3.6.64 \(\int \frac {x^5 (-7 b+10 a x^3)}{\sqrt [4]{-b x^3+a x^6} (-1-b x^7+a x^{10})} \, dx\)

Optimal. Leaf size=43 \[ 2 \tan ^{-1}\left (x \sqrt [4]{a x^6-b x^3}\right )-2 \tanh ^{-1}\left (x \sqrt [4]{a x^6-b x^3}\right ) \]

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Rubi [F]  time = 2.79, 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^5 \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b x^3+a x^6} \left (-1-b x^7+a x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5*(-7*b + 10*a*x^3))/((-(b*x^3) + a*x^6)^(1/4)*(-1 - b*x^7 + a*x^10)),x]

[Out]

(28*b*x^(3/4)*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^20/((-b + a*x^12)^(1/4)*(1 + b*x^28 - a*x^40)), x],
x, x^(1/4)])/(-(b*x^3) + a*x^6)^(1/4) + (40*a*x^(3/4)*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^32/((-b +
a*x^12)^(1/4)*(-1 - b*x^28 + a*x^40)), x], x, x^(1/4)])/(-(b*x^3) + a*x^6)^(1/4)

Rubi steps

\begin {align*} \int \frac {x^5 \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b x^3+a x^6} \left (-1-b x^7+a x^{10}\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-b+a x^3}\right ) \int \frac {x^{17/4} \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-1-b x^7+a x^{10}\right )} \, dx}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{20} \left (-7 b+10 a x^{12}\right )}{\sqrt [4]{-b+a x^{12}} \left (-1-b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{-b+a x^{12}} \left (1+b x^{28}-a x^{40}\right )}+\frac {10 a x^{32}}{\sqrt [4]{-b+a x^{12}} \left (-1-b x^{28}+a x^{40}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (40 a x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{32}}{\sqrt [4]{-b+a x^{12}} \left (-1-b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}+\frac {\left (28 b x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{-b+a x^{12}} \left (1+b x^{28}-a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b x^3+a x^6} \left (-1-b x^7+a x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^5*(-7*b + 10*a*x^3))/((-(b*x^3) + a*x^6)^(1/4)*(-1 - b*x^7 + a*x^10)),x]

[Out]

Integrate[(x^5*(-7*b + 10*a*x^3))/((-(b*x^3) + a*x^6)^(1/4)*(-1 - b*x^7 + a*x^10)), x]

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IntegrateAlgebraic [A]  time = 18.47, size = 43, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (x \sqrt [4]{-b x^3+a x^6}\right )-2 \tanh ^{-1}\left (x \sqrt [4]{-b x^3+a x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^5*(-7*b + 10*a*x^3))/((-(b*x^3) + a*x^6)^(1/4)*(-1 - b*x^7 + a*x^10)),x]

[Out]

2*ArcTan[x*(-(b*x^3) + a*x^6)^(1/4)] - 2*ArcTanh[x*(-(b*x^3) + a*x^6)^(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(x^5*(10*a*x^3-7*b)/(a*x^6-b*x^3)^(1/4)/(a*x^10-b*x^7-1),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 (10 \, a x^{3} - 7 \, b\right )} x^{5}}{{\left (a x^{10} - b x^{7} - 1\right )} {\left (a x^{6} - b x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(10*a*x^3-7*b)/(a*x^6-b*x^3)^(1/4)/(a*x^10-b*x^7-1),x, algorithm="giac")

[Out]

integrate((10*a*x^3 - 7*b)*x^5/((a*x^10 - b*x^7 - 1)*(a*x^6 - b*x^3)^(1/4)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (10 a \,x^{3}-7 b \right )}{\left (a \,x^{6}-b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{10}-b \,x^{7}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(10*a*x^3-7*b)/(a*x^6-b*x^3)^(1/4)/(a*x^10-b*x^7-1),x)

[Out]

int(x^5*(10*a*x^3-7*b)/(a*x^6-b*x^3)^(1/4)/(a*x^10-b*x^7-1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (10 \, a x^{3} - 7 \, b\right )} x^{5}}{{\left (a x^{10} - b x^{7} - 1\right )} {\left (a x^{6} - b x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(10*a*x^3-7*b)/(a*x^6-b*x^3)^(1/4)/(a*x^10-b*x^7-1),x, algorithm="maxima")

[Out]

integrate((10*a*x^3 - 7*b)*x^5/((a*x^10 - b*x^7 - 1)*(a*x^6 - b*x^3)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^5\,\left (7\,b-10\,a\,x^3\right )}{{\left (a\,x^6-b\,x^3\right )}^{1/4}\,\left (-a\,x^{10}+b\,x^7+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(7*b - 10*a*x^3))/((a*x^6 - b*x^3)^(1/4)*(b*x^7 - a*x^10 + 1)),x)

[Out]

int((x^5*(7*b - 10*a*x^3))/((a*x^6 - b*x^3)^(1/4)*(b*x^7 - a*x^10 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (10 a x^{3} - 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{3} - b\right )} \left (a x^{10} - b x^{7} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(10*a*x**3-7*b)/(a*x**6-b*x**3)**(1/4)/(a*x**10-b*x**7-1),x)

[Out]

Integral(x**5*(10*a*x**3 - 7*b)/((x**3*(a*x**3 - b))**(1/4)*(a*x**10 - b*x**7 - 1)), x)

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