3.12.8 \(\int \frac {x^5 (-4 b+5 a x^2)}{\sqrt [4]{-b+a x^2} (b-b x^8+a x^{10})} \, dx\) [1108]

Optimal. Leaf size=82 \[ \frac {1}{2} a \text {RootSum}\left [a^4 b+b^4 \text {$\#$1}^4+4 b^3 \text {$\#$1}^8+6 b^2 \text {$\#$1}^{12}+4 b \text {$\#$1}^{16}+\text {$\#$1}^{20}\& ,\frac {\log \left (\sqrt [4]{-b+a x^2}-\text {$\#$1}\right )}{b \text {$\#$1}+\text {$\#$1}^5}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.52, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^5 \left (-4 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-b x^8+a x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5*(-4*b + 5*a*x^2))/((-b + a*x^2)^(1/4)*(b - b*x^8 + a*x^10)),x]

[Out]

2*a*b^3*Defer[Subst][Defer[Int][x^2/(a^4*b + x^4*(b + x^4)^4), x], x, (-b + a*x^2)^(1/4)] + 14*a*b^2*Defer[Sub
st][Defer[Int][x^6/(a^4*b + x^4*(b + x^4)^4), x], x, (-b + a*x^2)^(1/4)] + 22*a*b*Defer[Subst][Defer[Int][x^10
/(a^4*b + x^4*(b + x^4)^4), x], x, (-b + a*x^2)^(1/4)] + 10*a*Defer[Subst][Defer[Int][x^14/(a^4*b + x^4*(b + x
^4)^4), x], x, (-b + a*x^2)^(1/4)]

Rubi steps

\begin {align*} \int \frac {x^5 \left (-4 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-b x^8+a x^{10}\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 (-4 b+5 a x)}{\sqrt [4]{-b+a x} \left (b-b x^4+a x^5\right )} \, dx,x,x^2\right )\\ &=(2 a) \text {Subst}\left (\int \frac {x^2 \left (b+x^4\right )^2 \left (b+5 x^4\right )}{a^4 b+x^4 \left (b+x^4\right )^4} \, dx,x,\sqrt [4]{-b+a x^2}\right )\\ &=(2 a) \text {Subst}\left (\int \left (\frac {b^3 x^2}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}}+\frac {7 b^2 x^6}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}}+\frac {11 b x^{10}}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}}+\frac {5 x^{14}}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}}\right ) \, dx,x,\sqrt [4]{-b+a x^2}\right )\\ &=(10 a) \text {Subst}\left (\int \frac {x^{14}}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}} \, dx,x,\sqrt [4]{-b+a x^2}\right )+(22 a b) \text {Subst}\left (\int \frac {x^{10}}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}} \, dx,x,\sqrt [4]{-b+a x^2}\right )+\left (14 a b^2\right ) \text {Subst}\left (\int \frac {x^6}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}} \, dx,x,\sqrt [4]{-b+a x^2}\right )+\left (2 a b^3\right ) \text {Subst}\left (\int \frac {x^2}{a^4 b+b^4 x^4+4 b^3 x^8+6 b^2 x^{12}+4 b x^{16}+x^{20}} \, dx,x,\sqrt [4]{-b+a x^2}\right )\\ &=(10 a) \text {Subst}\left (\int \frac {x^{14}}{a^4 b+x^4 \left (b+x^4\right )^4} \, dx,x,\sqrt [4]{-b+a x^2}\right )+(22 a b) \text {Subst}\left (\int \frac {x^{10}}{a^4 b+x^4 \left (b+x^4\right )^4} \, dx,x,\sqrt [4]{-b+a x^2}\right )+\left (14 a b^2\right ) \text {Subst}\left (\int \frac {x^6}{a^4 b+x^4 \left (b+x^4\right )^4} \, dx,x,\sqrt [4]{-b+a x^2}\right )+\left (2 a b^3\right ) \text {Subst}\left (\int \frac {x^2}{a^4 b+x^4 \left (b+x^4\right )^4} \, dx,x,\sqrt [4]{-b+a x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 8.59, size = 82, normalized size = 1.00 \begin {gather*} \frac {1}{2} a \text {RootSum}\left [a^4 b+b^4 \text {$\#$1}^4+4 b^3 \text {$\#$1}^8+6 b^2 \text {$\#$1}^{12}+4 b \text {$\#$1}^{16}+\text {$\#$1}^{20}\&,\frac {\log \left (\sqrt [4]{-b+a x^2}-\text {$\#$1}\right )}{b \text {$\#$1}+\text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(-4*b + 5*a*x^2))/((-b + a*x^2)^(1/4)*(b - b*x^8 + a*x^10)),x]

[Out]

(a*RootSum[a^4*b + b^4*#1^4 + 4*b^3*#1^8 + 6*b^2*#1^12 + 4*b*#1^16 + #1^20 & , Log[(-b + a*x^2)^(1/4) - #1]/(b
*#1 + #1^5) & ])/2

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (5 a \,x^{2}-4 b \right )}{\left (a \,x^{2}-b \right )^{\frac {1}{4}} \left (a \,x^{10}-b \,x^{8}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(5*a*x^2-4*b)/(a*x^2-b)^(1/4)/(a*x^10-b*x^8+b),x)

[Out]

int(x^5*(5*a*x^2-4*b)/(a*x^2-b)^(1/4)/(a*x^10-b*x^8+b),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(5*a*x^2-4*b)/(a*x^2-b)^(1/4)/(a*x^10-b*x^8+b),x, algorithm="maxima")

[Out]

integrate((5*a*x^2 - 4*b)*x^5/((a*x^10 - b*x^8 + b)*(a*x^2 - b)^(1/4)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.44, size = 142, normalized size = 1.73 \begin {gather*} -2 \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \arctan \left (-{\left (a x^{2} - b\right )}^{\frac {1}{4}} x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{4}} + \sqrt {\sqrt {a x^{2} - b} x^{4} - b \sqrt {-\frac {1}{b}}} \left (-\frac {1}{b}\right )^{\frac {1}{4}}\right ) + \frac {1}{2} \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} x^{2} + b \left (-\frac {1}{b}\right )^{\frac {3}{4}}\right ) - \frac {1}{2} \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} x^{2} - b \left (-\frac {1}{b}\right )^{\frac {3}{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(5*a*x^2-4*b)/(a*x^2-b)^(1/4)/(a*x^10-b*x^8+b),x, algorithm="fricas")

[Out]

-2*(-1/b)^(1/4)*arctan(-(a*x^2 - b)^(1/4)*x^2*(-1/b)^(1/4) + sqrt(sqrt(a*x^2 - b)*x^4 - b*sqrt(-1/b))*(-1/b)^(
1/4)) + 1/2*(-1/b)^(1/4)*log((a*x^2 - b)^(1/4)*x^2 + b*(-1/b)^(3/4)) - 1/2*(-1/b)^(1/4)*log((a*x^2 - b)^(1/4)*
x^2 - b*(-1/b)^(3/4))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \cdot \left (5 a x^{2} - 4 b\right )}{\sqrt [4]{a x^{2} - b} \left (a x^{10} - b x^{8} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(5*a*x**2-4*b)/(a*x**2-b)**(1/4)/(a*x**10-b*x**8+b),x)

[Out]

Integral(x**5*(5*a*x**2 - 4*b)/((a*x**2 - b)**(1/4)*(a*x**10 - b*x**8 + b)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(5*a*x^2-4*b)/(a*x^2-b)^(1/4)/(a*x^10-b*x^8+b),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 8.84, size = 2500, normalized size = 30.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^5*(4*b - 5*a*x^2))/((a*x^2 - b)^(1/4)*(b + a*x^10 - b*x^8)),x)

[Out]

symsum(log(root(3840000000000*a^8*b^8*f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 42949
67296*b^16*f^20 + 95367431640625*a^16*f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900
544000000*a^4*b^11*f^16 + 69793218560*b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 1
7280532480*b^14*f^12 - 58112000000*a^4*b^9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)
*(root(3840000000000*a^8*b^8*f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^1
6*f^20 + 95367431640625*a^16*f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000
*a^4*b^11*f^16 + 69793218560*b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 1728053248
0*b^14*f^12 - 58112000000*a^4*b^9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)^3*(root(
3840000000000*a^8*b^8*f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20
+ 95367431640625*a^16*f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^
11*f^16 + 69793218560*b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*
f^12 - 58112000000*a^4*b^9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)*(root(384000000
0000*a^8*b^8*f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 9536743
1640625*a^16*f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 +
69793218560*b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*f^12 - 58
112000000*a^4*b^9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)^3*(root(3840000000000*a^
8*b^8*f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 95367431640625
*a^16*f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 + 697932
18560*b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*f^12 - 581120000
00*a^4*b^9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)*(root(3840000000000*a^8*b^8*f^2
0 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 95367431640625*a^16*f^2
0 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 + 69793218560*b^1
5*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*f^12 - 58112000000*a^4*b^
9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)^3*(root(3840000000000*a^8*b^8*f^20 - 312
50000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 95367431640625*a^16*f^20 - 292
96875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 + 69793218560*b^15*f^16
+ 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*f^12 - 58112000000*a^4*b^9*f^8 +
1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)*((a*x^2 - b)^(1/4)*(2375018299490104770560*a^54
*b^30 + 223491131508260864000000*a^58*b^26 + 353905305190400000000000*a^62*b^22) + root(3840000000000*a^8*b^8*
f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 95367431640625*a^16*
f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 + 69793218560*
b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*f^12 - 58112000000*a^4
*b^9*f^8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)^3*(9223372036854775808000*a^55*b^30 -
225179981368524800000000*a^59*b^26 + 1374389534720000000000000*a^63*b^22 + root(3840000000000*a^8*b^8*f^20 -
31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 95367431640625*a^16*f^20 -
29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 + 69793218560*b^15*f^
16 + 1626112000000*a^4*b^10*f^12 + 2575000000000*a^8*b^6*f^12 + 17280532480*b^14*f^12 - 58112000000*a^4*b^9*f^
8 + 1614807040*b^13*f^8 + 67174400*b^12*f^4 + 1048576*b^11, f, k)*(a*x^2 - b)^(1/4)*(4796153459164483420160*a^
54*b^31 - 130604389193744384000000*a^58*b^27 + 1044536046387200000000000*a^62*b^23 - 2013265920000000000000000
*a^66*b^19))) - 1210567579837189324800*a^55*b^29 + 15481123719086080000000*a^59*b^25 - 8589934592000000000000*
a^63*b^21) + (332906084455227064320*a^54*b^29 - 11980278696247296000000*a^58*b^25)*(a*x^2 - b)^(1/4)) - 259407
338536540569600*a^55*b^28 + 967570232442880000000*a^59*b^24) + 18464758472219033600*a^54*b^28*(a*x^2 - b)^(1/4
)) - 9232379236109516800*a^55*b^27) + 360287970189639680*a^54*b^27*(a*x^2 - b)^(1/4)))*root(3840000000000*a^8*
b^8*f^20 - 31250000000000*a^12*b^4*f^20 - 209715200000*a^4*b^12*f^20 + 4294967296*b^16*f^20 + 95367431640625*a
^16*f^20 - 29296875000000*a^12*b^3*f^16 + 15200000000000*a^8*b^7*f^16 - 1900544000000*a^4*b^11*f^16 + 69793218
560*b^15*f^16 + 1626112000000*a^4*b^10*f^12 + 2...

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