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

Optimal. Leaf size=110 \[ -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{a x^6+b x}}{\sqrt {2} \sqrt {a x^6+b x}-x^2}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {a x^6+b x}}{\sqrt [4]{2}}+\frac {x^2}{2^{3/4}}}{x \sqrt [4]{a x^6+b x}}\right )}{\sqrt [4]{2}} \]

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Rubi [F]  time = 2.41, 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 (2 b+x^3+2 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)/((2*b + x^3 + 2*a*x^5)*(b*x + a*x^6)^(1/4)),x]

[Out]

(4*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 + a*x^20)^(1/4)*(2*b + x^12 + 2*a*x^20)), x], x, x^(
1/4)])/(b*x + a*x^6)^(1/4) - (4*x^(1/4)*(b + a*x^5)^(1/4)*Defer[Subst][Defer[Int][x^14/((b + a*x^20)^(1/4)*(2*
b + x^12 + 2*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 (2 b+x^3+2 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 (2 b+x^3+2 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 (2 b+x^{12}+2 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 {x^2}{\sqrt [4]{b+a x^{20}}}+\frac {x^2 \left (-5 b-x^{12}\right )}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 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}{\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 \frac {x^2 \left (-5 b-x^{12}\right )}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 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 {5 b x^2}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )}-\frac {x^{14}}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 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]{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 {4 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 (4 \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 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}{\sqrt [4]{b+a x^{20}} \left (2 b+x^{12}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 b+2 a x^5}{\left (2 b+x^3+2 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)/((2*b + x^3 + 2*a*x^5)*(b*x + a*x^6)^(1/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 3.00, size = 110, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{b x+a x^6}}{-x^2+\sqrt {2} \sqrt {b x+a x^6}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/4}}+\frac {\sqrt {b x+a x^6}}{\sqrt [4]{2}}}{x \sqrt [4]{b x+a x^6}}\right )}{\sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTan[(2^(3/4)*x*(b*x + a*x^6)^(1/4))/(-x^2 + Sqrt[2]*Sqrt[b*x + a*x^6])]/2^(1/4)) - ArcTanh[(x^2/2^(3/4) +
Sqrt[b*x + a*x^6]/2^(1/4))/(x*(b*x + a*x^6)^(1/4))]/2^(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)/(2*a*x^5+x^3+2*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 (2 \, a x^{5} + x^{3} + 2 \, b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((2*a*x^5-3*b)/(2*a*x^5+x^3+2*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 (2 \, a x^{5} + x^{3} + 2 \, b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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