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

Optimal. Leaf size=98 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \]

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Rubi [A]  time = 0.04, antiderivative size = 96, normalized size of antiderivative = 0.98, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 442

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> -Simp[(b*ArcTan[(Rt[-(b^2/a), 4]*
x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] + Simp[(b*ArcTanh[(Rt[-(b^2/a), 4]*x)/(Sq
rt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0
] && NegQ[b^2/a]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 68, normalized size = 0.69 \begin {gather*} -\frac {x^3 \left (1-\frac {a x^2}{b}\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};\frac {a x^2}{b},\frac {a x^2}{2 b}\right )}{6 b \left (a x^2-b\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(x^3*(1 - (a*x^2)/b)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (a*x^2)/b, (a*x^2)/(2*b)])/(b*(-b + a*x^2)^(3/4))

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IntegrateAlgebraic [A]  time = 2.47, size = 98, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 207, normalized size = 2.11 \begin {gather*} 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{4} b x \sqrt {\frac {a^{4} x^{2} \sqrt {\frac {1}{a^{6} b}} + 2 \, \sqrt {a x^{2} - b}}{x^{2}}} \left (\frac {1}{a^{6} b}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a x^{2} - b\right )}^{\frac {1}{4}} a^{4} b \left (\frac {1}{a^{6} b}\right )^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x^2-2*b)/(a*x^2-b)^(3/4),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a \,x^{2}-2 b \right ) \left (a \,x^{2}-b \right )^{\frac {3}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x^2-2*b)/(a*x^2-b)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^2/((a*x^2 - b)^(3/4)*(a*x^2 - 2*b)), 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^2-b\right )}^{3/4}\,\left (2\,b-a\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a x^{2} - 2 b\right ) \left (a x^{2} - b\right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a*x**2-2*b)/(a*x**2-b)**(3/4),x)

[Out]

Integral(x**2/((a*x**2 - 2*b)*(a*x**2 - b)**(3/4)), x)

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