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

Optimal. Leaf size=108 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^5-b x}}{\sqrt {a x^5-b x}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5-b x}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5-b x}}\right ) \]

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

Verification is not applicable to the result.

[In]

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

[Out]

(4*x*(1 - (a*x^4)/b)^(1/4)*Hypergeometric2F1[3/16, 1/4, 19/16, (a*x^4)/b])/(3*(-(b*x) + a*x^5)^(1/4)) - (16*b*
x^(1/4)*(-b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^2/((b - x^12 - a*x^16)*(-b + a*x^16)^(1/4)), x], x, x^(1/
4)])/(-(b*x) + a*x^5)^(1/4) - (4*x^(1/4)*(-b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((-b + a*x^16)^(1/4)*
(-b + x^12 + a*x^16)), x], x, x^(1/4)])/(-(b*x) + a*x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {3 b+a x^4}{\left (-b+x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \int \frac {3 b+a x^4}{\sqrt [4]{x} \sqrt [4]{-b+a x^4} \left (-b+x^3+a x^4\right )} \, dx}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3 b+a x^{16}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt [4]{-b+a x^{16}}}+\frac {x^2 \left (4 b-x^{12}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (4 b-x^{12}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 b x^2}{\left (b-x^{12}-a x^{16}\right ) \sqrt [4]{-b+a x^{16}}}-\frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1-\frac {a x^{16}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ &=\frac {4 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {3}{16},\frac {1}{4};\frac {19}{16};\frac {a x^4}{b}\right )}{3 \sqrt [4]{-b x+a x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (-b+x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}-\frac {\left (16 b \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b-x^{12}-a x^{16}\right ) \sqrt [4]{-b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 b+a x^4}{\left (-b+x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.23, size = 108, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^5}}{-x^2+\sqrt {-b x+a x^5}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^5}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x) + a*x^5)^(1/4))/(-x^2 + Sqrt[-(b*x) + a*x^5])]) - Sqrt[2]*ArcTanh[(x^2/Sqr
t[2] + Sqrt[-(b*x) + a*x^5]/Sqrt[2])/(x*(-(b*x) + a*x^5)^(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((a*x^4+3*b)/(a*x^4+x^3-b)/(a*x^5-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 {a x^{4} + 3 \, b}{{\left (a x^{5} - b x\right )}^{\frac {1}{4}} {\left (a x^{4} + x^{3} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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