3.5.47 \(\int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} (b-c^4 x^4+a x^6)} \, dx\)

Optimal. Leaf size=35 \[ -\tan ^{-1}\left (\frac {c x}{\sqrt [4]{a x^6+b}}\right )-\tanh ^{-1}\left (\frac {c x}{\sqrt [4]{a x^6+b}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.14, 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 {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(c*x*(1 + (a*x^6)/b)^(1/4)*Hypergeometric2F1[1/6, 1/4, 7/6, -((a*x^6)/b)])/(b + a*x^6)^(1/4) - c^5*Defer[Int][
x^4/((-b + c^4*x^4 - a*x^6)*(b + a*x^6)^(1/4)), x] - 3*b*c*Defer[Int][1/((b + a*x^6)^(1/4)*(b - c^4*x^4 + a*x^
6)), x]

Rubi steps

\begin {align*} \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx &=\int \left (\frac {c}{\sqrt [4]{b+a x^6}}-\frac {3 b c-c^5 x^4}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )}\right ) \, dx\\ &=c \int \frac {1}{\sqrt [4]{b+a x^6}} \, dx-\int \frac {3 b c-c^5 x^4}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx\\ &=\frac {\left (c \sqrt [4]{1+\frac {a x^6}{b}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {a x^6}{b}}} \, dx}{\sqrt [4]{b+a x^6}}-\int \left (\frac {c^5 x^4}{\left (-b+c^4 x^4-a x^6\right ) \sqrt [4]{b+a x^6}}+\frac {3 b c}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )}\right ) \, dx\\ &=\frac {c x \sqrt [4]{1+\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-\frac {a x^6}{b}\right )}{\sqrt [4]{b+a x^6}}-(3 b c) \int \frac {1}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx-c^5 \int \frac {x^4}{\left (-b+c^4 x^4-a x^6\right ) \sqrt [4]{b+a x^6}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b c+a c x^6}{\sqrt [4]{b+a x^6} \left (b-c^4 x^4+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.87, size = 37, normalized size = 1.06 \begin {gather*} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^6}}{c x}\right )-\tanh ^{-1}\left (\frac {c x}{\sqrt [4]{b+a x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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((a*c*x^6-2*b*c)/(a*x^6+b)^(1/4)/(-c^4*x^4+a*x^6+b),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,b\,c-a\,c\,x^6}{{\left (a\,x^6+b\right )}^{1/4}\,\left (-c^4\,x^4+a\,x^6+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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((a*c*x**6-2*b*c)/(a*x**6+b)**(1/4)/(-c**4*x**4+a*x**6+b),x)

[Out]

Timed out

________________________________________________________________________________________