3.19.62 \(\int \frac {1-x^5}{\sqrt {a+b x} (1+x^5)} \, dx\)

Optimal. Leaf size=128 \[ \frac {2}{5} b^4 \text {RootSum}\left [-\text {$\#$1}^{10}+5 \text {$\#$1}^8 a-10 \text {$\#$1}^6 a^2+10 \text {$\#$1}^4 a^3-5 \text {$\#$1}^2 a^4+a^5-b^5\& ,\frac {\log \left (\sqrt {a+b x}-\text {$\#$1}\right )}{\text {$\#$1}^9-4 \text {$\#$1}^7 a+6 \text {$\#$1}^5 a^2-4 \text {$\#$1}^3 a^3+\text {$\#$1} a^4}\& \right ]-\frac {2 \sqrt {a+b x}}{b} \]

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Rubi [F]  time = 1.95, 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 {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - x^5)/(Sqrt[a + b*x]*(1 + x^5)),x]

[Out]

(-2*Sqrt[a + b*x])/b - (4*ArcTanh[Sqrt[a + b*x]/Sqrt[a - b]])/(5*Sqrt[a - b]) + (4*Defer[Subst][Defer[Int][x^6
/(-(a^4*(1 + (b*(a^3 + a^2*b + a*b^2 + b^3))/a^4)) + 4*a^3*(1 + (b*(3*a^2 + 2*a*b + b^2))/(4*a^3))*x^2 - 6*a^2
*(1 + (b*(3*a + b))/(6*a^2))*x^4 + 4*a*(1 + b/(4*a))*x^6 - x^8), x], x, Sqrt[a + b*x]])/5 + (4*(a^3 + 2*a^2*b
+ 3*a*b^2 + 4*b^3)*Defer[Subst][Defer[Int][(a^4*(1 + (b*(a^3 + a^2*b + a*b^2 + b^3))/a^4) - 4*a^3*(1 + (b*(3*a
^2 + 2*a*b + b^2))/(4*a^3))*x^2 + 6*a^2*(1 + (b*(3*a + b))/(6*a^2))*x^4 - 4*a*(1 + b/(4*a))*x^6 + x^8)^(-1), x
], x, Sqrt[a + b*x]])/5 - (4*(3*a^2 + 4*a*b + 3*b^2)*Defer[Subst][Defer[Int][x^2/(a^4*(1 + (b*(a^3 + a^2*b + a
*b^2 + b^3))/a^4) - 4*a^3*(1 + (b*(3*a^2 + 2*a*b + b^2))/(4*a^3))*x^2 + 6*a^2*(1 + (b*(3*a + b))/(6*a^2))*x^4
- 4*a*(1 + b/(4*a))*x^6 + x^8), x], x, Sqrt[a + b*x]])/5 + (4*(3*a + 2*b)*Defer[Subst][Defer[Int][x^4/(a^4*(1
+ (b*(a^3 + a^2*b + a*b^2 + b^3))/a^4) - 4*a^3*(1 + (b*(3*a^2 + 2*a*b + b^2))/(4*a^3))*x^2 + 6*a^2*(1 + (b*(3*
a + b))/(6*a^2))*x^4 - 4*a*(1 + b/(4*a))*x^6 + x^8), x], x, Sqrt[a + b*x]])/5

Rubi steps

\begin {align*} \int \frac {1-x^5}{\sqrt {a+b x} \left (1+x^5\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1+\frac {\left (a-x^2\right )^5}{b^5}}{1-\frac {\left (a-x^2\right )^5}{b^5}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-1+\frac {2 b}{5 \left (-a+b+x^2\right )}+\frac {2 b \left (a^3 \left (1+\frac {b \left (2 a^2+3 a b+4 b^2\right )}{a^3}\right )-3 a^2 \left (1+\frac {b (4 a+3 b)}{3 a^2}\right ) x^2+3 a \left (1+\frac {2 b}{3 a}\right ) x^4-x^6\right )}{5 \left (a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8\right )}\right ) \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 \sqrt {a+b x}}{b}+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b x}\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {a^3 \left (1+\frac {b \left (2 a^2+3 a b+4 b^2\right )}{a^3}\right )-3 a^2 \left (1+\frac {b (4 a+3 b)}{3 a^2}\right ) x^2+3 a \left (1+\frac {2 b}{3 a}\right ) x^4-x^6}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {2 \sqrt {a+b x}}{b}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b}}\right )}{5 \sqrt {a-b}}+\frac {4}{5} \operatorname {Subst}\left (\int \left (\frac {x^6}{-a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2-6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4+4 a \left (1+\frac {b}{4 a}\right ) x^6-x^8}+\frac {a^3+2 a^2 b+3 a b^2+4 b^3}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8}+\frac {\left (-3 a^2-4 a b-3 b^2\right ) x^2}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8}+\frac {(3 a+2 b) x^4}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8}\right ) \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {2 \sqrt {a+b x}}{b}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b}}\right )}{5 \sqrt {a-b}}+\frac {4}{5} \operatorname {Subst}\left (\int \frac {x^6}{-a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )+4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2-6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4+4 a \left (1+\frac {b}{4 a}\right ) x^6-x^8} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{5} (4 (3 a+2 b)) \operatorname {Subst}\left (\int \frac {x^4}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right )-\frac {1}{5} \left (4 \left (3 a^2+4 a b+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{5} \left (4 \left (a^3+2 a^2 b+3 a b^2+4 b^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^4 \left (1+\frac {b \left (a^3+a^2 b+a b^2+b^3\right )}{a^4}\right )-4 a^3 \left (1+\frac {b \left (3 a^2+2 a b+b^2\right )}{4 a^3}\right ) x^2+6 a^2 \left (1+\frac {b (3 a+b)}{6 a^2}\right ) x^4-4 a \left (1+\frac {b}{4 a}\right ) x^6+x^8} \, dx,x,\sqrt {a+b x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 230, normalized size = 1.80 \begin {gather*} \frac {2}{5} \left (-\frac {5 \sqrt {a+b x}}{b}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {2 \sqrt [5]{-1} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+\sqrt [5]{-1} b}}\right )}{\sqrt {a+\sqrt [5]{-1} b}}-\frac {2 (-1)^{2/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-(-1)^{2/5} b}}\right )}{\sqrt {a-(-1)^{2/5} b}}+\frac {2 (-1)^{3/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+(-1)^{3/5} b}}\right )}{\sqrt {a+(-1)^{3/5} b}}-\frac {2 (-1)^{4/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-(-1)^{4/5} b}}\right )}{\sqrt {a-(-1)^{4/5} b}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 - x^5)/(Sqrt[a + b*x]*(1 + x^5)),x]

[Out]

(2*((-5*Sqrt[a + b*x])/b - (2*ArcTanh[Sqrt[a + b*x]/Sqrt[a - b]])/Sqrt[a - b] + (2*(-1)^(1/5)*ArcTanh[Sqrt[a +
b*x]/Sqrt[a + (-1)^(1/5)*b]])/Sqrt[a + (-1)^(1/5)*b] - (2*(-1)^(2/5)*ArcTanh[Sqrt[a + b*x]/Sqrt[a - (-1)^(2/5
)*b]])/Sqrt[a - (-1)^(2/5)*b] + (2*(-1)^(3/5)*ArcTanh[Sqrt[a + b*x]/Sqrt[a + (-1)^(3/5)*b]])/Sqrt[a + (-1)^(3/
5)*b] - (2*(-1)^(4/5)*ArcTanh[Sqrt[a + b*x]/Sqrt[a - (-1)^(4/5)*b]])/Sqrt[a - (-1)^(4/5)*b]))/5

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IntegrateAlgebraic [B]  time = 0.32, size = 464, normalized size = 3.62 \begin {gather*} -\frac {2 \sqrt {a+b x}}{b}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {-a+b} \sqrt {a+b x}}{a-b}\right )}{5 \sqrt {-a+b}}+\frac {2}{5} \text {RootSum}\left [a^4+a^3 b+a^2 b^2+a b^3+b^4-4 a^3 \text {$\#$1}^2-3 a^2 b \text {$\#$1}^2-2 a b^2 \text {$\#$1}^2-b^3 \text {$\#$1}^2+6 a^2 \text {$\#$1}^4+3 a b \text {$\#$1}^4+b^2 \text {$\#$1}^4-4 a \text {$\#$1}^6-b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-a^3 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-2 a^2 b \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-3 a b^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )-4 b^3 \log \left (\sqrt {a+b x}-\text {$\#$1}\right )+3 a^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2+4 a b \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2+3 b^2 \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^2-3 a \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {a+b x}-\text {$\#$1}\right ) \text {$\#$1}^6}{4 a^3 \text {$\#$1}+3 a^2 b \text {$\#$1}+2 a b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 a^2 \text {$\#$1}^3-6 a b \text {$\#$1}^3-2 b^2 \text {$\#$1}^3+12 a \text {$\#$1}^5+3 b \text {$\#$1}^5-4 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - x^5)/(Sqrt[a + b*x]*(1 + x^5)),x]

[Out]

(-2*Sqrt[a + b*x])/b - (4*ArcTan[(Sqrt[-a + b]*Sqrt[a + b*x])/(a - b)])/(5*Sqrt[-a + b]) + (2*RootSum[a^4 + a^
3*b + a^2*b^2 + a*b^3 + b^4 - 4*a^3*#1^2 - 3*a^2*b*#1^2 - 2*a*b^2*#1^2 - b^3*#1^2 + 6*a^2*#1^4 + 3*a*b*#1^4 +
b^2*#1^4 - 4*a*#1^6 - b*#1^6 + #1^8 & , (-(a^3*Log[Sqrt[a + b*x] - #1]) - 2*a^2*b*Log[Sqrt[a + b*x] - #1] - 3*
a*b^2*Log[Sqrt[a + b*x] - #1] - 4*b^3*Log[Sqrt[a + b*x] - #1] + 3*a^2*Log[Sqrt[a + b*x] - #1]*#1^2 + 4*a*b*Log
[Sqrt[a + b*x] - #1]*#1^2 + 3*b^2*Log[Sqrt[a + b*x] - #1]*#1^2 - 3*a*Log[Sqrt[a + b*x] - #1]*#1^4 - 2*b*Log[Sq
rt[a + b*x] - #1]*#1^4 + Log[Sqrt[a + b*x] - #1]*#1^6)/(4*a^3*#1 + 3*a^2*b*#1 + 2*a*b^2*#1 + b^3*#1 - 12*a^2*#
1^3 - 6*a*b*#1^3 - 2*b^2*#1^3 + 12*a*#1^5 + 3*b*#1^5 - 4*#1^7) & ])/5

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fricas [F(-1)]  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+1)/(b*x+a)^(1/2)/(x^5+1),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {b x + a}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(x^5 - 1)/((x^5 + 1)*sqrt(b*x + a)), x)

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maple [B]  time = 0.19, size = 265, normalized size = 2.07

method	result	size
risch	\(-\frac {2 \sqrt {b x +a}}{b}+\frac {4 \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +12 \textit {\_R}^{3} a^{2}+6 \textit {\_R}^{3} a b +2 \textit {\_R}^{3} b^{2}-4 \textit {\_R} \,a^{3}-3 \textit {\_R} \,a^{2} b -2 \textit {\_R} a \,b^{2}-\textit {\_R} \,b^{3}}\right )}{5}\)	\(265\)
derivativedivides	\(-\frac {2 \left (\sqrt {b x +a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +12 \textit {\_R}^{3} a^{2}+6 \textit {\_R}^{3} a b +2 \textit {\_R}^{3} b^{2}-4 \textit {\_R} \,a^{3}-3 \textit {\_R} \,a^{2} b -2 \textit {\_R} a \,b^{2}-\textit {\_R} \,b^{3}}\right )}{5}-\frac {2 b \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}\right )}{b}\)	\(267\)
default	\(-\frac {2 \left (\sqrt {b x +a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+\left (-4 a -b \right ) \textit {\_Z}^{6}+\left (6 a^{2}+3 a b +b^{2}\right ) \textit {\_Z}^{4}+\left (-4 a^{3}-3 a^{2} b -2 a \,b^{2}-b^{3}\right ) \textit {\_Z}^{2}+a^{4}+a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+\left (3 a +2 b \right ) \textit {\_R}^{4}+\left (-3 a^{2}-4 a b -3 b^{2}\right ) \textit {\_R}^{2}+a^{3}+2 a^{2} b +3 a \,b^{2}+4 b^{3}\right ) \ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-12 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +12 \textit {\_R}^{3} a^{2}+6 \textit {\_R}^{3} a b +2 \textit {\_R}^{3} b^{2}-4 \textit {\_R} \,a^{3}-3 \textit {\_R} \,a^{2} b -2 \textit {\_R} a \,b^{2}-\textit {\_R} \,b^{3}}\right )}{5}-\frac {2 b \arctan \left (\frac {\sqrt {b x +a}}{\sqrt {-a +b}}\right )}{5 \sqrt {-a +b}}\right )}{b}\)	\(267\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^5+1)/(b*x+a)^(1/2)/(x^5+1),x,method=_RETURNVERBOSE)

[Out]

-2*(b*x+a)^(1/2)/b+4/5/(-a+b)^(1/2)*arctan((b*x+a)^(1/2)/(-a+b)^(1/2))+2/5*sum((-_R^6+(3*a+2*b)*_R^4+(-3*a^2-4
*a*b-3*b^2)*_R^2+a^3+2*a^2*b+3*a*b^2+4*b^3)/(4*_R^7-12*_R^5*a-3*_R^5*b+12*_R^3*a^2+6*_R^3*a*b+2*_R^3*b^2-4*_R*
a^3-3*_R*a^2*b-2*_R*a*b^2-_R*b^3)*ln((b*x+a)^(1/2)-_R),_R=RootOf(_Z^8+(-4*a-b)*_Z^6+(6*a^2+3*a*b+b^2)*_Z^4+(-4
*a^3-3*a^2*b-2*a*b^2-b^3)*_Z^2+a^4+a^3*b+a^2*b^2+a*b^3+b^4))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for
more details)Is 4*a-4*b positive or negative?

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mupad [B]  time = 1.06, size = 1163, normalized size = 9.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*log(655360*b^32*(a + b*x)^(1/2) + (2*(1638400*b^33 - (16*((2*((2*(400000000*a^2*b^34 - (400000000*a*b^35*(a
+ b*x)^(1/2))/(a - b)^(1/2)))/(5*(a - b)^(1/2)) - 320000000*a^2*b^33*(a + b*x)^(1/2)))/(5*(a - b)^(1/2)) + 64
000000*a^3*b^32))/(625*(a - b)^2)))/(5*(a - b)^(1/2))))/(5*(a - b)^(1/2)) - (2*log(655360*b^32*(a + b*x)^(1/2)
- (2*(1638400*b^33 - (16*((2*((2*(400000000*a^2*b^34 + (400000000*a*b^35*(a + b*x)^(1/2))/(a - b)^(1/2)))/(5*
(a - b)^(1/2)) + 320000000*a^2*b^33*(a + b*x)^(1/2)))/(5*(a - b)^(1/2)) + 64000000*a^3*b^32))/(625*(a - b)^2))
)/(5*(a - b)^(1/2))))/(5*(a - b)^(1/2)) - (2*(a + b*x)^(1/2))/b + symsum(log(root(390625*a^2*b^2*z^8 + 390625*
a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6 + 125000*a^2*b*z^6 - 62500*b
^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)*
(root(390625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^
2*z^6 + 125000*a^2*b*z^6 - 62500*b^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 160
0*b*z^2 + 1600*a*z^2 + 256, z, k)*(root(390625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*
z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6 + 125000*a^2*b*z^6 - 62500*b^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 1
0000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)^4*(root(390625*a^2*b^2*z^8 + 390625*a^3*b*
z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6 + 125000*a^2*b*z^6 - 62500*b^3*z^6
+ 62500*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)*(root(
390625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6
+ 125000*a^2*b*z^6 - 62500*b^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^
2 + 1600*a*z^2 + 256, z, k)*(400000000*a^2*b^34 - 1000000000*root(390625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 3906
25*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*a*b^2*z^6 + 125000*a^2*b*z^6 - 62500*b^3*z^6 + 62500*a
^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 - 1600*b*z^2 + 1600*a*z^2 + 256, z, k)*a*b^35*(a + b*x)
^(1/2)) - 320000000*a^2*b^33*(a + b*x)^(1/2)) + 64000000*a^3*b^32) - 1638400*b^33) - 655360*b^32*(a + b*x)^(1/
2)))*root(390625*a^2*b^2*z^8 + 390625*a^3*b*z^8 + 390625*a*b^3*z^8 + 390625*b^4*z^8 + 390625*a^4*z^8 + 187500*
a*b^2*z^6 + 125000*a^2*b*z^6 - 62500*b^3*z^6 + 62500*a^3*z^6 - 20000*a*b*z^4 + 10000*b^2*z^4 + 10000*a^2*z^4 -
1600*b*z^2 + 1600*a*z^2 + 256, z, k), k, 1, 8)

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sympy [F(-1)]  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+1)/(b*x+a)**(1/2)/(x**5+1),x)

[Out]

Timed out

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