3.31.13 \(\int \frac {x^5 (-7 b+9 a x^2)}{\sqrt [4]{-b x^3+a x^5} (1-b x^7+a x^9)} \, dx\)

Optimal. Leaf size=413 \[ -\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^5-b x^3}-2^{2/3} x \sqrt [4]{a x^5-b x^3}}{2^{2/3} x \sqrt [4]{a x^5-b x^3}+x^2 \left (-\sqrt [4]{a x^5-b x^3}\right )-\sqrt {2} x+2 \sqrt [6]{2}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^5-b x^3}-2^{2/3} x \sqrt [4]{a x^5-b x^3}}{2^{2/3} x \sqrt [4]{a x^5-b x^3}+x^2 \left (-\sqrt [4]{a x^5-b x^3}\right )+\sqrt {2} x-2 \sqrt [6]{2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-\sqrt {2} x^3 \sqrt [4]{a x^5-b x^3}-2\ 2^{5/6} x \sqrt [4]{a x^5-b x^3}+4 \sqrt [6]{2} x^2 \sqrt [4]{a x^5-b x^3}}{2\ 2^{2/3} x^3 \sqrt {a x^5-b x^3}+x^4 \left (-\sqrt {a x^5-b x^3}\right )-2 \sqrt [3]{2} x^2 \sqrt {a x^5-b x^3}-x^2+2\ 2^{2/3} x-2 \sqrt [3]{2}}\right ) \]

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Rubi [F]  time = 2.71, 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+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a*x^9)),x]

[Out]

(28*b*x^(3/4)*(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^20/((-b + a*x^8)^(1/4)*(-1 + b*x^28 - a*x^36)), x],
x, x^(1/4)])/(-(b*x^3) + a*x^5)^(1/4) + (36*a*x^(3/4)*(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^28/((-b +
a*x^8)^(1/4)*(1 - b*x^28 + a*x^36)), x], x, x^(1/4)])/(-(b*x^3) + a*x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{17/4} \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (1-b x^7+a x^9\right )} \, dx}{\sqrt [4]{-b x^3+a x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{20} \left (-7 b+9 a x^8\right )}{\sqrt [4]{-b+a x^8} \left (1-b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{-b+a x^8} \left (-1+b x^{28}-a x^{36}\right )}+\frac {9 a x^{28}}{\sqrt [4]{-b+a x^8} \left (1-b x^{28}+a x^{36}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}\\ &=\frac {\left (36 a x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{28}}{\sqrt [4]{-b+a x^8} \left (1-b x^{28}+a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}+\frac {\left (28 b x^{3/4} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{-b+a x^8} \left (-1+b x^{28}-a x^{36}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a*x^9)),x]

[Out]

Integrate[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a*x^9)), x]

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IntegrateAlgebraic [F]  time = 78.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a*x^9)),x]

[Out]

Could not integrate

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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*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-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 (9 \, a x^{2} - 7 \, b\right )} x^{5}}{{\left (a x^{9} - b x^{7} + 1\right )} {\left (a x^{5} - 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*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x, algorithm="giac")

[Out]

integrate((9*a*x^2 - 7*b)*x^5/((a*x^9 - b*x^7 + 1)*(a*x^5 - b*x^3)^(1/4)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (9 a \,x^{2}-7 b \right )}{\left (a \,x^{5}-b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{9}-b \,x^{7}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x)

[Out]

int(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-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 (9 \, a x^{2} - 7 \, b\right )} x^{5}}{{\left (a x^{9} - b x^{7} + 1\right )} {\left (a x^{5} - 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*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x, algorithm="maxima")

[Out]

integrate((9*a*x^2 - 7*b)*x^5/((a*x^9 - b*x^7 + 1)*(a*x^5 - b*x^3)^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^5\,\left (7\,b-9\,a\,x^2\right )}{{\left (a\,x^5-b\,x^3\right )}^{1/4}\,\left (a\,x^9-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 - 9*a*x^2))/((a*x^5 - b*x^3)^(1/4)*(a*x^9 - b*x^7 + 1)),x)

[Out]

-int((x^5*(7*b - 9*a*x^2))/((a*x^5 - b*x^3)^(1/4)*(a*x^9 - b*x^7 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (9 a x^{2} - 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{2} - b\right )} \left (a x^{9} - b x^{7} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(9*a*x**2-7*b)/(a*x**5-b*x**3)**(1/4)/(a*x**9-b*x**7+1),x)

[Out]

Integral(x**5*(9*a*x**2 - 7*b)/((x**3*(a*x**2 - b))**(1/4)*(a*x**9 - b*x**7 + 1)), x)

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