3.5.32 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{25}} \, dx\)

Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )} \]

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Rubi [A]  time = 0.15, 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}}{24 x^{24} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \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^25,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(24*x^24*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(22*x
^22*(a + b*x^2)) - (a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^20*(a + b*x^2)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4])/(9*x^18*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*x^16*(a + b*x^2)) - (b^
5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*x^14*(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^{25}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, 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^{13}} \, 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^{13}}+\frac {5 a^4 b^6}{x^{12}}+\frac {10 a^3 b^7}{x^{11}}+\frac {10 a^2 b^8}{x^{10}}+\frac {5 a b^9}{x^9}+\frac {b^{10}}{x^8}\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}}{24 x^{24} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \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 (462 a^5+2520 a^4 b x^2+5544 a^3 b^2 x^4+6160 a^2 b^3 x^6+3465 a b^4 x^8+792 b^5 x^{10}\right )}{11088 x^{24} \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^25,x]

[Out]

-1/11088*(Sqrt[(a + b*x^2)^2]*(462*a^5 + 2520*a^4*b*x^2 + 5544*a^3*b^2*x^4 + 6160*a^2*b^3*x^6 + 3465*a*b^4*x^8
+ 792*b^5*x^10))/(x^24*(a + b*x^2))

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IntegrateAlgebraic [B]  time = 1.90, size = 708, normalized size = 2.78 \begin {gather*} \frac {128 b^{11} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-462 a^{16} b-7602 a^{15} b^2 x^2-58674 a^{14} b^3 x^4-281974 a^{13} b^4 x^6-944405 a^{12} b^5 x^8-2337511 a^{11} b^6 x^{10}-4422891 a^{10} b^7 x^{12}-6526113 a^9 b^8 x^{14}-7589208 a^8 b^9 x^{16}-6978840 a^7 b^{10} x^{18}-5057976 a^6 b^{11} x^{20}-2858856 a^5 b^{12} x^{22}-1235389 a^4 b^{13} x^{24}-394559 a^3 b^{14} x^{26}-87835 a^2 b^{15} x^{28}-12177 a b^{16} x^{30}-792 b^{17} x^{32}\right )+128 \sqrt {b^2} b^{11} \left (462 a^{17}+8064 a^{16} b x^2+66276 a^{15} b^2 x^4+340648 a^{14} b^3 x^6+1226379 a^{13} b^4 x^8+3281916 a^{12} b^5 x^{10}+6760402 a^{11} b^6 x^{12}[phone] a^{10} b^7 x^{14}[phone] a^9 b^8 x^{16}[phone] a^8 b^9 x^{18}[phone] a^7 b^{10} x^{20}+7916832 a^6 b^{11} x^{22}+4094245 a^5 b^{12} x^{24}+1629948 a^4 b^{13} x^{26}+482394 a^3 b^{14} x^{28}+100012 a^2 b^{15} x^{30}+12969 a b^{16} x^{32}+792 b^{17} x^{34}\right )}{693 \sqrt {b^2} x^{24} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-2048 a^{11} b^{11}-22528 a^{10} b^{12} x^2-112640 a^9 b^{13} x^4-337920 a^8 b^{14} x^6-675840 a^7 b^{15} x^8-946176 a^6 b^{16} x^{10}-946176 a^5 b^{17} x^{12}-675840 a^4 b^{18} x^{14}-337920 a^3 b^{19} x^{16}-112640 a^2 b^{20} x^{18}-22528 a b^{21} x^{20}-2048 b^{22} x^{22}\right )+693 x^{24} \left (2048 a^{12} b^{12}+24576 a^{11} b^{13} x^2+135168 a^{10} b^{14} x^4+450560 a^9 b^{15} x^6+1013760 a^8 b^{16} x^8+1622016 a^7 b^{17} x^{10}+1892352 a^6 b^{18} x^{12}+1622016 a^5 b^{19} x^{14}+1013760 a^4 b^{20} x^{16}+450560 a^3 b^{21} x^{18}+135168 a^2 b^{22} x^{20}+24576 a b^{23} x^{22}+2048 b^{24} x^{24}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^25,x]

[Out]

(128*b^11*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-462*a^16*b - 7602*a^15*b^2*x^2 - 58674*a^14*b^3*x^4 - 281974*a^13*
b^4*x^6 - 944405*a^12*b^5*x^8 - 2337511*a^11*b^6*x^10 - 4422891*a^10*b^7*x^12 - 6526113*a^9*b^8*x^14 - 7589208
*a^8*b^9*x^16 - 6978840*a^7*b^10*x^18 - 5057976*a^6*b^11*x^20 - 2858856*a^5*b^12*x^22 - 1235389*a^4*b^13*x^24
- 394559*a^3*b^14*x^26 - 87835*a^2*b^15*x^28 - 12177*a*b^16*x^30 - 792*b^17*x^32) + 128*b^11*Sqrt[b^2]*(462*a^
17 + 8064*a^16*b*x^2 + 66276*a^15*b^2*x^4 + 340648*a^14*b^3*x^6 + 1226379*a^13*b^4*x^8 + 3281916*a^12*b^5*x^10
+ 6760402*a^11*b^6*x^12 + 10949004*a^10*b^7*x^14 + 14115321*a^9*b^8*x^16 + 14568048*a^8*b^9*x^18 + 12036816*a
^7*b^10*x^20 + 7916832*a^6*b^11*x^22 + 4094245*a^5*b^12*x^24 + 1629948*a^4*b^13*x^26 + 482394*a^3*b^14*x^28 +
100012*a^2*b^15*x^30 + 12969*a*b^16*x^32 + 792*b^17*x^34))/(693*Sqrt[b^2]*x^24*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
*(-2048*a^11*b^11 - 22528*a^10*b^12*x^2 - 112640*a^9*b^13*x^4 - 337920*a^8*b^14*x^6 - 675840*a^7*b^15*x^8 - 94
6176*a^6*b^16*x^10 - 946176*a^5*b^17*x^12 - 675840*a^4*b^18*x^14 - 337920*a^3*b^19*x^16 - 112640*a^2*b^20*x^18
- 22528*a*b^21*x^20 - 2048*b^22*x^22) + 693*x^24*(2048*a^12*b^12 + 24576*a^11*b^13*x^2 + 135168*a^10*b^14*x^4
+ 450560*a^9*b^15*x^6 + 1013760*a^8*b^16*x^8 + 1622016*a^7*b^17*x^10 + 1892352*a^6*b^18*x^12 + 1622016*a^5*b^
19*x^14 + 1013760*a^4*b^20*x^16 + 450560*a^3*b^21*x^18 + 135168*a^2*b^22*x^20 + 24576*a*b^23*x^22 + 2048*b^24*
x^24))

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fricas [A]  time = 0.76, size = 59, normalized size = 0.23 \begin {gather*} -\frac {792 \, b^{5} x^{10} + 3465 \, a b^{4} x^{8} + 6160 \, a^{2} b^{3} x^{6} + 5544 \, a^{3} b^{2} x^{4} + 2520 \, a^{4} b x^{2} + 462 \, a^{5}}{11088 \, x^{24}} \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^25,x, algorithm="fricas")

[Out]

-1/11088*(792*b^5*x^10 + 3465*a*b^4*x^8 + 6160*a^2*b^3*x^6 + 5544*a^3*b^2*x^4 + 2520*a^4*b*x^2 + 462*a^5)/x^24

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giac [A]  time = 0.17, size = 107, normalized size = 0.42 \begin {gather*} -\frac {792 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 3465 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 6160 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 5544 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 2520 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 462 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{11088 \, x^{24}} \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^25,x, algorithm="giac")

[Out]

-1/11088*(792*b^5*x^10*sgn(b*x^2 + a) + 3465*a*b^4*x^8*sgn(b*x^2 + a) + 6160*a^2*b^3*x^6*sgn(b*x^2 + a) + 5544
*a^3*b^2*x^4*sgn(b*x^2 + a) + 2520*a^4*b*x^2*sgn(b*x^2 + a) + 462*a^5*sgn(b*x^2 + a))/x^24

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maple [A]  time = 0.01, size = 80, normalized size = 0.31 \begin {gather*} -\frac {\left (792 b^{5} x^{10}+3465 a \,b^{4} x^{8}+6160 a^{2} b^{3} x^{6}+5544 a^{3} b^{2} x^{4}+2520 a^{4} b \,x^{2}+462 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{11088 \left (b \,x^{2}+a \right )^{5} x^{24}} \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^25,x)

[Out]

-1/11088*(792*b^5*x^10+3465*a*b^4*x^8+6160*a^2*b^3*x^6+5544*a^3*b^2*x^4+2520*a^4*b*x^2+462*a^5)*((b*x^2+a)^2)^
(5/2)/x^24/(b*x^2+a)^5

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maxima [A]  time = 1.38, size = 57, normalized size = 0.22 \begin {gather*} -\frac {b^{5}}{14 \, x^{14}} - \frac {5 \, a b^{4}}{16 \, x^{16}} - \frac {5 \, a^{2} b^{3}}{9 \, x^{18}} - \frac {a^{3} b^{2}}{2 \, x^{20}} - \frac {5 \, a^{4} b}{22 \, x^{22}} - \frac {a^{5}}{24 \, x^{24}} \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^25,x, algorithm="maxima")

[Out]

-1/14*b^5/x^14 - 5/16*a*b^4/x^16 - 5/9*a^2*b^3/x^18 - 1/2*a^3*b^2/x^20 - 5/22*a^4*b/x^22 - 1/24*a^5/x^24

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mupad [B]  time = 4.22, size = 231, normalized size = 0.91 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{24\,x^{24}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{16\,x^{16}\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{22\,x^{22}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{9\,x^{18}\,\left (b\,x^2+a\right )}-\frac {a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^{20}\,\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^25,x)

[Out]

- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(24*x^24*(a + b*x^2)) - (b^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(14*
x^14*(a + b*x^2)) - (5*a*b^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(16*x^16*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^
4 + 2*a*b*x^2)^(1/2))/(22*x^22*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(9*x^18*(a + b*x^2
)) - (a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(2*x^20*(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^{25}}\, 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**25,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**25, x)

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