3.7.33 \(\int \frac {\sqrt {1+x^5} (-2+3 x^5)}{a-x^4+2 a x^5+a x^{10}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5+1}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^5+1}}\right )}{a^{3/4}} \]

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Rubi [F]  time = 0.55, 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 {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(a - x^4 + 2*a*x^5 + a*x^10),x]

[Out]

-2*Defer[Int][Sqrt[1 + x^5]/(a - x^4 + 2*a*x^5 + a*x^10), x] + 3*Defer[Int][(x^5*Sqrt[1 + x^5])/(a - x^4 + 2*a
*x^5 + a*x^10), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx &=\int \left (-\frac {2 \sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}}+\frac {3 x^5 \sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}} \, dx\right )+3 \int \frac {x^5 \sqrt {1+x^5}}{a-x^4+2 a x^5+a x^{10}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{a-x^4+2 a x^5+a x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(a - x^4 + 2*a*x^5 + a*x^10),x]

[Out]

Integrate[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(a - x^4 + 2*a*x^5 + a*x^10), x]

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IntegrateAlgebraic [A]  time = 7.28, size = 49, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {1+x^5}}\right )}{a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(a - x^4 + 2*a*x^5 + a*x^10),x]

[Out]

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

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fricas [B]  time = 0.71, size = 238, normalized size = 4.86 \begin {gather*} -\frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x}{\sqrt {x^{5} + 1}}\right ) - \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} + 2 \, \sqrt {x^{5} + 1} {\left (a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {a x^{10} + 2 \, a x^{5} + x^{4} - 2 \, \sqrt {x^{5} + 1} {\left (a \frac {1}{a^{3}}^{\frac {1}{4}} x^{3} + {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a^{2} x^{7} + a^{2} x^{2}\right )} \sqrt {\frac {1}{a^{3}}} + a}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)^(1/2)*(3*x^5-2)/(a*x^10+2*a*x^5-x^4+a),x, algorithm="fricas")

[Out]

-(a^(-3))^(1/4)*arctan(a^2*(a^(-3))^(3/4)*x/sqrt(x^5 + 1)) - 1/4*(a^(-3))^(1/4)*log((a*x^10 + 2*a*x^5 + x^4 +
2*sqrt(x^5 + 1)*(a*(a^(-3))^(1/4)*x^3 + (a^3*x^6 + a^3*x)*(a^(-3))^(3/4)) + 2*(a^2*x^7 + a^2*x^2)*sqrt(a^(-3))
+ a)/(a*x^10 + 2*a*x^5 - x^4 + a)) + 1/4*(a^(-3))^(1/4)*log((a*x^10 + 2*a*x^5 + x^4 - 2*sqrt(x^5 + 1)*(a*(a^(
-3))^(1/4)*x^3 + (a^3*x^6 + a^3*x)*(a^(-3))^(3/4)) + 2*(a^2*x^7 + a^2*x^2)*sqrt(a^(-3)) + a)/(a*x^10 + 2*a*x^5
- x^4 + a))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)^(1/2)*(3*x^5-2)/(a*x^10+2*a*x^5-x^4+a),x, algorithm="giac")

[Out]

integrate((3*x^5 - 2)*sqrt(x^5 + 1)/(a*x^10 + 2*a*x^5 - x^4 + a), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{5}+1}\, \left (3 x^{5}-2\right )}{a \,x^{10}+2 a \,x^{5}-x^{4}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+1)^(1/2)*(3*x^5-2)/(a*x^10+2*a*x^5-x^4+a),x)

[Out]

int((x^5+1)^(1/2)*(3*x^5-2)/(a*x^10+2*a*x^5-x^4+a),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{a x^{10} + 2 \, a x^{5} - x^{4} + a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)^(1/2)*(3*x^5-2)/(a*x^10+2*a*x^5-x^4+a),x, algorithm="maxima")

[Out]

integrate((3*x^5 - 2)*sqrt(x^5 + 1)/(a*x^10 + 2*a*x^5 - x^4 + a), x)

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mupad [B]  time = 14.59, size = 309, normalized size = 6.31 \begin {gather*} \frac {\ln \left (\frac {\left (x\,\sqrt {a^3}+a^2\,x^4-2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}\right )\,\left (a\,x^3-2\,x^6\,\sqrt {a^3}-3\,x\,\sqrt {a^3}+a^2\,x^4+a^2\,x^9+2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}\right )}{\left (a^2-x^2\,\sqrt {a^3}+a^2\,x^5\right )\,\left (4\,\sqrt {a^3}+2\,x^5\,\sqrt {a^3}-a\,x^2-a^2\,x^8\right )}\right )}{2\,{\left (a^3\right )}^{1/4}}+\frac {\ln \left (\frac {\left (2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}+x\,\sqrt {a^3}\,1{}\mathrm {i}-a^2\,x^4\,1{}\mathrm {i}\right )\,\left (2\,a\,\sqrt {x^5+1}\,{\left (a^3\right )}^{1/4}+x^6\,\sqrt {a^3}\,2{}\mathrm {i}+a\,x^3\,1{}\mathrm {i}+x\,\sqrt {a^3}\,3{}\mathrm {i}+a^2\,x^4\,1{}\mathrm {i}+a^2\,x^9\,1{}\mathrm {i}\right )}{\left (x^2\,\sqrt {a^3}+a^2+a^2\,x^5\right )\,\left (4\,\sqrt {a^3}+2\,x^5\,\sqrt {a^3}+a\,x^2+a^2\,x^8\right )}\right )\,1{}\mathrm {i}}{2\,{\left (a^3\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 1)^(1/2)*(3*x^5 - 2))/(a + 2*a*x^5 + a*x^10 - x^4),x)

[Out]

log(((x*(a^3)^(1/2) + a^2*x^4 - 2*a*(x^5 + 1)^(1/2)*(a^3)^(1/4))*(a*x^3 - 2*x^6*(a^3)^(1/2) - 3*x*(a^3)^(1/2)
+ a^2*x^4 + a^2*x^9 + 2*a*(x^5 + 1)^(1/2)*(a^3)^(1/4)))/((a^2 - x^2*(a^3)^(1/2) + a^2*x^5)*(4*(a^3)^(1/2) + 2*
x^5*(a^3)^(1/2) - a*x^2 - a^2*x^8)))/(2*(a^3)^(1/4)) + (log(((x*(a^3)^(1/2)*1i - a^2*x^4*1i + 2*a*(x^5 + 1)^(1
/2)*(a^3)^(1/4))*(x^6*(a^3)^(1/2)*2i + a*x^3*1i + x*(a^3)^(1/2)*3i + a^2*x^4*1i + a^2*x^9*1i + 2*a*(x^5 + 1)^(
1/2)*(a^3)^(1/4)))/((x^2*(a^3)^(1/2) + a^2 + a^2*x^5)*(4*(a^3)^(1/2) + 2*x^5*(a^3)^(1/2) + a*x^2 + a^2*x^8)))*
1i)/(2*(a^3)^(1/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (3 x^{5} - 2\right )}{a x^{10} + 2 a x^{5} + a - x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+1)**(1/2)*(3*x**5-2)/(a*x**10+2*a*x**5-x**4+a),x)

[Out]

Integral(sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1))*(3*x**5 - 2)/(a*x**10 + 2*a*x**5 + a - x**4), x)

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