3.16.81 \(\int \frac {3 b+2 a x^5}{(-b+x^3+a x^5) \sqrt [4]{-b x+a x^6}} \, dx\)

Optimal. Leaf size=108 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^6-b x}}{\sqrt {a x^6-b x}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^6-b x}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^6-b x}}\right ) \]

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Rubi [F]  time = 2.62, 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 {3 b+2 a x^5}{\left (-b+x^3+a x^5\right ) \sqrt [4]{-b x+a x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(8*x*(1 - (a*x^5)/b)^(1/4)*Hypergeometric2F1[3/20, 1/4, 23/20, (a*x^5)/b])/(3*(-(b*x) + a*x^6)^(1/4)) - (20*b*
x^(1/4)*(-b + a*x^5)^(1/4)*Defer[Subst][Defer[Int][x^2/((b - x^12 - a*x^20)*(-b + a*x^20)^(1/4)), x], x, x^(1/
4)])/(-(b*x) + a*x^6)^(1/4) - (8*x^(1/4)*(-b + a*x^5)^(1/4)*Defer[Subst][Defer[Int][x^14/((-b + a*x^20)^(1/4)*
(-b + x^12 + a*x^20)), x], x, x^(1/4)])/(-(b*x) + a*x^6)^(1/4)

Rubi steps

\begin {align*} \int \frac {3 b+2 a x^5}{\left (-b+x^3+a x^5\right ) \sqrt [4]{-b x+a x^6}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {3 b+2 a x^5}{\sqrt [4]{x} \sqrt [4]{-b+a x^5} \left (-b+x^3+a x^5\right )} \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3 b+2 a x^{20}\right )}{\sqrt [4]{-b+a x^{20}} \left (-b+x^{12}+a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x^2}{\sqrt [4]{-b+a x^{20}}}+\frac {x^2 \left (5 b-2 x^{12}\right )}{\sqrt [4]{-b+a x^{20}} \left (-b+x^{12}+a x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (5 b-2 x^{12}\right )}{\sqrt [4]{-b+a x^{20}} \left (-b+x^{12}+a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {5 b x^2}{\left (b-x^{12}-a x^{20}\right ) \sqrt [4]{-b+a x^{20}}}-\frac {2 x^{14}}{\sqrt [4]{-b+a x^{20}} \left (-b+x^{12}+a x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1-\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1-\frac {a x^{20}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {8 x \sqrt [4]{1-\frac {a x^5}{b}} \, _2F_1\left (\frac {3}{20},\frac {1}{4};\frac {23}{20};\frac {a x^5}{b}\right )}{3 \sqrt [4]{-b x+a x^6}}-\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{20}} \left (-b+x^{12}+a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}-\frac {\left (20 b \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b-x^{12}-a x^{20}\right ) \sqrt [4]{-b+a x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 b+2 a x^5}{\left (-b+x^3+a x^5\right ) \sqrt [4]{-b x+a x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(3*b + 2*a*x^5)/((-b + x^3 + a*x^5)*(-(b*x) + a*x^6)^(1/4)),x]

[Out]

Integrate[(3*b + 2*a*x^5)/((-b + x^3 + a*x^5)*(-(b*x) + a*x^6)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 2.94, size = 108, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^6}}{-x^2+\sqrt {-b x+a x^6}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^6}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3*b + 2*a*x^5)/((-b + x^3 + a*x^5)*(-(b*x) + a*x^6)^(1/4)),x]

[Out]

-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x) + a*x^6)^(1/4))/(-x^2 + Sqrt[-(b*x) + a*x^6])]) - Sqrt[2]*ArcTanh[(x^2/Sqr
t[2] + Sqrt[-(b*x) + a*x^6]/Sqrt[2])/(x*(-(b*x) + a*x^6)^(1/4))]

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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((2*a*x^5+3*b)/(a*x^5+x^3-b)/(a*x^6-b*x)^(1/4),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 {2 \, a x^{5} + 3 \, b}{{\left (a x^{6} - b x\right )}^{\frac {1}{4}} {\left (a x^{5} + x^{3} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^5+3*b)/(a*x^5+x^3-b)/(a*x^6-b*x)^(1/4),x, algorithm="giac")

[Out]

integrate((2*a*x^5 + 3*b)/((a*x^6 - b*x)^(1/4)*(a*x^5 + x^3 - b)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{5}+3 b}{\left (a \,x^{5}+x^{3}-b \right ) \left (a \,x^{6}-b x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x^5+3*b)/(a*x^5+x^3-b)/(a*x^6-b*x)^(1/4),x)

[Out]

int((2*a*x^5+3*b)/(a*x^5+x^3-b)/(a*x^6-b*x)^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{5} + 3 \, b}{{\left (a x^{6} - b x\right )}^{\frac {1}{4}} {\left (a x^{5} + x^{3} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^5+3*b)/(a*x^5+x^3-b)/(a*x^6-b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*a*x^5 + 3*b)/((a*x^6 - b*x)^(1/4)*(a*x^5 + x^3 - b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,a\,x^5+3\,b}{{\left (a\,x^6-b\,x\right )}^{1/4}\,\left (a\,x^5+x^3-b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*b + 2*a*x^5)/((a*x^6 - b*x)^(1/4)*(a*x^5 - b + x^3)),x)

[Out]

int((3*b + 2*a*x^5)/((a*x^6 - b*x)^(1/4)*(a*x^5 - b + x^3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x**5+3*b)/(a*x**5+x**3-b)/(a*x**6-b*x)**(1/4),x)

[Out]

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

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