3.31.80 \(\int \frac {(-3+x^2) (1-2 x^2+x^4+x^6)}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx\)

Optimal. Leaf size=514 \[ \frac {\left (\frac {a x^2-a+b x^3}{c x^2-c+d x^3}\right )^{3/4} \left (-96 a^2 c^3 x^8+64 a^2 c^3 x^6+96 a^2 c^3 x^4-96 a^2 c^3 x^2+32 a^2 c^3-96 a^2 c^2 d x^9-36 a^2 c^2 d x^7+72 a^2 c^2 d x^5-36 a^2 c^2 d x^3+3 a^2 c d^2 x^8-3 a^2 c d^2 x^6+7 a^2 d^3 x^9+36 a b c^3 x^7-72 a b c^3 x^5+36 a b c^3 x^3+42 a b c^2 d x^8-42 a b c^2 d x^6+6 a b c d^2 x^9-45 b^2 c^3 x^8+45 b^2 c^3 x^6-45 b^2 c^2 d x^9\right )}{96 a^3 c^2 x^9}+\frac {\left (32 a^3 c^2 d+7 a^3 d^3-32 a^2 b c^3+3 a^2 b c d^2+5 a b^2 c^2 d-15 b^3 c^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x^2-a+b x^3}{c x^2-c+d x^3}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {\left (-32 a^3 c^2 d-7 a^3 d^3+32 a^2 b c^3-3 a^2 b c d^2-5 a b^2 c^2 d+15 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x^2-a+b x^3}{c x^2-c+d x^3}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}} \]

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Rubi [F]  time = 5.38, 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 {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-3 + x^2)*(1 - 2*x^2 + x^4 + x^6))/(x^10*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4)),x]

[Out]

(-3*(-a + a*x^2 + b*x^3)^(1/4)*Defer[Int][(-c + c*x^2 + d*x^3)^(1/4)/(x^10*(-a + a*x^2 + b*x^3)^(1/4)), x])/((
(a - a*x^2 - b*x^3)/(c - c*x^2 - d*x^3))^(1/4)*(-c + c*x^2 + d*x^3)^(1/4)) + (7*(-a + a*x^2 + b*x^3)^(1/4)*Def
er[Int][(-c + c*x^2 + d*x^3)^(1/4)/(x^8*(-a + a*x^2 + b*x^3)^(1/4)), x])/(((a - a*x^2 - b*x^3)/(c - c*x^2 - d*
x^3))^(1/4)*(-c + c*x^2 + d*x^3)^(1/4)) - (5*(-a + a*x^2 + b*x^3)^(1/4)*Defer[Int][(-c + c*x^2 + d*x^3)^(1/4)/
(x^6*(-a + a*x^2 + b*x^3)^(1/4)), x])/(((a - a*x^2 - b*x^3)/(c - c*x^2 - d*x^3))^(1/4)*(-c + c*x^2 + d*x^3)^(1
/4)) - (2*(-a + a*x^2 + b*x^3)^(1/4)*Defer[Int][(-c + c*x^2 + d*x^3)^(1/4)/(x^4*(-a + a*x^2 + b*x^3)^(1/4)), x
])/(((a - a*x^2 - b*x^3)/(c - c*x^2 - d*x^3))^(1/4)*(-c + c*x^2 + d*x^3)^(1/4)) + ((-a + a*x^2 + b*x^3)^(1/4)*
Defer[Int][(-c + c*x^2 + d*x^3)^(1/4)/(x^2*(-a + a*x^2 + b*x^3)^(1/4)), x])/(((a - a*x^2 - b*x^3)/(c - c*x^2 -
d*x^3))^(1/4)*(-c + c*x^2 + d*x^3)^(1/4))

Rubi steps

\begin {align*} \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx &=\frac {\sqrt [4]{-a+a x^2+b x^3} \int \frac {\left (-3+x^2\right ) \sqrt [4]{-c+c x^2+d x^3} \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}\\ &=\frac {\sqrt [4]{-a+a x^2+b x^3} \int \left (-\frac {3 \sqrt [4]{-c+c x^2+d x^3}}{x^{10} \sqrt [4]{-a+a x^2+b x^3}}+\frac {7 \sqrt [4]{-c+c x^2+d x^3}}{x^8 \sqrt [4]{-a+a x^2+b x^3}}-\frac {5 \sqrt [4]{-c+c x^2+d x^3}}{x^6 \sqrt [4]{-a+a x^2+b x^3}}-\frac {2 \sqrt [4]{-c+c x^2+d x^3}}{x^4 \sqrt [4]{-a+a x^2+b x^3}}+\frac {\sqrt [4]{-c+c x^2+d x^3}}{x^2 \sqrt [4]{-a+a x^2+b x^3}}\right ) \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}\\ &=\frac {\sqrt [4]{-a+a x^2+b x^3} \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^2 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}-\frac {\left (2 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^4 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}-\frac {\left (3 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^{10} \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}-\frac {\left (5 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^6 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}+\frac {\left (7 \sqrt [4]{-a+a x^2+b x^3}\right ) \int \frac {\sqrt [4]{-c+c x^2+d x^3}}{x^8 \sqrt [4]{-a+a x^2+b x^3}} \, dx}{\sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}} \sqrt [4]{-c+c x^2+d x^3}}\\ \end {align*}

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Mathematica [F]  time = 1.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^2\right ) \left (1-2 x^2+x^4+x^6\right )}{x^{10} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-3 + x^2)*(1 - 2*x^2 + x^4 + x^6))/(x^10*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4)),x]

[Out]

Integrate[((-3 + x^2)*(1 - 2*x^2 + x^4 + x^6))/(x^10*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4)), x]

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IntegrateAlgebraic [A]  time = 1.82, size = 514, normalized size = 1.00 \begin {gather*} \frac {\left (\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}\right )^{3/4} \left (32 a^2 c^3-96 a^2 c^3 x^2+36 a b c^3 x^3-36 a^2 c^2 d x^3+96 a^2 c^3 x^4-72 a b c^3 x^5+72 a^2 c^2 d x^5+64 a^2 c^3 x^6+45 b^2 c^3 x^6-42 a b c^2 d x^6-3 a^2 c d^2 x^6+36 a b c^3 x^7-36 a^2 c^2 d x^7-96 a^2 c^3 x^8-45 b^2 c^3 x^8+42 a b c^2 d x^8+3 a^2 c d^2 x^8-96 a^2 c^2 d x^9-45 b^2 c^2 d x^9+6 a b c d^2 x^9+7 a^2 d^3 x^9\right )}{96 a^3 c^2 x^9}+\frac {\left (-32 a^2 b c^3-15 b^3 c^3+32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {\left (32 a^2 b c^3+15 b^3 c^3-32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-3 + x^2)*(1 - 2*x^2 + x^4 + x^6))/(x^10*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4
)),x]

[Out]

(((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(3/4)*(32*a^2*c^3 - 96*a^2*c^3*x^2 + 36*a*b*c^3*x^3 - 36*a^2*c^2*
d*x^3 + 96*a^2*c^3*x^4 - 72*a*b*c^3*x^5 + 72*a^2*c^2*d*x^5 + 64*a^2*c^3*x^6 + 45*b^2*c^3*x^6 - 42*a*b*c^2*d*x^
6 - 3*a^2*c*d^2*x^6 + 36*a*b*c^3*x^7 - 36*a^2*c^2*d*x^7 - 96*a^2*c^3*x^8 - 45*b^2*c^3*x^8 + 42*a*b*c^2*d*x^8 +
3*a^2*c*d^2*x^8 - 96*a^2*c^2*d*x^9 - 45*b^2*c^2*d*x^9 + 6*a*b*c*d^2*x^9 + 7*a^2*d^3*x^9))/(96*a^3*c^2*x^9) +
((-32*a^2*b*c^3 - 15*b^3*c^3 + 32*a^3*c^2*d + 5*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 7*a^3*d^3)*ArcTan[(c^(1/4)*((-a
+ a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4)) + ((32*a^2*b*c^3 + 15*b^3*c^3 -
32*a^3*c^2*d - 5*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 7*a^3*d^3)*ArcTanh[(c^(1/4)*((-a + a*x^2 + b*x^3)/(-c + c*x^2
+ d*x^3))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4))

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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^2-3)*(x^6+x^4-2*x^2+1)/x^10/((b*x^3+a*x^2-a)/(d*x^3+c*x^2-c))^(1/4),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 (x^{6} + x^{4} - 2 \, x^{2} + 1\right )} {\left (x^{2} - 3\right )}}{x^{10} \left (\frac {b x^{3} + a x^{2} - a}{d x^{3} + c x^{2} - c}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-3)*(x^6+x^4-2*x^2+1)/x^10/((b*x^3+a*x^2-a)/(d*x^3+c*x^2-c))^(1/4),x, algorithm="giac")

[Out]

integrate((x^6 + x^4 - 2*x^2 + 1)*(x^2 - 3)/(x^10*((b*x^3 + a*x^2 - a)/(d*x^3 + c*x^2 - c))^(1/4)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-3\right ) \left (x^{6}+x^{4}-2 x^{2}+1\right )}{x^{10} \left (\frac {b \,x^{3}+a \,x^{2}-a}{d \,x^{3}+c \,x^{2}-c}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-3)*(x^6+x^4-2*x^2+1)/x^10/((b*x^3+a*x^2-a)/(d*x^3+c*x^2-c))^(1/4),x)

[Out]

int((x^2-3)*(x^6+x^4-2*x^2+1)/x^10/((b*x^3+a*x^2-a)/(d*x^3+c*x^2-c))^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{4} - 2 \, x^{2} + 1\right )} {\left (x^{2} - 3\right )}}{x^{10} \left (\frac {b x^{3} + a x^{2} - a}{d x^{3} + c x^{2} - c}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-3)*(x^6+x^4-2*x^2+1)/x^10/((b*x^3+a*x^2-a)/(d*x^3+c*x^2-c))^(1/4),x, algorithm="maxima")

[Out]

integrate((x^6 + x^4 - 2*x^2 + 1)*(x^2 - 3)/(x^10*((b*x^3 + a*x^2 - a)/(d*x^3 + c*x^2 - c))^(1/4)), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 3)*(x^4 - 2*x^2 + x^6 + 1))/(x^10*((a*x^2 - a + b*x^3)/(c*x^2 - c + d*x^3))^(1/4)),x)

[Out]

\text{Hanged}

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sympy [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**2-3)*(x**6+x**4-2*x**2+1)/x**10/((b*x**3+a*x**2-a)/(d*x**3+c*x**2-c))**(1/4),x)

[Out]

Timed out

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