3.5.29 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{19}} \, dx\)

Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{10} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^{12} \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \begin {gather*} -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^{12} \left (a+b x^2\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{10} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^19,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(18*x^18*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*x
^16*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^14*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4])/(6*x^12*(a + b*x^2)) - (a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^10*(a + b*x^2)) - (b^5
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a + b*x^2))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 646

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

Rule 1111

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && Integ
erQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{19}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^{10}} \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (\frac {a^5 b^5}{x^{10}}+\frac {5 a^4 b^6}{x^9}+\frac {10 a^3 b^7}{x^8}+\frac {10 a^2 b^8}{x^7}+\frac {5 a b^9}{x^6}+\frac {b^{10}}{x^5}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{18 x^{18} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 x^{14} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^{12} \left (a+b x^2\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{10} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (56 a^5+315 a^4 b x^2+720 a^3 b^2 x^4+840 a^2 b^3 x^6+504 a b^4 x^8+126 b^5 x^{10}\right )}{1008 x^{18} \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^19,x]

[Out]

-1/1008*(Sqrt[(a + b*x^2)^2]*(56*a^5 + 315*a^4*b*x^2 + 720*a^3*b^2*x^4 + 840*a^2*b^3*x^6 + 504*a*b^4*x^8 + 126
*b^5*x^10))/(x^18*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 1.59, size = 576, normalized size = 2.26 \begin {gather*} \frac {16 b^8 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-56 a^{13} b-763 a^{12} b^2 x^2-4808 a^{11} b^3 x^4-18556 a^{10} b^4 x^6-48944 a^9 b^5 x^8-93184 a^8 b^6 x^{10}-131768 a^7 b^7 x^{12}-140140 a^6 b^8 x^{14}-112112 a^5 b^9 x^{16}-66639 a^4 b^{10} x^{18}-28608 a^3 b^{11} x^{20}-8400 a^2 b^{12} x^{22}-1512 a b^{13} x^{24}-126 b^{14} x^{26}\right )+16 \sqrt {b^2} b^8 \left (56 a^{14}+819 a^{13} b x^2+5571 a^{12} b^2 x^4+23364 a^{11} b^3 x^6+67500 a^{10} b^4 x^8+142128 a^9 b^5 x^{10}+224952 a^8 b^6 x^{12}+271908 a^7 b^7 x^{14}+252252 a^6 b^8 x^{16}+178751 a^5 b^9 x^{18}+95247 a^4 b^{10} x^{20}+37008 a^3 b^{11} x^{22}+9912 a^2 b^{12} x^{24}+1638 a b^{13} x^{26}+126 b^{14} x^{28}\right )}{63 \sqrt {b^2} x^{18} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-256 a^8 b^8-2048 a^7 b^9 x^2-7168 a^6 b^{10} x^4-14336 a^5 b^{11} x^6-17920 a^4 b^{12} x^8-14336 a^3 b^{13} x^{10}-7168 a^2 b^{14} x^{12}-2048 a b^{15} x^{14}-256 b^{16} x^{16}\right )+63 x^{18} \left (256 a^9 b^9+2304 a^8 b^{10} x^2+9216 a^7 b^{11} x^4+21504 a^6 b^{12} x^6+32256 a^5 b^{13} x^8+32256 a^4 b^{14} x^{10}+21504 a^3 b^{15} x^{12}+9216 a^2 b^{16} x^{14}+2304 a b^{17} x^{16}+256 b^{18} x^{18}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^19,x]

[Out]

(16*b^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-56*a^13*b - 763*a^12*b^2*x^2 - 4808*a^11*b^3*x^4 - 18556*a^10*b^4*x^
6 - 48944*a^9*b^5*x^8 - 93184*a^8*b^6*x^10 - 131768*a^7*b^7*x^12 - 140140*a^6*b^8*x^14 - 112112*a^5*b^9*x^16 -
66639*a^4*b^10*x^18 - 28608*a^3*b^11*x^20 - 8400*a^2*b^12*x^22 - 1512*a*b^13*x^24 - 126*b^14*x^26) + 16*b^8*S
qrt[b^2]*(56*a^14 + 819*a^13*b*x^2 + 5571*a^12*b^2*x^4 + 23364*a^11*b^3*x^6 + 67500*a^10*b^4*x^8 + 142128*a^9*
b^5*x^10 + 224952*a^8*b^6*x^12 + 271908*a^7*b^7*x^14 + 252252*a^6*b^8*x^16 + 178751*a^5*b^9*x^18 + 95247*a^4*b
^10*x^20 + 37008*a^3*b^11*x^22 + 9912*a^2*b^12*x^24 + 1638*a*b^13*x^26 + 126*b^14*x^28))/(63*Sqrt[b^2]*x^18*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-256*a^8*b^8 - 2048*a^7*b^9*x^2 - 7168*a^6*b^10*x^4 - 14336*a^5*b^11*x^6 - 1792
0*a^4*b^12*x^8 - 14336*a^3*b^13*x^10 - 7168*a^2*b^14*x^12 - 2048*a*b^15*x^14 - 256*b^16*x^16) + 63*x^18*(256*a
^9*b^9 + 2304*a^8*b^10*x^2 + 9216*a^7*b^11*x^4 + 21504*a^6*b^12*x^6 + 32256*a^5*b^13*x^8 + 32256*a^4*b^14*x^10
+ 21504*a^3*b^15*x^12 + 9216*a^2*b^16*x^14 + 2304*a*b^17*x^16 + 256*b^18*x^18))

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fricas [A]  time = 0.82, size = 59, normalized size = 0.23 \begin {gather*} -\frac {126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \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^19,x, algorithm="fricas")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18

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giac [A]  time = 0.17, size = 107, normalized size = 0.42 \begin {gather*} -\frac {126 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 504 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 840 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 720 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 315 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 56 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{1008 \, x^{18}} \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^19,x, algorithm="giac")

[Out]

-1/1008*(126*b^5*x^10*sgn(b*x^2 + a) + 504*a*b^4*x^8*sgn(b*x^2 + a) + 840*a^2*b^3*x^6*sgn(b*x^2 + a) + 720*a^3
*b^2*x^4*sgn(b*x^2 + a) + 315*a^4*b*x^2*sgn(b*x^2 + a) + 56*a^5*sgn(b*x^2 + a))/x^18

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maple [A]  time = 0.01, size = 80, normalized size = 0.31 \begin {gather*} -\frac {\left (126 b^{5} x^{10}+504 a \,b^{4} x^{8}+840 a^{2} b^{3} x^{6}+720 a^{3} b^{2} x^{4}+315 a^{4} b \,x^{2}+56 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{1008 \left (b \,x^{2}+a \right )^{5} x^{18}} \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^19,x)

[Out]

-1/1008*(126*b^5*x^10+504*a*b^4*x^8+840*a^2*b^3*x^6+720*a^3*b^2*x^4+315*a^4*b*x^2+56*a^5)*((b*x^2+a)^2)^(5/2)/
x^18/(b*x^2+a)^5

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maxima [A]  time = 1.33, size = 57, normalized size = 0.22 \begin {gather*} -\frac {b^{5}}{8 \, x^{8}} - \frac {a b^{4}}{2 \, x^{10}} - \frac {5 \, a^{2} b^{3}}{6 \, x^{12}} - \frac {5 \, a^{3} b^{2}}{7 \, x^{14}} - \frac {5 \, a^{4} b}{16 \, x^{16}} - \frac {a^{5}}{18 \, x^{18}} \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^19,x, algorithm="maxima")

[Out]

-1/8*b^5/x^8 - 1/2*a*b^4/x^10 - 5/6*a^2*b^3/x^12 - 5/7*a^3*b^2/x^14 - 5/16*a^4*b/x^16 - 1/18*a^5/x^18

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mupad [B]  time = 4.27, size = 231, normalized size = 0.91 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{18\,x^{18}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^8\,\left (b\,x^2+a\right )}-\frac {a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^{10}\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{16\,x^{16}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{6\,x^{12}\,\left (b\,x^2+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{7\,x^{14}\,\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^19,x)

[Out]

- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(18*x^18*(a + b*x^2)) - (b^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(8*x
^8*(a + b*x^2)) - (a*b^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(2*x^10*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^4 + 2
*a*b*x^2)^(1/2))/(16*x^16*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(6*x^12*(a + b*x^2)) -
(5*a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(7*x^14*(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^{19}}\, 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**19,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**19, x)

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