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

Optimal. Leaf size=53 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b x^2}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {\left (a x^3-b x^2\right )^{3/4}}{a x-b}\right ) \]

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Rubi [F]  time = 1.99, 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 {x (-6 b+5 a x)}{\sqrt [4]{-b x^2+a x^3} \left (b-a x+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-10*a*Sqrt[x]*(-b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^4/((-b + a*x^2)^(1/4)*(-b + a*x^2 - x^12)), x], x, S
qrt[x]])/(-(b*x^2) + a*x^3)^(1/4) - (12*b*Sqrt[x]*(-b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^2/((-b + a*x^2)^(
1/4)*(b - a*x^2 + x^12)), x], x, Sqrt[x]])/(-(b*x^2) + a*x^3)^(1/4)

Rubi steps

\begin {align*} \int \frac {x (-6 b+5 a x)}{\sqrt [4]{-b x^2+a x^3} \left (b-a x+x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {\sqrt {x} (-6 b+5 a x)}{\sqrt [4]{-b+a x} \left (b-a x+x^6\right )} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-6 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-a x^2+x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \left (-\frac {5 a x^4}{\sqrt [4]{-b+a x^2} \left (-b+a x^2-x^{12}\right )}-\frac {6 b x^2}{\sqrt [4]{-b+a x^2} \left (b-a x^2+x^{12}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (10 a \sqrt {x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-b+a x^2} \left (-b+a x^2-x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^3}}-\frac {\left (12 b \sqrt {x} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^2} \left (b-a x^2+x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ \end {align*}

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Mathematica [C]  time = 18.23, size = 58426, normalized size = 1102.38 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.66, size = 53, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{-b x^2+a x^3}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {\left (-b x^2+a x^3\right )^{3/4}}{-b+a x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (5 a x -6 b \right )}{\left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}} \left (x^{6}-a x +b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {x\,\left (6\,b-5\,a\,x\right )}{{\left (a\,x^3-b\,x^2\right )}^{1/4}\,\left (x^6-a\,x+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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