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

Optimal. Leaf size=49 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b x^2}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {\left (a x^3+b x^2\right )^{3/4}}{a x+b}\right ) \]

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Rubi [F]  time = 1.84, 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]

(-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) - (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, 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 {6 b x^2}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{12}\right )}-\frac {5 a x^4}{\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 = 17.68, size = 57707, normalized size = 1177.69 \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 = 0.61, size = 49, 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 = 0.03, 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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