3.7.59 \(\int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} (b-a x^2+x^7)} \, dx\)

Optimal. Leaf size=52 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b x}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x \left (a x^3-b x\right )^{3/4}}{a x^2-b}\right ) \]

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Rubi [F]  time = 2.26, 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 {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-20*a*x^(1/4)*(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^14/((-b + a*x^8)^(1/4)*(-b + a*x^8 - x^28)), x], x
, x^(1/4)])/(-(b*x) + a*x^3)^(1/4) - (28*b*x^(1/4)*(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^6/((-b + a*x^8
)^(1/4)*(b - a*x^8 + x^28)), x], x, x^(1/4)])/(-(b*x) + a*x^3)^(1/4)

Rubi steps

\begin {align*} \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx &=\int \frac {x \left (-7 b+5 a x^2\right )}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{3/4} \left (-7 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-a x^2+x^7\right )} \, dx}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (-7 b+5 a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8+x^{28}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {5 a x^{14}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8-x^{28}\right )}-\frac {7 b x^6}{\sqrt [4]{-b+a x^8} \left (b-a x^8+x^{28}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=-\frac {\left (20 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8-x^{28}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (28 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{-b+a x^8} \left (b-a x^8+x^{28}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ \end {align*}

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-7*b*x + 5*a*x^3)/((-(b*x) + a*x^3)^(1/4)*(b - a*x^2 + x^7)),x]

[Out]

Integrate[(-7*b*x + 5*a*x^3)/((-(b*x) + a*x^3)^(1/4)*(b - a*x^2 + x^7)), x]

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IntegrateAlgebraic [A]  time = 2.24, size = 52, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{-b x+a x^3}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x \left (-b x+a x^3\right )^{3/4}}{-b+a x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-7*b*x + 5*a*x^3)/((-(b*x) + a*x^3)^(1/4)*(b - a*x^2 + x^7)),x]

[Out]

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

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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((5*a*x^3-7*b*x)/(a*x^3-b*x)^(1/4)/(x^7-a*x^2+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 {5 \, a x^{3} - 7 \, b x}{{\left (x^{7} - a x^{2} + b\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*a*x^3-7*b*x)/(a*x^3-b*x)^(1/4)/(x^7-a*x^2+b),x, algorithm="giac")

[Out]

integrate((5*a*x^3 - 7*b*x)/((x^7 - a*x^2 + b)*(a*x^3 - b*x)^(1/4)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {5 a \,x^{3}-7 b x}{\left (a \,x^{3}-b x \right )^{\frac {1}{4}} \left (x^{7}-a \,x^{2}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*a*x^3-7*b*x)/(a*x^3-b*x)^(1/4)/(x^7-a*x^2+b),x)

[Out]

int((5*a*x^3-7*b*x)/(a*x^3-b*x)^(1/4)/(x^7-a*x^2+b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, a x^{3} - 7 \, b x}{{\left (x^{7} - a x^{2} + b\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*a*x^3-7*b*x)/(a*x^3-b*x)^(1/4)/(x^7-a*x^2+b),x, algorithm="maxima")

[Out]

integrate((5*a*x^3 - 7*b*x)/((x^7 - a*x^2 + b)*(a*x^3 - b*x)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {7\,b\,x-5\,a\,x^3}{{\left (a\,x^3-b\,x\right )}^{1/4}\,\left (x^7-a\,x^2+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*b*x - 5*a*x^3)/((a*x^3 - b*x)^(1/4)*(b - a*x^2 + x^7)),x)

[Out]

-int((7*b*x - 5*a*x^3)/((a*x^3 - b*x)^(1/4)*(b - a*x^2 + x^7)), 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^{2} - 7 b\right )}{\sqrt [4]{x \left (a x^{2} - b\right )} \left (- a x^{2} + b + x^{7}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*a*x**3-7*b*x)/(a*x**3-b*x)**(1/4)/(x**7-a*x**2+b),x)

[Out]

Integral(x*(5*a*x**2 - 7*b)/((x*(a*x**2 - b))**(1/4)*(-a*x**2 + b + x**7)), x)

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