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

Optimal. Leaf size=136 \[ -\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{b} \sqrt {a x^2+b}}{\sqrt {a}}-\frac {\sqrt {a} x^2}{2 \sqrt [4]{b}}}{x \sqrt [4]{a x^2+b}}\right )}{2 a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{a x^2+b}}{2 \sqrt {b} \sqrt {a x^2+b}+a x^2}\right )}{2 a^{3/2} \sqrt [4]{b}} \]

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Rubi [A]  time = 0.04, antiderivative size = 115, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {441} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b^{3/4} \left (1-\frac {\sqrt {a x^2+b}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{a x^2+b}}\right )}{a^{3/2} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {b^{3/4} \left (\frac {\sqrt {a x^2+b}}{\sqrt {b}}+1\right )}{\sqrt {a} x \sqrt [4]{a x^2+b}}\right )}{a^{3/2} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 441

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

Rubi steps

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

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Mathematica [C]  time = 0.06, size = 67, normalized size = 0.49 \begin {gather*} \frac {x^3 \left (\frac {a x^2+b}{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/((b + a*x^2)^(3/4)*(2*b + a*x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.39, size = 136, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-\frac {\sqrt {a} x^2}{2 \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {b+a x^2}}{\sqrt {a}}}{x \sqrt [4]{b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{b+a x^2}}{a x^2+2 \sqrt {b} \sqrt {b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.59, size = 207, normalized size = 1.52 \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+b)^(3/4)/(a*x^2+2*b),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*s
qrt(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} + 2 \, b\right )} {\left (a x^{2} + 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^2+b)^(3/4)/(a*x^2+2*b),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/(a*x^2+b)^(3/4)/(a*x^2+2*b),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} + 2 \, b\right )} {\left (a x^{2} + 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^2+b)^(3/4)/(a*x^2+2*b),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/((b + a*x^2)^(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} + b\right )^{\frac {3}{4}} \left (a x^{2} + 2 b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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