3.18.19 \(\int \frac {2 b+a x^3}{(-b+x^2+a x^3) \sqrt [4]{-b x^2+a x^5}} \, dx\) [1719]

Optimal. Leaf size=116 \[ -\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^5}}{-x^2+\sqrt {-b x^2+a x^5}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^5}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^5}}\right ) \]

[Out]

-2^(1/2)*arctan(2^(1/2)*x*(a*x^5-b*x^2)^(1/4)/(-x^2+(a*x^5-b*x^2)^(1/2)))-2^(1/2)*arctanh((1/2*x^2*2^(1/2)+1/2
*(a*x^5-b*x^2)^(1/2)*2^(1/2))/x/(a*x^5-b*x^2)^(1/4))

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Rubi [F]
time = 1.17, 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+a x^3}{\left (-b+x^2+a x^3\right ) \sqrt [4]{-b x^2+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*x*(1 - (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, (a*x^3)/b])/(-(b*x^2) + a*x^5)^(1/4) - (6*b*Sqrt[x
]*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][1/((b - x^4 - a*x^6)*(-b + a*x^6)^(1/4)), x], x, Sqrt[x]])/(-(b*x
^2) + a*x^5)^(1/4) - (2*Sqrt[x]*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^4/((-b + a*x^6)^(1/4)*(-b + x^4 +
a*x^6)), x], x, Sqrt[x]])/(-(b*x^2) + a*x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {2 b+a x^3}{\left (-b+x^2+a x^3\right ) \sqrt [4]{-b x^2+a x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^3}\right ) \int \frac {2 b+a x^3}{\sqrt {x} \sqrt [4]{-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx}{\sqrt [4]{-b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-b+a x^6}}+\frac {3 b-x^4}{\sqrt [4]{-b+a x^6} \left (-b+x^4+a x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}+\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {3 b-x^4}{\sqrt [4]{-b+a x^6} \left (-b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \left (-\frac {3 b}{\left (b-x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}-\frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b+x^4+a x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1-\frac {a x^3}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^6}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}\\ &=\frac {2 x \sqrt [4]{1-\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^3}{b}\right )}{\sqrt [4]{-b x^2+a x^5}}-\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{-b+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (b-x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^5}}\\ \end {align*}

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Mathematica [A]
time = 2.79, size = 126, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x^3} \left (\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x^3}}{-x+\sqrt {-b+a x^3}}\right )+\tanh ^{-1}\left (\frac {x+\sqrt {-b+a x^3}}{\sqrt {2} \sqrt {x} \sqrt [4]{-b+a x^3}}\right )\right )}{\sqrt [4]{-b x^2+a x^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}+2 b}{\left (a \,x^{3}+x^{2}-b \right ) \left (a \,x^{5}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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^3+2*b)/(a*x^3+x^2-b)/(a*x^5-b*x^2)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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