3.16.59 \(\int \frac {x^4 (-2 b+a x^2)}{(-b+a x^2)^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx\) [1559]

Optimal. Leaf size=107 \[ -\frac {x \left (-b+a x^2+c x^4\right )^{3/4}}{2 c \left (b-a x^2\right )}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \]

[Out]

-1/2*x*(c*x^4+a*x^2-b)^(3/4)/c/(-a*x^2+b)-1/4*arctan(c^(1/4)*x/(c*x^4+a*x^2-b)^(1/4))/c^(5/4)-1/4*arctanh(c^(1
/4)*x/(c*x^4+a*x^2-b)^(1/4))/c^(5/4)

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Rubi [F]
time = 0.10, 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^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(x^4*(-2*b + a*x^2))/((-b + a*x^2)^2*(-b + a*x^2 + c*x^4)^(1/4)), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx &=\int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.73, size = 102, normalized size = 0.95 \begin {gather*} \frac {-\frac {2 \sqrt [4]{c} x \left (-b+a x^2+c x^4\right )^{3/4}}{b-a x^2}-\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*c^(1/4)*x*(-b + a*x^2 + c*x^4)^(3/4))/(b - a*x^2) - ArcTan[(c^(1/4)*x)/(-b + a*x^2 + c*x^4)^(1/4)] - ArcT
anh[(c^(1/4)*x)/(-b + a*x^2 + c*x^4)^(1/4)])/(4*c^(5/4))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{2}-2 b \right )}{\left (a \,x^{2}-b \right )^{2} \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (88) = 176\).
time = 0.39, size = 228, normalized size = 2.13 \begin {gather*} -\frac {4 \, {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \arctan \left (\frac {c \sqrt {\frac {c^{3} \sqrt {\frac {1}{c^{5}}} x^{2} + \sqrt {c x^{4} + a x^{2} - b}}{x^{2}}} \frac {1}{c^{5}}^{\frac {1}{4}} x - {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} c \frac {1}{c^{5}}^{\frac {1}{4}}}{x}\right ) + {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (-\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x - {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (c x^{4} + a x^{2} - b\right )}^{\frac {3}{4}} x}{8 \, {\left (a c x^{2} - b c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a*x^2-2*b)/(a*x^2-b)^2/(c*x^4+a*x^2-b)^(1/4),x, algorithm="fricas")

[Out]

-1/8*(4*(a*c*x^2 - b*c)*(c^(-5))^(1/4)*arctan((c*sqrt((c^3*sqrt(c^(-5))*x^2 + sqrt(c*x^4 + a*x^2 - b))/x^2)*(c
^(-5))^(1/4)*x - (c*x^4 + a*x^2 - b)^(1/4)*c*(c^(-5))^(1/4))/x) + (a*c*x^2 - b*c)*(c^(-5))^(1/4)*log((c^4*(c^(
-5))^(3/4)*x + (c*x^4 + a*x^2 - b)^(1/4))/x) - (a*c*x^2 - b*c)*(c^(-5))^(1/4)*log(-(c^4*(c^(-5))^(3/4)*x - (c*
x^4 + a*x^2 - b)^(1/4))/x) - 4*(c*x^4 + a*x^2 - b)^(3/4)*x)/(a*c*x^2 - b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a*x**2-2*b)/(a*x**2-b)**2/(c*x**4+a*x**2-b)**(1/4),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^4\,\left (2\,b-a\,x^2\right )}{{\left (b-a\,x^2\right )}^2\,{\left (c\,x^4+a\,x^2-b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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