3.19.43 \(\int \frac {x^2 (-2 b+a x^6)}{(b+a x^6)^{3/4} (b+c x^4+a x^6)} \, dx\) [1843]

Optimal. Leaf size=125 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{-\sqrt {c} x^2+\sqrt {b+a x^6}}\right )}{\sqrt {2} c^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \]

[Out]

-1/2*arctan(2^(1/2)*c^(1/4)*x*(a*x^6+b)^(1/4)/(-x^2*c^(1/2)+(a*x^6+b)^(1/2)))*2^(1/2)/c^(3/4)+1/2*arctanh((1/2
*c^(1/4)*x^2*2^(1/2)+1/2*(a*x^6+b)^(1/2)*2^(1/2)/c^(1/4))/x/(a*x^6+b)^(1/4))*2^(1/2)/c^(3/4)

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Rubi [F]
time = 1.23, 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^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[b]*(1 + (a*x^6)/b)^(3/4)*EllipticF[ArcTan[(Sqrt[a]*x^3)/Sqrt[b]]/2, 2])/(3*Sqrt[a]*(b + a*x^6)^(3/4))
- (c*x*(1 + (a*x^6)/b)^(3/4)*Hypergeometric2F1[1/6, 3/4, 7/6, -((a*x^6)/b)])/(a*(b + a*x^6)^(3/4)) + (b*c*Defe
r[Int][1/((b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)), x])/a - 3*b*Defer[Int][x^2/((b + a*x^6)^(3/4)*(b + c*x^4 + a
*x^6)), x] + (c^2*Defer[Int][x^4/((b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)), x])/a

Rubi steps

\begin {align*} \int \frac {x^2 \left (-2 b+a x^6\right )}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx &=\int \left (-\frac {c}{a \left (b+a x^6\right )^{3/4}}+\frac {x^2}{\left (b+a x^6\right )^{3/4}}+\frac {b c-3 a b x^2+c^2 x^4}{a \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {b c-3 a b x^2+c^2 x^4}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx}{a}-\frac {c \int \frac {1}{\left (b+a x^6\right )^{3/4}} \, dx}{a}+\int \frac {x^2}{\left (b+a x^6\right )^{3/4}} \, dx\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (b+a x^2\right )^{3/4}} \, dx,x,x^3\right )+\frac {\int \left (\frac {b c}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )}-\frac {3 a b x^2}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )}+\frac {c^2 x^4}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )}\right ) \, dx}{a}-\frac {\left (c \left (1+\frac {a x^6}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {a x^6}{b}\right )^{3/4}} \, dx}{a \left (b+a x^6\right )^{3/4}}\\ &=-\frac {c x \left (1+\frac {a x^6}{b}\right )^{3/4} \, _2F_1\left (\frac {1}{6},\frac {3}{4};\frac {7}{6};-\frac {a x^6}{b}\right )}{a \left (b+a x^6\right )^{3/4}}-(3 b) \int \frac {x^2}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx+\frac {(b c) \int \frac {1}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx}{a}+\frac {c^2 \int \frac {x^4}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx}{a}+\frac {\left (1+\frac {a x^6}{b}\right )^{3/4} \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,x^3\right )}{3 \left (b+a x^6\right )^{3/4}}\\ &=\frac {2 \sqrt {b} \left (1+\frac {a x^6}{b}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a} x^3}{\sqrt {b}}\right )\right |2\right )}{3 \sqrt {a} \left (b+a x^6\right )^{3/4}}-\frac {c x \left (1+\frac {a x^6}{b}\right )^{3/4} \, _2F_1\left (\frac {1}{6},\frac {3}{4};\frac {7}{6};-\frac {a x^6}{b}\right )}{a \left (b+a x^6\right )^{3/4}}-(3 b) \int \frac {x^2}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx+\frac {(b c) \int \frac {1}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx}{a}+\frac {c^2 \int \frac {x^4}{\left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx}{a}\\ \end {align*}

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Mathematica [A]
time = 7.98, size = 108, normalized size = 0.86 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{\sqrt {c} x^2-\sqrt {b+a x^6}}\right )+\tanh ^{-1}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}\right )}{\sqrt {2} c^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^6)^(1/4))/(Sqrt[c]*x^2 - Sqrt[b + a*x^6])] + ArcTanh[(Sqrt[c]*x^2 + Sqrt[b
+ a*x^6])/(Sqrt[2]*c^(1/4)*x*(b + a*x^6)^(1/4))])/(Sqrt[2]*c^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-1)] Timed out
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*(a*x^6-2*b)/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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*(a*x**6-2*b)/(a*x**6+b)**(3/4)/(a*x**6+c*x**4+b),x)

[Out]

Timed out

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^2\,\left (2\,b-a\,x^6\right )}{{\left (a\,x^6+b\right )}^{3/4}\,\left (a\,x^6+c\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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