3.1.99 \(\int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} (b+x^2+a x^5)} \, dx\)

Optimal. Leaf size=16 \[ -2 \tan ^{-1}\left (\frac {x}{\sqrt {a x^5+b}}\right ) \]

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Rubi [F]  time = 0.84, 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 {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*x*Sqrt[1 + (a*x^5)/b]*Hypergeometric2F1[1/5, 1/2, 6/5, -((a*x^5)/b)])/Sqrt[b + a*x^5] - 5*b*Defer[Int][1/(S
qrt[b + a*x^5]*(b + x^2 + a*x^5)), x] - 3*Defer[Int][x^2/(Sqrt[b + a*x^5]*(b + x^2 + a*x^5)), x]

Rubi steps

\begin {align*} \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx &=\int \left (\frac {3}{\sqrt {b+a x^5}}-\frac {5 b+3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}\right ) \, dx\\ &=3 \int \frac {1}{\sqrt {b+a x^5}} \, dx-\int \frac {5 b+3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx\\ &=\frac {\left (3 \sqrt {1+\frac {a x^5}{b}}\right ) \int \frac {1}{\sqrt {1+\frac {a x^5}{b}}} \, dx}{\sqrt {b+a x^5}}-\int \left (\frac {5 b}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}+\frac {3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}\right ) \, dx\\ &=\frac {3 x \sqrt {1+\frac {a x^5}{b}} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {6}{5};-\frac {a x^5}{b}\right )}{\sqrt {b+a x^5}}-3 \int \frac {x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx-(5 b) \int \frac {1}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.46, size = 16, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt {b+a x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcTan[x/Sqrt[b + a*x^5]]

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fricas [B]  time = 0.57, size = 35, normalized size = 2.19 \begin {gather*} \arctan \left (\frac {{\left (a x^{5} - x^{2} + b\right )} \sqrt {a x^{5} + b}}{2 \, {\left (a x^{6} + b x\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(1/2*(a*x^5 - x^2 + b)*sqrt(a*x^5 + b)/(a*x^6 + b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((3*a*x^5 - 2*b)/((a*x^5 + x^2 + b)*sqrt(a*x^5 + b)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {3 a \,x^{5}-2 b}{\sqrt {a \,x^{5}+b}\, \left (a \,x^{5}+x^{2}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((3*a*x^5 - 2*b)/((a*x^5 + x^2 + b)*sqrt(a*x^5 + b)), x)

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mupad [B]  time = 2.26, size = 42, normalized size = 2.62 \begin {gather*} \ln \left (\frac {b+a\,x^5-x^2+x\,\sqrt {a\,x^5+b}\,2{}\mathrm {i}}{a\,x^5+x^2+b}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((b + x*(b + a*x^5)^(1/2)*2i + a*x^5 - x^2)/(b + a*x^5 + x^2))*1i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((3*a*x**5 - 2*b)/(sqrt(a*x**5 + b)*(a*x**5 + b + x**2)), x)

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