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

Optimal. Leaf size=138 \[ \frac {1}{5} \left (a^2 b^5+b^5\right ) \text {RootSum}\left [-\text {$\#$1}^{10}+5 \text {$\#$1}^8 a-10 \text {$\#$1}^6 a^2+10 \text {$\#$1}^4 a^3-5 \text {$\#$1}^2 a^4+a^5-a b^6\& ,\frac {\log \left (\sqrt {a+b x}-\text {$\#$1}\right )}{\text {$\#$1}^9-4 \text {$\#$1}^7 a+6 \text {$\#$1}^5 a^2-4 \text {$\#$1}^3 a^3+\text {$\#$1} a^4}\& \right ]-\frac {2 a \sqrt {a+b x}}{b} \]

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Rubi [B]  time = 1.27, antiderivative size = 417, normalized size of antiderivative = 3.02, number of steps used = 14, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6740, 6739, 12, 208, 206} \begin {gather*} -\frac {2 \left (a^2+1\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}-b^{6/5}}}+\frac {2 \sqrt [5]{-1} \left (a^2+1\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}+\sqrt [5]{-1} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}+\sqrt [5]{-1} b^{6/5}}}-\frac {2 (-1)^{2/5} \left (a^2+1\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-(-1)^{2/5} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}-(-1)^{2/5} b^{6/5}}}+\frac {2 (-1)^{3/5} \left (a^2+1\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}+(-1)^{3/5} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}+(-1)^{3/5} b^{6/5}}}-\frac {2 (-1)^{4/5} \left (a^2+1\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-(-1)^{4/5} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}-(-1)^{4/5} b^{6/5}}}-\frac {2 a \sqrt {a+b x}}{b} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*a*Sqrt[a + b*x])/b - (2*(1 + a^2)*b^(1/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) - b^(6/5)])])/(5*a^
(9/10)*Sqrt[a^(4/5) - b^(6/5)]) + (2*(-1)^(1/5)*(1 + a^2)*b^(1/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5)
+ (-1)^(1/5)*b^(6/5)])])/(5*a^(9/10)*Sqrt[a^(4/5) + (-1)^(1/5)*b^(6/5)]) - (2*(-1)^(2/5)*(1 + a^2)*b^(1/5)*Ar
cTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) - (-1)^(2/5)*b^(6/5)])])/(5*a^(9/10)*Sqrt[a^(4/5) - (-1)^(2/5)*b^(6
/5)]) + (2*(-1)^(3/5)*(1 + a^2)*b^(1/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) + (-1)^(3/5)*b^(6/5)])])/
(5*a^(9/10)*Sqrt[a^(4/5) + (-1)^(3/5)*b^(6/5)]) - (2*(-1)^(4/5)*(1 + a^2)*b^(1/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/
10)*Sqrt[a^(4/5) - (-1)^(4/5)*b^(6/5)])])/(5*a^(9/10)*Sqrt[a^(4/5) - (-1)^(4/5)*b^(6/5)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 6739

Int[((a_) + (b_.)*(v_)^(n_))^(-1), x_Symbol] :> Dist[1/(a*n), Sum[Int[Together[1/(1 - v/((-1)^((2*k)/n)*Rt[-(a
/b), n]))], x], {k, 1, n}], x] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0]

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rubi steps

\begin {align*} \int \frac {b-a x^5}{\sqrt {a+b x} \left (a b+x^5\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {b^6+a \left (a-x^2\right )^5}{a b-\frac {\left (a-x^2\right )^5}{b^5}} \, dx,x,\sqrt {a+b x}\right )}{b^6}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-a b^5-\frac {b^5 \left (-b-a^2 b\right )}{a b-\frac {\left (a-x^2\right )^5}{b^5}}\right ) \, dx,x,\sqrt {a+b x}\right )}{b^6}\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\left (2 \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {\left (a-x^2\right )^5}{b^5}} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {2 a \sqrt {a+b x}}{b}-\frac {\left (2 \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [5]{a} b^{6/5}}{a-\sqrt [5]{a} b^{6/5}-x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a b}+\frac {\left (2 \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [5]{a} b^{6/5}}{\sqrt [5]{-1} a+\sqrt [5]{a} b^{6/5}-\sqrt [5]{-1} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a b}-\frac {\left (2 \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [5]{a} b^{6/5}}{(-1)^{2/5} a-\sqrt [5]{a} b^{6/5}-(-1)^{2/5} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a b}+\frac {\left (2 \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [5]{a} b^{6/5}}{(-1)^{3/5} a+\sqrt [5]{a} b^{6/5}-(-1)^{3/5} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a b}-\frac {\left (2 \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [5]{a} b^{6/5}}{(-1)^{4/5} a-\sqrt [5]{a} b^{6/5}-(-1)^{4/5} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a b}\\ &=-\frac {2 a \sqrt {a+b x}}{b}-\frac {\left (2 \left (1+a^2\right ) \sqrt [5]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt [5]{a} b^{6/5}-x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a^{4/5}}+\frac {\left (2 \left (1+a^2\right ) \sqrt [5]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [5]{-1} a+\sqrt [5]{a} b^{6/5}-\sqrt [5]{-1} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a^{4/5}}-\frac {\left (2 \left (1+a^2\right ) \sqrt [5]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1)^{2/5} a-\sqrt [5]{a} b^{6/5}-(-1)^{2/5} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a^{4/5}}+\frac {\left (2 \left (1+a^2\right ) \sqrt [5]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1)^{3/5} a+\sqrt [5]{a} b^{6/5}-(-1)^{3/5} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a^{4/5}}-\frac {\left (2 \left (1+a^2\right ) \sqrt [5]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1)^{4/5} a-\sqrt [5]{a} b^{6/5}-(-1)^{4/5} x^2} \, dx,x,\sqrt {a+b x}\right )}{5 a^{4/5}}\\ &=-\frac {2 a \sqrt {a+b x}}{b}-\frac {2 \left (1+a^2\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}-b^{6/5}}}+\frac {2 \sqrt [5]{-1} \left (1+a^2\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}+\sqrt [5]{-1} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}+\sqrt [5]{-1} b^{6/5}}}-\frac {2 (-1)^{2/5} \left (1+a^2\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-(-1)^{2/5} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}-(-1)^{2/5} b^{6/5}}}+\frac {2 (-1)^{3/5} \left (1+a^2\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}+(-1)^{3/5} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}+(-1)^{3/5} b^{6/5}}}-\frac {2 (-1)^{4/5} \left (1+a^2\right ) \sqrt [5]{b} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-(-1)^{4/5} b^{6/5}}}\right )}{5 a^{9/10} \sqrt {a^{4/5}-(-1)^{4/5} b^{6/5}}}\\ \end {align*}

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Mathematica [B]  time = 1.64, size = 393, normalized size = 2.85 \begin {gather*} \frac {2 \left (-5 a^{19/10} \sqrt {a+b x}-\frac {\left (a^2+1\right ) b^{6/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-b^{6/5}}}\right )}{\sqrt {a^{4/5}-b^{6/5}}}+\frac {\sqrt [5]{-1} \left (a^2+1\right ) b^{6/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}+\sqrt [5]{-1} b^{6/5}}}\right )}{\sqrt {a^{4/5}+\sqrt [5]{-1} b^{6/5}}}-\frac {(-1)^{2/5} \left (a^2+1\right ) b^{6/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-(-1)^{2/5} b^{6/5}}}\right )}{\sqrt {a^{4/5}-(-1)^{2/5} b^{6/5}}}+\frac {(-1)^{3/5} \left (a^2+1\right ) b^{6/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}+(-1)^{3/5} b^{6/5}}}\right )}{\sqrt {a^{4/5}+(-1)^{3/5} b^{6/5}}}-\frac {(-1)^{4/5} \left (a^2+1\right ) b^{6/5} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt [10]{a} \sqrt {a^{4/5}-(-1)^{4/5} b^{6/5}}}\right )}{\sqrt {a^{4/5}-(-1)^{4/5} b^{6/5}}}\right )}{5 a^{9/10} b} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(-5*a^(19/10)*Sqrt[a + b*x] - ((1 + a^2)*b^(6/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) - b^(6/5)])])
/Sqrt[a^(4/5) - b^(6/5)] + ((-1)^(1/5)*(1 + a^2)*b^(6/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) + (-1)^(
1/5)*b^(6/5)])])/Sqrt[a^(4/5) + (-1)^(1/5)*b^(6/5)] - ((-1)^(2/5)*(1 + a^2)*b^(6/5)*ArcTanh[Sqrt[a + b*x]/(a^(
1/10)*Sqrt[a^(4/5) - (-1)^(2/5)*b^(6/5)])])/Sqrt[a^(4/5) - (-1)^(2/5)*b^(6/5)] + ((-1)^(3/5)*(1 + a^2)*b^(6/5)
*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) + (-1)^(3/5)*b^(6/5)])])/Sqrt[a^(4/5) + (-1)^(3/5)*b^(6/5)] - ((
-1)^(4/5)*(1 + a^2)*b^(6/5)*ArcTanh[Sqrt[a + b*x]/(a^(1/10)*Sqrt[a^(4/5) - (-1)^(4/5)*b^(6/5)])])/Sqrt[a^(4/5)
- (-1)^(4/5)*b^(6/5)]))/(5*a^(9/10)*b)

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IntegrateAlgebraic [A]  time = 0.11, size = 138, normalized size = 1.00 \begin {gather*} -\frac {2 a \sqrt {a+b x}}{b}+\frac {1}{5} \left (b^5+a^2 b^5\right ) \text {RootSum}\left [a^5-a b^6-5 a^4 \text {$\#$1}^2+10 a^3 \text {$\#$1}^4-10 a^2 \text {$\#$1}^6+5 a \text {$\#$1}^8-\text {$\#$1}^{10}\&,\frac {\log \left (\sqrt {a+b x}-\text {$\#$1}\right )}{a^4 \text {$\#$1}-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^5-4 a \text {$\#$1}^7+\text {$\#$1}^9}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sqrt[a + b*x])/b + ((b^5 + a^2*b^5)*RootSum[a^5 - a*b^6 - 5*a^4*#1^2 + 10*a^3*#1^4 - 10*a^2*#1^6 + 5*a*#
1^8 - #1^10 & , Log[Sqrt[a + b*x] - #1]/(a^4*#1 - 4*a^3*#1^3 + 6*a^2*#1^5 - 4*a*#1^7 + #1^9) & ])/5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> no explicit roots found

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%{[-15360,46080]:[1,0,-5]%%},[0,6]%%%}+%%%{%%{[-1280,14080]:[1
,0,-5]%%},[0,5]%%%}+%%%{%%{[11904,640]:[1,0,-5]%%},[0,4]%%%}+%%%{%%{[3168,23520]:[1,0,-5]%%},[0,3]%%%}+%%%{%%{
[660,9220]:[1,0,-5]%%},[0,2]%%%}+%%%{%%{[495,635]:[1,0,-5]%%},[0,1]%%%}+%%%{150,[0,0]%%%} / %%%{30720,[0,6]%%%
}+%%%{%%{[5120,17920]:[1,0,-5]%%},[0,5]%%%}+%%%{%%{[18176,30720]:[1,0,-5]%%},[0,4]%%%}+%%%{%%{[16512,43200]:[1
,0,-5]%%},[0,3]%%%}+%%%{%%{[5600,15480]:[1,0,-5]%%},[0,2]%%%}+%%%{%%{[1060,2190]:[1,0,-5]%%},[0,1]%%%}+%%%{%%{
[75,225]:[1,0,-5]%%},[0,0]%%%} Error: Bad Argument Value

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maple [B]  time = 0.95, size = 123, normalized size = 0.89

method	result	size
derivativedivides	\(-\frac {2 \left (a \sqrt {b x +a}+\frac {b^{6} \left (a^{2}+1\right ) \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{10}-5 a \,\textit {\_Z}^{8}+10 a^{2} \textit {\_Z}^{6}-10 a^{3} \textit {\_Z}^{4}+a \,b^{6}+5 a^{4} \textit {\_Z}^{2}-a^{5}\right )}{\sum }\frac {\ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{-\textit {\_R}^{9}+4 \textit {\_R}^{7} a -6 \textit {\_R}^{5} a^{2}+4 \textit {\_R}^{3} a^{3}-\textit {\_R} \,a^{4}}\right )}{10}\right )}{b}\)	\(123\)
default	\(-\frac {2 \left (a \sqrt {b x +a}+\frac {b^{6} \left (a^{2}+1\right ) \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{10}-5 a \,\textit {\_Z}^{8}+10 a^{2} \textit {\_Z}^{6}-10 a^{3} \textit {\_Z}^{4}+a \,b^{6}+5 a^{4} \textit {\_Z}^{2}-a^{5}\right )}{\sum }\frac {\ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{-\textit {\_R}^{9}+4 \textit {\_R}^{7} a -6 \textit {\_R}^{5} a^{2}+4 \textit {\_R}^{3} a^{3}-\textit {\_R} \,a^{4}}\right )}{10}\right )}{b}\)	\(123\)
risch	\(-\frac {2 a \sqrt {b x +a}}{b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{10}-5 a \,\textit {\_Z}^{8}+10 a^{2} \textit {\_Z}^{6}-10 a^{3} \textit {\_Z}^{4}+a \,b^{6}+5 a^{4} \textit {\_Z}^{2}-a^{5}\right )}{\sum }\frac {\ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{\textit {\_R}^{9}-4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a^{2}-4 \textit {\_R}^{3} a^{3}+\textit {\_R} \,a^{4}}\right ) a^{2} b^{5}}{5}+\frac {b^{5} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{10}-5 a \,\textit {\_Z}^{8}+10 a^{2} \textit {\_Z}^{6}-10 a^{3} \textit {\_Z}^{4}+a \,b^{6}+5 a^{4} \textit {\_Z}^{2}-a^{5}\right )}{\sum }\frac {\ln \left (\sqrt {b x +a}-\textit {\_R} \right )}{\textit {\_R}^{9}-4 \textit {\_R}^{7} a +6 \textit {\_R}^{5} a^{2}-4 \textit {\_R}^{3} a^{3}+\textit {\_R} \,a^{4}}\right )}{5}\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b*(a*(b*x+a)^(1/2)+1/10*b^6*(a^2+1)*sum(1/(-_R^9+4*_R^7*a-6*_R^5*a^2+4*_R^3*a^3-_R*a^4)*ln((b*x+a)^(1/2)-_R
),_R=RootOf(_Z^10-5*_Z^8*a+10*_Z^6*a^2-10*_Z^4*a^3+a*b^6+5*_Z^2*a^4-a^5)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.83, size = 1622, normalized size = 11.75

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-root(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 + 3906250*a^10*b^4*d^8 + 1953125*a^12*b^4*d^8 + 1953
125*a^8*b^4*d^8 + 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6 + 312500*a^9*b^2*d^6 + 78125*a^15*b^2*d^6 + 78125*
a^7*b^2*d^6 + 252*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 120*a^14*b^2 + 120*a^6*b^2 + 45*a^16*b^2 + 45*a^4*b^
2 + 10*a^18*b^2 + 10*a^2*b^2 + a^20*b^2 + b^2, d, k)*((a + b*x)^(1/2)*(2560*b^40 + 20480*a^2*b^40 + 71680*a^4*
b^40 + 143360*a^6*b^40 + 179200*a^8*b^40 + 143360*a^10*b^40 + 71680*a^12*b^40 + 20480*a^14*b^40 + 2560*a^16*b^
40) + root(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 + 3906250*a^10*b^4*d^8 + 1953125*a^12*b^4*d^8 + 1953125*a^
8*b^4*d^8 + 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6 + 312500*a^9*b^2*d^6 + 78125*a^15*b^2*d^6 + 78125*a^7*b^
2*d^6 + 252*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 120*a^14*b^2 + 120*a^6*b^2 + 45*a^16*b^2 + 45*a^4*b^2 + 10
*a^18*b^2 + 10*a^2*b^2 + a^20*b^2 + b^2, d, k)*(12800*a*b^41 - root(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 +
3906250*a^10*b^4*d^8 + 1953125*a^12*b^4*d^8 + 1953125*a^8*b^4*d^8 + 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6
+ 312500*a^9*b^2*d^6 + 78125*a^15*b^2*d^6 + 78125*a^7*b^2*d^6 + 252*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 1
20*a^14*b^2 + 120*a^6*b^2 + 45*a^16*b^2 + 45*a^4*b^2 + 10*a^18*b^2 + 10*a^2*b^2 + a^20*b^2 + b^2, d, k)^4*(roo
t(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 + 3906250*a^10*b^4*d^8 + 1953125*a^12*b^4*d^8 + 1953125*a^8*b^4*d^8
+ 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6 + 312500*a^9*b^2*d^6 + 78125*a^15*b^2*d^6 + 78125*a^7*b^2*d^6 + 2
52*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 120*a^14*b^2 + 120*a^6*b^2 + 45*a^16*b^2 + 45*a^4*b^2 + 10*a^18*b^2
+ 10*a^2*b^2 + a^20*b^2 + b^2, d, k)*(root(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 + 3906250*a^10*b^4*d^8 +
1953125*a^12*b^4*d^8 + 1953125*a^8*b^4*d^8 + 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6 + 312500*a^9*b^2*d^6 +
78125*a^15*b^2*d^6 + 78125*a^7*b^2*d^6 + 252*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 120*a^14*b^2 + 120*a^6*b^
2 + 45*a^16*b^2 + 45*a^4*b^2 + 10*a^18*b^2 + 10*a^2*b^2 + a^20*b^2 + b^2, d, k)*(200000000*a^8*b^41 + 20000000
0*a^10*b^41 - 1000000000*root(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 + 3906250*a^10*b^4*d^8 + 1953125*a^12*b
^4*d^8 + 1953125*a^8*b^4*d^8 + 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6 + 312500*a^9*b^2*d^6 + 78125*a^15*b^2
*d^6 + 78125*a^7*b^2*d^6 + 252*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 120*a^14*b^2 + 120*a^6*b^2 + 45*a^16*b^
2 + 45*a^4*b^2 + 10*a^18*b^2 + 10*a^2*b^2 + a^20*b^2 + b^2, d, k)*a^8*b^42*(a + b*x)^(1/2)) - (a + b*x)^(1/2)*
(80000000*a^7*b^40 + 160000000*a^9*b^40 + 80000000*a^11*b^40)) + 8000000*a^7*b^39 + 24000000*a^9*b^39 + 240000
00*a^11*b^39 + 8000000*a^13*b^39) + 89600*a^3*b^41 + 268800*a^5*b^41 + 448000*a^7*b^41 + 448000*a^9*b^41 + 268
800*a^11*b^41 + 89600*a^13*b^41 + 12800*a^15*b^41)))*root(9765625*a^9*b^6*d^10 - 9765625*a^13*d^10 + 3906250*a
^10*b^4*d^8 + 1953125*a^12*b^4*d^8 + 1953125*a^8*b^4*d^8 + 468750*a^11*b^2*d^6 + 312500*a^13*b^2*d^6 + 312500*
a^9*b^2*d^6 + 78125*a^15*b^2*d^6 + 78125*a^7*b^2*d^6 + 252*a^10*b^2 + 210*a^12*b^2 + 210*a^8*b^2 + 120*a^14*b^
2 + 120*a^6*b^2 + 45*a^16*b^2 + 45*a^4*b^2 + 10*a^18*b^2 + 10*a^2*b^2 + a^20*b^2 + b^2, d, k), k, 1, 10) - (2*
a*(a + b*x)^(1/2))/b

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

[Out]

Timed out

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