2534400*a^7*b^11*x^16 + 342
4256*a^6*b^12*x^18 + 3548160*a^5*b^13*x^20 + 2767872*a^4*b^14*x^22 + 1572864*a^3*b^15*x^24 + 614400*a^2*b^16*x
^26 + 147456*a*b^17*x^28 + 16384*b^18*x^30))

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fricas [A]  time = 0.87, size = 15, normalized size = 0.21 \begin {gather*} -\frac {3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/x^7,x, algorithm="fricas")

[Out]

-1/12*(3*b*x^2 + 2*a)/x^6

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giac [A]  time = 0.16, size = 31, normalized size = 0.43 \begin {gather*} -\frac {3 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/x^7,x, algorithm="giac")

[Out]

-1/12*(3*b*x^2*sgn(b*x^2 + a) + 2*a*sgn(b*x^2 + a))/x^6

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maple [A]  time = 0.00, size = 36, normalized size = 0.50 \begin {gather*} -\frac {\left (3 b \,x^{2}+2 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{12 \left (b \,x^{2}+a \right ) x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^2+a)^2)^(1/2)/x^7,x)

[Out]

-1/12*(3*b*x^2+2*a)*((b*x^2+a)^2)^(1/2)/x^6/(b*x^2+a)

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maxima [A]  time = 1.30, size = 15, normalized size = 0.21 \begin {gather*} -\frac {3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/x^7,x, algorithm="maxima")

[Out]

-1/12*(3*b*x^2 + 2*a)/x^6

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mupad [B]  time = 4.24, size = 35, normalized size = 0.49 \begin {gather*} -\frac {\left (3\,b\,x^2+2\,a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{12\,x^6\,\left (b\,x^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2)^(1/2)/x^7,x)

[Out]

-((2*a + 3*b*x^2)*((a + b*x^2)^2)^(1/2))/(12*x^6*(a + b*x^2))

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sympy [A]  time = 0.19, size = 15, normalized size = 0.21 \begin {gather*} \frac {- 2 a - 3 b x^{2}}{12 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**2+a)**2)**(1/2)/x**7,x)

[Out]

(-2*a - 3*b*x**2)/(12*x**6)

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