3.1059 \(\int \frac{x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx\)

Optimal. Leaf size=115 \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.132644, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.2143, size = 53, normalized size = 0.46 \[ \frac{x^{3} \sqrt [4]{a + b x^{2}} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- \frac{b x^{2}}{a},- \frac{b x^{2}}{2 a} \right )}}{6 a^{2} \sqrt [4]{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2+a)**(3/4)/(b*x**2+2*a),x)

[Out]

x**3*(a + b*x**2)**(1/4)*appellf1(3/2, 3/4, 1, 5/2, -b*x**2/a, -b*x**2/(2*a))/(6
*a**2*(1 + b*x**2/a)**(1/4))

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Mathematica [C]  time = 0.287666, size = 171, normalized size = 1.49 \[ \frac{10 a x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{3 \left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right ) \left (10 a F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(10*a*x^3*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -(b*x^2)/(2*a)])/(3*(a + b*x^
2)^(3/4)*(2*a + b*x^2)*(10*a*AppellF1[3/2, 3/4, 1, 5/2, -((b*x^2)/a), -(b*x^2)/(
2*a)] - b*x^2*(2*AppellF1[5/2, 3/4, 2, 7/2, -((b*x^2)/a), -(b*x^2)/(2*a)] + 3*Ap
pellF1[5/2, 7/4, 1, 7/2, -((b*x^2)/a), -(b*x^2)/(2*a)])))

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{b{x}^{2}+2\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + 2*a)*(b*x^2 + a)^(3/4)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.240366, size = 251, normalized size = 2.18 \[ 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}}}{\sqrt{\frac{1}{2}} x \sqrt{\frac{b^{4} x^{2} \sqrt{-\frac{1}{a b^{6}}} + 2 \, \sqrt{b x^{2} + a}}{x^{2}}} +{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} +{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{2} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} b^{2} x \left (-\frac{1}{a b^{6}}\right )^{\frac{1}{4}} -{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + 2*a)*(b*x^2 + a)^(3/4)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{2} + 2 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + 2*a)*(b*x^2 + a)^(3/4)),x, algorithm="giac")

[Out]

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