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

Optimal. Leaf size=133 \[ \frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {a x^6-b}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^6-b}}\right )}{\sqrt {2} c^{3/4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^6-b}}{\sqrt {a x^6-b}-\sqrt {c} x^2}\right )}{\sqrt {2} c^{3/4}} \]

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Rubi [F]  time = 2.02, 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]

(Sqrt[(a*x^6)/(Sqrt[b] + Sqrt[-b + a*x^6])^2]*(Sqrt[b] + Sqrt[-b + a*x^6])*EllipticF[2*ArcTan[(-b + a*x^6)^(1/
4)/b^(1/4)], 1/2])/(3*a*b^(1/4)*x^3) - (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*Defer[Int][1/((b - c*x^4 - a*x^6)*(-b + a*x^6)^(3/4)), 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} \operatorname {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-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}}-\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-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, 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 (2 \sqrt {\frac {a x^6}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^6}\right )}{3 a x^3}\\ &=\frac {\sqrt {\frac {a x^6}{\left (\sqrt {b}+\sqrt {-b+a x^6}\right )^2}} \left (\sqrt {b}+\sqrt {-b+a x^6}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^6}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 a \sqrt [4]{b} x^3}-\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-c x^4-a x^6\right ) \left (-b+a x^6\right )^{3/4}} \, 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 [F]  time = 0.41, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 15.74, size = 133, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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])]/(Sqrt[2]*c^(3/4))) + ArcT
anh[((c^(1/4)*x^2)/Sqrt[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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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*(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} + c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}}\,{d x} \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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maple [F]  time = 0.03, 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*} \int \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} + c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}}\,{d x} \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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (a\,x^6+2\,b\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))/((a*x^6 - b)^(3/4)*(a*x^6 - b + c*x^4)),x)

[Out]

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

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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*(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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